MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgcgreq Structured version   Visualization version   GIF version

Theorem tgcgreq 25311
Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-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))
tgcgreq.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
tgcgreq (𝜑𝐶 = 𝐷)

Proof of Theorem tgcgreq
StepHypRef Expression
1 tgcgreq.1 . 2 (𝜑𝐴 = 𝐵)
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)
8 tgcgrcomlr.c . . 3 (𝜑𝐶𝑃)
9 tgcgrcomlr.d . . 3 (𝜑𝐷𝑃)
10 tgcgrcomlr.6 . . 3 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
112, 3, 4, 5, 6, 7, 8, 9, 10tgcgreqb 25310 . 2 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
121, 11mpbid 222 1 (𝜑𝐶 = 𝐷)
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  tgidinside  25400  tgbtwnconn1lem3  25403  krippenlem  25519  ragcgr  25536  lmiisolem  25622  cgrg3col4  25668
  Copyright terms: Public domain W3C validator