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Theorem axtgcgrrflx 28470
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 28461 . . . . 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 4237 . . . . . 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 4237 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
42, 3sstri 3993 . . . . 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 4030 . . . 4 TarskiG ⊆ TarskiGC
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sselid 3981 . . 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 28462 . . . . 5 (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
1211simprbi 496 . . . 4 (𝐺 ∈ TarskiGC → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦)))
1312simpld 494 . . 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 7438 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
18 oveq2 7439 . . . . 5 (𝑥 = 𝑋 → (𝑦 𝑥) = (𝑦 𝑋))
1917, 18eqeq12d 2753 . . . 4 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑦 𝑥) ↔ (𝑋 𝑦) = (𝑦 𝑋)))
20 oveq2 7439 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
21 oveq1 7438 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑋) = (𝑌 𝑋))
2220, 21eqeq12d 2753 . . . 4 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑦 𝑋) ↔ (𝑋 𝑌) = (𝑌 𝑋)))
2319, 22rspc2v 3633 . . 3 ((𝑋𝑃𝑌𝑃) → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) → (𝑋 𝑌) = (𝑌 𝑋)))
2415, 16, 23syl2anc 584 . 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 395  w3o 1086   = wceq 1540  wcel 2108  {cab 2714  wral 3061  {crab 3436  Vcvv 3480  [wsbc 3788  cdif 3948  cin 3950  {csn 4626  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  TarskiGCcstrkgc 28436  TarskiGBcstrkgb 28437  TarskiGCBcstrkgcb 28438  Itvcitv 28441  LineGclng 28442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-trkgc 28456  df-trkg 28461
This theorem is referenced by:  tgcgrcomimp  28485  tgcgrcomr  28486  tgcgrcoml  28487  tgcgrcomlr  28488  tgbtwnconn1lem1  28580  tgbtwnconn1lem2  28581  tgbtwnconn1lem3  28582  miriso  28678  symquadlem  28697  midexlem  28700  footexALT  28726  footexlem1  28727  footexlem2  28728  colperpexlem1  28738  opphllem  28743  cgraswap  28828  isoas  28872  f1otrg  28879
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