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Theorem tgcgrcomlr 25309
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 (𝜑𝐺 ∈ TarskiG)
tgcgrcomlr.a (𝜑𝐴𝑃)
tgcgrcomlr.b (𝜑𝐵𝑃)
tgcgrcomlr.c (𝜑𝐶𝑃)
tgcgrcomlr.d (𝜑𝐷𝑃)
tgcgrcomlr.6 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
Assertion
Ref Expression
tgcgrcomlr (𝜑 → (𝐵 𝐴) = (𝐷 𝐶))

Proof of Theorem tgcgrcomlr
StepHypRef Expression
1 tgcgrcomlr.6 . 2 (𝜑 → (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 (𝜑𝐺 ∈ TarskiG)
6 tgcgrcomlr.a . . 3 (𝜑𝐴𝑃)
7 tgcgrcomlr.b . . 3 (𝜑𝐵𝑃)
82, 3, 4, 5, 6, 7axtgcgrrflx 25295 . 2 (𝜑 → (𝐴 𝐵) = (𝐵 𝐴))
9 tgcgrcomlr.c . . 3 (𝜑𝐶𝑃)
10 tgcgrcomlr.d . . 3 (𝜑𝐷𝑃)
112, 3, 4, 5, 9, 10axtgcgrrflx 25295 . 2 (𝜑 → (𝐶 𝐷) = (𝐷 𝐶))
121, 8, 113eqtr3d 2663 1 (𝜑 → (𝐵 𝐴) = (𝐷 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cfv 5857  (class class class)co 6615  Basecbs 15800  distcds 15890  TarskiGcstrkg 25263  Itvcitv 25269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-trkgc 25281  df-trkg 25286
This theorem is referenced by:  tgcgrextend  25314  tgifscgr  25337  tgcgrsub  25338  iscgrglt  25343  trgcgrg  25344  tgcgrxfr  25347  cgr3swap12  25352  cgr3swap23  25353  tgbtwnxfr  25359  lnext  25396  tgbtwnconn1lem1  25401  tgbtwnconn1lem2  25402  tgbtwnconn1lem3  25403  tgbtwnconn1  25404  legov2  25415  legtri3  25419  legbtwn  25423  tgcgrsub2  25424  miriso  25499  mircgrextend  25511  mirtrcgr  25512  miduniq  25514  colmid  25517  symquadlem  25518  krippenlem  25519  midexlem  25521  ragcom  25527  ragflat  25533  ragcgr  25536  footex  25547  colperpexlem1  25556  mideulem2  25560  opphllem  25561  opphllem3  25575  lmiisolem  25622  hypcgrlem1  25625  trgcopy  25630  trgcopyeulem  25631  iscgra1  25636  cgracgr  25644  cgraswap  25646  cgrcgra  25647  cgracom  25648  cgratr  25649  dfcgra2  25655  sacgr  25656  acopy  25658  acopyeu  25659  cgrg3col4  25668  tgsas1  25669  tgsas3  25672  tgasa1  25673
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