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

Theorem tgcgreqb 25732
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))
Assertion
Ref Expression
tgcgreqb (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))

Proof of Theorem tgcgreqb
StepHypRef Expression
1 tkgeom.p . . 3 𝑃 = (Base‘𝐺)
2 tkgeom.d . . 3 = (dist‘𝐺)
3 tkgeom.i . . 3 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54adantr 473 . . 3 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
6 tgcgrcomlr.c . . . 4 (𝜑𝐶𝑃)
76adantr 473 . . 3 ((𝜑𝐴 = 𝐵) → 𝐶𝑃)
8 tgcgrcomlr.d . . . 4 (𝜑𝐷𝑃)
98adantr 473 . . 3 ((𝜑𝐴 = 𝐵) → 𝐷𝑃)
10 tgcgrcomlr.b . . . 4 (𝜑𝐵𝑃)
1110adantr 473 . . 3 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
12 tgcgrcomlr.6 . . . . 5 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
1312adantr 473 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐶 𝐷))
14 simpr 478 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
1514oveq1d 6893 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐵 𝐵))
1613, 15eqtr3d 2835 . . 3 ((𝜑𝐴 = 𝐵) → (𝐶 𝐷) = (𝐵 𝐵))
171, 2, 3, 5, 7, 9, 11, 16axtgcgrid 25714 . 2 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐷)
184adantr 473 . . 3 ((𝜑𝐶 = 𝐷) → 𝐺 ∈ TarskiG)
19 tgcgrcomlr.a . . . 4 (𝜑𝐴𝑃)
2019adantr 473 . . 3 ((𝜑𝐶 = 𝐷) → 𝐴𝑃)
2110adantr 473 . . 3 ((𝜑𝐶 = 𝐷) → 𝐵𝑃)
228adantr 473 . . 3 ((𝜑𝐶 = 𝐷) → 𝐷𝑃)
2312adantr 473 . . . 4 ((𝜑𝐶 = 𝐷) → (𝐴 𝐵) = (𝐶 𝐷))
24 simpr 478 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝐶 = 𝐷)
2524oveq1d 6893 . . . 4 ((𝜑𝐶 = 𝐷) → (𝐶 𝐷) = (𝐷 𝐷))
2623, 25eqtrd 2833 . . 3 ((𝜑𝐶 = 𝐷) → (𝐴 𝐵) = (𝐷 𝐷))
271, 2, 3, 18, 20, 21, 22, 26axtgcgrid 25714 . 2 ((𝜑𝐶 = 𝐷) → 𝐴 = 𝐵)
2817, 27impbida 836 1 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  Basecbs 16184  distcds 16276  TarskiGcstrkg 25681  Itvcitv 25687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-trkgc 25699  df-trkg 25704
This theorem is referenced by:  tgcgreq  25733  tgcgrneq  25734
  Copyright terms: Public domain W3C validator