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Theorem axtgcgrid 25407
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 25397 . . . . 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 3866 . . . . . 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 3866 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
42, 3sstri 3645 . . . . 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 3668 . . . 4 TarskiG ⊆ TarskiGC
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3634 . . 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 25398 . . . . 5 (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
1211simprbi 479 . . . 4 (𝐺 ∈ TarskiGC → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦)))
1312simprd 478 . . 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 6697 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
2019eqeq1d 2653 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑧 𝑧) ↔ (𝑋 𝑦) = (𝑧 𝑧)))
21 eqeq1 2655 . . . . 5 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
2220, 21imbi12d 333 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ((𝑋 𝑦) = (𝑧 𝑧) → 𝑋 = 𝑦)))
23 oveq2 6698 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
2423eqeq1d 2653 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑧 𝑧) ↔ (𝑋 𝑌) = (𝑧 𝑧)))
25 eqeq2 2662 . . . . 5 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
2624, 25imbi12d 333 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦) = (𝑧 𝑧) → 𝑋 = 𝑦) ↔ ((𝑋 𝑌) = (𝑧 𝑧) → 𝑋 = 𝑌)))
27 id 22 . . . . . . 7 (𝑧 = 𝑍𝑧 = 𝑍)
2827, 27oveq12d 6708 . . . . . 6 (𝑧 = 𝑍 → (𝑧 𝑧) = (𝑍 𝑍))
2928eqeq2d 2661 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌) = (𝑧 𝑧) ↔ (𝑋 𝑌) = (𝑍 𝑍)))
3029imbi1d 330 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌) = (𝑧 𝑧) → 𝑋 = 𝑌) ↔ ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3122, 26, 30rspc3v 3356 . . 3 ((𝑋𝑃𝑌𝑃𝑍𝑃) → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) → ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3216, 17, 18, 31syl3anc 1366 . 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 383  w3o 1053   = wceq 1523  wcel 2030  {cab 2637  wral 2941  {crab 2945  Vcvv 3231  [wsbc 3468  cdif 3604  cin 3606  {csn 4210  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  TarskiGCcstrkgc 25375  TarskiGBcstrkgb 25376  TarskiGCBcstrkgcb 25377  Itvcitv 25380  LineGclng 25381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-trkgc 25392  df-trkg 25397
This theorem is referenced by:  tgcgreqb  25421  tgcgrtriv  25424  tgsegconeq  25426  tgbtwntriv2  25427  tgbtwndiff  25446  tgifscgr  25448  tgbtwnxfr  25470  lnid  25510  tgbtwnconn1lem2  25513  tgbtwnconn1lem3  25514  legtri3  25530  legeq  25533  legbtwn  25534  mirreu3  25594  colmid  25628  krippenlem  25630  lmiisolem  25733  hypcgrlem1  25736  hypcgrlem2  25737  f1otrg  25796
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