(* Basic_2A1: uses: teqg_fpb_trans lleq_fpb_trans fleq_fpb_trans *)
lemma feqg_fpbc_trans (S) (G) (L) (T):
reflexive … S → symmetric … S → Transitive … S →
- â\88\80G1,L1,T1. â\9dªG1,L1,T1â\9d« â\89\9b[S] â\9dªG,L,Tâ\9d« →
- â\88\80G2,L2,T2. â\9dªG,L,Tâ\9d« â\89» â\9dªG2,L2,T2â\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89» â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,L1,T1. â\9d¨G1,L1,T1â\9d© â\89\9b[S] â\9d¨G,L,Tâ\9d© →
+ â\88\80G2,L2,T2. â\9d¨G,L,Tâ\9d© â\89» â\9d¨G2,L2,T2â\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89» â\9d¨G2,L2,T2â\9d©.
#S #G #L #T #H1S #H2S #H3S #G1 #L1 #T1 #H1 #G2 #L2 #T2 #H2
elim (fpbc_inv_gen sfull … H2) -H2 #H2 #Hn2
/6 width=9 by fpbc_intro, feqg_fpb_trans, feqg_canc_sn, feqg_feqx/
(* Basic_2A1: uses: fpb_inv_fleq *)
lemma fpbc_inv_feqg (S):
- â\88\80G1,G2,L1,L2,T1,T2. â\9dªG1,L1,T1â\9d« â\89» â\9dªG2,L2,T2â\9d« →
- â\9dªG1,L1,T1â\9d« â\89\9b[S] â\9dªG2,L2,T2â\9d« → ⊥.
+ â\88\80G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â\89» â\9d¨G2,L2,T2â\9d© →
+ â\9d¨G1,L1,T1â\9d© â\89\9b[S] â\9d¨G2,L2,T2â\9d© → ⊥.
#S #G1 #G2 #L1 #L2 #T1 #T2 #H #H12
elim (fpbc_inv_gen S … H) -H #_ #Hn2
/2 width=1 by/