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Theorem tgcgreqb 25283
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 481 . . 3 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
6 tgcgrcomlr.c . . . 4 (𝜑𝐶𝑃)
76adantr 481 . . 3 ((𝜑𝐴 = 𝐵) → 𝐶𝑃)
8 tgcgrcomlr.d . . . 4 (𝜑𝐷𝑃)
98adantr 481 . . 3 ((𝜑𝐴 = 𝐵) → 𝐷𝑃)
10 tgcgrcomlr.b . . . 4 (𝜑𝐵𝑃)
1110adantr 481 . . 3 ((𝜑𝐴 = 𝐵) → 𝐵𝑃)
12 tgcgrcomlr.6 . . . . 5 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
1312adantr 481 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐶 𝐷))
14 simpr 477 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
1514oveq1d 6622 . . . 4 ((𝜑𝐴 = 𝐵) → (𝐴 𝐵) = (𝐵 𝐵))
1613, 15eqtr3d 2657 . . 3 ((𝜑𝐴 = 𝐵) → (𝐶 𝐷) = (𝐵 𝐵))
171, 2, 3, 5, 7, 9, 11, 16axtgcgrid 25269 . 2 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐷)
184adantr 481 . . 3 ((𝜑𝐶 = 𝐷) → 𝐺 ∈ TarskiG)
19 tgcgrcomlr.a . . . 4 (𝜑𝐴𝑃)
2019adantr 481 . . 3 ((𝜑𝐶 = 𝐷) → 𝐴𝑃)
2110adantr 481 . . 3 ((𝜑𝐶 = 𝐷) → 𝐵𝑃)
228adantr 481 . . 3 ((𝜑𝐶 = 𝐷) → 𝐷𝑃)
2312adantr 481 . . . 4 ((𝜑𝐶 = 𝐷) → (𝐴 𝐵) = (𝐶 𝐷))
24 simpr 477 . . . . 5 ((𝜑𝐶 = 𝐷) → 𝐶 = 𝐷)
2524oveq1d 6622 . . . 4 ((𝜑𝐶 = 𝐷) → (𝐶 𝐷) = (𝐷 𝐷))
2623, 25eqtrd 2655 . . 3 ((𝜑𝐶 = 𝐷) → (𝐴 𝐵) = (𝐷 𝐷))
271, 2, 3, 18, 20, 21, 22, 26axtgcgrid 25269 . 2 ((𝜑𝐶 = 𝐷) → 𝐴 = 𝐵)
2817, 27impbida 876 1 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cfv 5849  (class class class)co 6607  Basecbs 15784  distcds 15874  TarskiGcstrkg 25236  Itvcitv 25242
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 4751
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 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-iota 5812  df-fv 5857  df-ov 6610  df-trkgc 25254  df-trkg 25259
This theorem is referenced by:  tgcgreq  25284  tgcgrneq  25285
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