ndefinition mm_image:
∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma B.
#A; #B; #Ma; #Mb; #f;
- napply (mk_magma …)
+ napply mk_magma
[ napply (image … f Ma)
| #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
- napply (ex_intro …)
+ napply ex_intro
[ napply (op … x0 y0)
- | napply (conj …)
- [ napply (op_closed …); nassumption
+ | napply conj
+ [ napply op_closed; nassumption
| nrewrite < Hx1;
nrewrite < Hy1;
- napply (mmprop … f …)]##]
+ napply (mmprop … f)]##]
nqed.
ndefinition counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
ndefinition mm_counter_image:
∀A,B. ∀Ma: magma A. ∀Mb: magma B. magma_morphism … Ma Mb → magma A.
#A; #B; #Ma; #Mb; #f;
- napply (mk_magma …)
+ napply mk_magma
[ napply (counter_image … f Mb)
| #x; #y; nwhd in ⊢ (% → % → ?); *; #x0; *; #Hx0; #Hx1; *; #y0; *; #Hy0; #Hy1; nwhd;
- napply (ex_intro …)
+ napply ex_intro
[ napply (op … x0 y0)
- | napply (conj …)
- [ napply (op_closed …); nassumption
+ | napply conj
+ [ napply op_closed; nassumption
| nrewrite < Hx1;
nrewrite < Hy1;
- napply (mmprop … f …)]##]
+ napply (mmprop … f)]##]
nqed.
ndefinition m_intersect: ∀A. magma A → magma A → magma A.
napply (mk_magma …)
[ napply (M1 ∩ M2)
| #x; #y; nwhd in ⊢ (% → % → %); *; #Hx1; #Hx2; *; #Hy1; #Hy2;
- napply (conj …); napply (op_closed …); nassumption ]
+ napply conj; napply op_closed; nassumption ]
nqed.
\ No newline at end of file