Theorem tgcgrcomlr 25375

Description: Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-2019.)

Hypotheses

Ref	Expression
tkgeom.p	|- P = ( Base ` G )
tkgeom.d	|- .- = ( dist ` G )
tkgeom.i	|- I = (Itv ` G )
tkgeom.g	|- ( ph -> G e. TarskiG )
tgcgrcomlr.a	|- ( ph -> A e. P )
tgcgrcomlr.b	|- ( ph -> B e. P )
tgcgrcomlr.c	|- ( ph -> C e. P )
tgcgrcomlr.d	|- ( ph -> D e. P )
tgcgrcomlr.6	|- ( ph -> ( A .- B ) = ( C .- D ) )

Assertion

Ref	Expression
tgcgrcomlr	|- ( ph -> ( B .- A ) = ( D .- C ) )

Proof of Theorem tgcgrcomlr

Step	Hyp	Ref	Expression
1		tgcgrcomlr.6	. 2 |- ( ph -> ( A .- B ) = ( C .- D ) )
2		tkgeom.p	. . 3 |- P = ( Base ` G )
3		tkgeom.d	. . 3 |- .- = ( dist ` G )
4		tkgeom.i	. . 3 |- I = (Itv ` G )
5		tkgeom.g	. . 3 |- ( ph -> G e. TarskiG )
6		tgcgrcomlr.a	. . 3 |- ( ph -> A e. P )
7		tgcgrcomlr.b	. . 3 |- ( ph -> B e. P )
8	2, 3, 4, 5, 6, 7	axtgcgrrflx 25361	. 2 |- ( ph -> ( A .- B ) = ( B .- A ) )
9		tgcgrcomlr.c	. . 3 |- ( ph -> C e. P )
10		tgcgrcomlr.d	. . 3 |- ( ph -> D e. P )
11	2, 3, 4, 5, 9, 10	axtgcgrrflx 25361	. 2 |- ( ph -> ( C .- D ) = ( D .- C ) )
12	1, 8, 11	3eqtr3d 2664	1 |- ( ph -> ( B .- A ) = ( D .- C ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-trkgc 25347  df-trkg 25352

This theorem is referenced by:  tgcgrextend  25380  tgifscgr  25403  tgcgrsub  25404  iscgrglt  25409  trgcgrg  25410  tgcgrxfr  25413  cgr3swap12  25418  cgr3swap23  25419  tgbtwnxfr  25425  lnext  25462  tgbtwnconn1lem1  25467  tgbtwnconn1lem2  25468  tgbtwnconn1lem3  25469  tgbtwnconn1  25470  legov2  25481  legtri3  25485  legbtwn  25489  tgcgrsub2  25490  miriso  25565  mircgrextend  25577  mirtrcgr  25578  miduniq  25580  colmid  25583  symquadlem  25584  krippenlem  25585  midexlem  25587  ragcom  25593  ragflat  25599  ragcgr  25602  footex  25613  colperpexlem1  25622  mideulem2  25626  opphllem  25627  opphllem3  25641  lmiisolem  25688  hypcgrlem1  25691  trgcopy  25696  trgcopyeulem  25697  iscgra1  25702  cgracgr  25710  cgraswap  25712  cgrcgra  25713  cgracom  25714  cgratr  25715  dfcgra2  25721  sacgr  25722  acopy  25724  acopyeu  25725  cgrg3col4  25734  tgsas1  25735  tgsas3  25738  tgasa1  25739