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Theorem axtgcgrid 26554
Description: Axiom of identity of congruence, Axiom A3 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
axtrkg.p 𝑃 = (Base‘𝐺)
axtrkg.d = (dist‘𝐺)
axtrkg.i 𝐼 = (Itv‘𝐺)
axtrkg.g (𝜑𝐺 ∈ TarskiG)
axtgcgrid.1 (𝜑𝑋𝑃)
axtgcgrid.2 (𝜑𝑌𝑃)
axtgcgrid.3 (𝜑𝑍𝑃)
axtgcgrid.4 (𝜑 → (𝑋 𝑌) = (𝑍 𝑍))
Assertion
Ref Expression
axtgcgrid (𝜑𝑋 = 𝑌)

Proof of Theorem axtgcgrid
Dummy variables 𝑓 𝑖 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-trkg 26544 . . . . 5 TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
2 inss1 4143 . . . . . 6 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGC ∩ TarskiGB)
3 inss1 4143 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
42, 3sstri 3910 . . . . 5 ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGC
51, 4eqsstri 3935 . . . 4 TarskiG ⊆ TarskiGC
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3899 . . 3 (𝜑𝐺 ∈ TarskiGC)
8 axtrkg.p . . . . . 6 𝑃 = (Base‘𝐺)
9 axtrkg.d . . . . . 6 = (dist‘𝐺)
10 axtrkg.i . . . . . 6 𝐼 = (Itv‘𝐺)
118, 9, 10istrkgc 26545 . . . . 5 (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
1211simprbi 500 . . . 4 (𝐺 ∈ TarskiGC → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦)))
1312simprd 499 . . 3 (𝐺 ∈ TarskiGC → ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))
147, 13syl 17 . 2 (𝜑 → ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))
15 axtgcgrid.4 . 2 (𝜑 → (𝑋 𝑌) = (𝑍 𝑍))
16 axtgcgrid.1 . . 3 (𝜑𝑋𝑃)
17 axtgcgrid.2 . . 3 (𝜑𝑌𝑃)
18 axtgcgrid.3 . . 3 (𝜑𝑍𝑃)
19 oveq1 7220 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
2019eqeq1d 2739 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑧 𝑧) ↔ (𝑋 𝑦) = (𝑧 𝑧)))
21 eqeq1 2741 . . . . 5 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
2220, 21imbi12d 348 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ((𝑋 𝑦) = (𝑧 𝑧) → 𝑋 = 𝑦)))
23 oveq2 7221 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
2423eqeq1d 2739 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑧 𝑧) ↔ (𝑋 𝑌) = (𝑧 𝑧)))
25 eqeq2 2749 . . . . 5 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
2624, 25imbi12d 348 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦) = (𝑧 𝑧) → 𝑋 = 𝑦) ↔ ((𝑋 𝑌) = (𝑧 𝑧) → 𝑋 = 𝑌)))
27 id 22 . . . . . . 7 (𝑧 = 𝑍𝑧 = 𝑍)
2827, 27oveq12d 7231 . . . . . 6 (𝑧 = 𝑍 → (𝑧 𝑧) = (𝑍 𝑍))
2928eqeq2d 2748 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌) = (𝑧 𝑧) ↔ (𝑋 𝑌) = (𝑍 𝑍)))
3029imbi1d 345 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌) = (𝑧 𝑧) → 𝑋 = 𝑌) ↔ ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3122, 26, 30rspc3v 3550 . . 3 ((𝑋𝑃𝑌𝑃𝑍𝑃) → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) → ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3216, 17, 18, 31syl3anc 1373 . 2 (𝜑 → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) → ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3314, 15, 32mp2d 49 1 (𝜑𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3o 1088   = wceq 1543  wcel 2110  {cab 2714  wral 3061  {crab 3065  Vcvv 3408  [wsbc 3694  cdif 3863  cin 3865  {csn 4541  cfv 6380  (class class class)co 7213  cmpo 7215  Basecbs 16760  distcds 16811  TarskiGcstrkg 26521  TarskiGCcstrkgc 26522  TarskiGBcstrkgb 26523  TarskiGCBcstrkgcb 26524  Itvcitv 26527  LineGclng 26528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-trkgc 26539  df-trkg 26544
This theorem is referenced by:  tgcgreqb  26572  tgcgrtriv  26575  tgsegconeq  26577  tgbtwntriv2  26578  tgbtwndiff  26597  tgifscgr  26599  tgbtwnxfr  26621  lnid  26661  tgbtwnconn1lem2  26664  tgbtwnconn1lem3  26665  legtri3  26681  legeq  26684  legbtwn  26685  mirreu3  26745  colmid  26779  krippenlem  26781  lmiisolem  26887  hypcgrlem1  26890  hypcgrlem2  26891  f1otrg  26962
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