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Theorem axtgcgrrflx 25406
Description: Axiom of reflexivity of congruence, Axiom A1 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)
axtgcgrrflx.1 (𝜑𝑋𝑃)
axtgcgrrflx.2 (𝜑𝑌𝑃)
Assertion
Ref Expression
axtgcgrrflx (𝜑 → (𝑋 𝑌) = (𝑌 𝑋))

Proof of Theorem axtgcgrrflx
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)))
1312simpld 474 . . 3 (𝐺 ∈ TarskiGC → ∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥))
147, 13syl 17 . 2 (𝜑 → ∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥))
15 axtgcgrrflx.1 . . 3 (𝜑𝑋𝑃)
16 axtgcgrrflx.2 . . 3 (𝜑𝑌𝑃)
17 oveq1 6697 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
18 oveq2 6698 . . . . 5 (𝑥 = 𝑋 → (𝑦 𝑥) = (𝑦 𝑋))
1917, 18eqeq12d 2666 . . . 4 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑦 𝑥) ↔ (𝑋 𝑦) = (𝑦 𝑋)))
20 oveq2 6698 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
21 oveq1 6697 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑋) = (𝑌 𝑋))
2220, 21eqeq12d 2666 . . . 4 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑦 𝑋) ↔ (𝑋 𝑌) = (𝑌 𝑋)))
2319, 22rspc2v 3353 . . 3 ((𝑋𝑃𝑌𝑃) → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) → (𝑋 𝑌) = (𝑌 𝑋)))
2415, 16, 23syl2anc 694 . 2 (𝜑 → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) → (𝑋 𝑌) = (𝑌 𝑋)))
2514, 24mpd 15 1 (𝜑 → (𝑋 𝑌) = (𝑌 𝑋))
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:  tgcgrcomimp  25417  tgcgrcomr  25418  tgcgrcoml  25419  tgcgrcomlr  25420  tgbtwnconn1lem1  25512  tgbtwnconn1lem2  25513  tgbtwnconn1lem3  25514  miriso  25610  symquadlem  25629  midexlem  25632  footex  25658  colperpexlem1  25667  opphllem  25672  cgraswap  25757  isoas  25789  f1otrg  25796
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