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Theorem axtgcgrrflx 26175
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 26166 . . . . 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 4202 . . . . . 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 4202 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
42, 3sstri 3973 . . . . 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 3998 . . . 4 TarskiG ⊆ TarskiGC
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3962 . . 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 26167 . . . . 5 (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
1211simprbi 497 . . . 4 (𝐺 ∈ TarskiGC → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦)))
1312simpld 495 . . 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 7152 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
18 oveq2 7153 . . . . 5 (𝑥 = 𝑋 → (𝑦 𝑥) = (𝑦 𝑋))
1917, 18eqeq12d 2834 . . . 4 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑦 𝑥) ↔ (𝑋 𝑦) = (𝑦 𝑋)))
20 oveq2 7153 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
21 oveq1 7152 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑋) = (𝑌 𝑋))
2220, 21eqeq12d 2834 . . . 4 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑦 𝑋) ↔ (𝑋 𝑌) = (𝑌 𝑋)))
2319, 22rspc2v 3630 . . 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 396  w3o 1078   = wceq 1528  wcel 2105  {cab 2796  wral 3135  {crab 3139  Vcvv 3492  [wsbc 3769  cdif 3930  cin 3932  {csn 4557  cfv 6348  (class class class)co 7145  cmpo 7147  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  TarskiGCcstrkgc 26144  TarskiGBcstrkgb 26145  TarskiGCBcstrkgcb 26146  Itvcitv 26149  LineGclng 26150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-trkgc 26161  df-trkg 26166
This theorem is referenced by:  tgcgrcomimp  26190  tgcgrcomr  26191  tgcgrcoml  26192  tgcgrcomlr  26193  tgbtwnconn1lem1  26285  tgbtwnconn1lem2  26286  tgbtwnconn1lem3  26287  miriso  26383  symquadlem  26402  midexlem  26405  footexALT  26431  footexlem1  26432  footexlem2  26433  colperpexlem1  26443  opphllem  26448  cgraswap  26533  isoas  26577  f1otrg  26584
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