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Theorem axtgcgrrflx 28484
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 28475 . . . . 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 4244 . . . . . 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 4244 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
42, 3sstri 4004 . . . . 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 4029 . . . 4 TarskiG ⊆ TarskiGC
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sselid 3992 . . 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 28476 . . . . 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 7437 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
18 oveq2 7438 . . . . 5 (𝑥 = 𝑋 → (𝑦 𝑥) = (𝑦 𝑋))
1917, 18eqeq12d 2750 . . . 4 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑦 𝑥) ↔ (𝑋 𝑦) = (𝑦 𝑋)))
20 oveq2 7438 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
21 oveq1 7437 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑋) = (𝑌 𝑋))
2220, 21eqeq12d 2750 . . . 4 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑦 𝑋) ↔ (𝑋 𝑌) = (𝑌 𝑋)))
2319, 22rspc2v 3632 . . 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 1085   = wceq 1536  wcel 2105  {cab 2711  wral 3058  {crab 3432  Vcvv 3477  [wsbc 3790  cdif 3959  cin 3961  {csn 4630  cfv 6562  (class class class)co 7430  cmpo 7432  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  TarskiGCcstrkgc 28450  TarskiGBcstrkgb 28451  TarskiGCBcstrkgcb 28452  Itvcitv 28455  LineGclng 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-trkgc 28470  df-trkg 28475
This theorem is referenced by:  tgcgrcomimp  28499  tgcgrcomr  28500  tgcgrcoml  28501  tgcgrcomlr  28502  tgbtwnconn1lem1  28594  tgbtwnconn1lem2  28595  tgbtwnconn1lem3  28596  miriso  28692  symquadlem  28711  midexlem  28714  footexALT  28740  footexlem1  28741  footexlem2  28742  colperpexlem1  28752  opphllem  28757  cgraswap  28842  isoas  28886  f1otrg  28893
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