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Theorem axtgcgrid 25075
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 25065 . . . . 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 3790 . . . . . 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 3790 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
42, 3sstri 3572 . . . . 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 3593 . . . 4 TarskiG ⊆ TarskiGC
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3561 . . 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 25066 . . . . 5 (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
1211simprbi 478 . . . 4 (𝐺 ∈ TarskiGC → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦)))
1312simprd 477 . . 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 6530 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
2019eqeq1d 2607 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑧 𝑧) ↔ (𝑋 𝑦) = (𝑧 𝑧)))
21 eqeq1 2609 . . . . 5 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
2220, 21imbi12d 332 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ((𝑋 𝑦) = (𝑧 𝑧) → 𝑋 = 𝑦)))
23 oveq2 6531 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
2423eqeq1d 2607 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑧 𝑧) ↔ (𝑋 𝑌) = (𝑧 𝑧)))
25 eqeq2 2616 . . . . 5 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
2624, 25imbi12d 332 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦) = (𝑧 𝑧) → 𝑋 = 𝑦) ↔ ((𝑋 𝑌) = (𝑧 𝑧) → 𝑋 = 𝑌)))
27 id 22 . . . . . . 7 (𝑧 = 𝑍𝑧 = 𝑍)
2827, 27oveq12d 6541 . . . . . 6 (𝑧 = 𝑍 → (𝑧 𝑧) = (𝑍 𝑍))
2928eqeq2d 2615 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌) = (𝑧 𝑧) ↔ (𝑋 𝑌) = (𝑍 𝑍)))
3029imbi1d 329 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌) = (𝑧 𝑧) → 𝑋 = 𝑌) ↔ ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3122, 26, 30rspc3v 3291 . . 3 ((𝑋𝑃𝑌𝑃𝑍𝑃) → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) → ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3216, 17, 18, 31syl3anc 1317 . 2 (𝜑 → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) → ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3314, 15, 32mp2d 46 1 (𝜑𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3o 1029   = wceq 1474  wcel 1975  {cab 2591  wral 2891  {crab 2895  Vcvv 3168  [wsbc 3397  cdif 3532  cin 3534  {csn 4120  cfv 5786  (class class class)co 6523  cmpt2 6525  Basecbs 15637  distcds 15719  TarskiGcstrkg 25042  TarskiGCcstrkgc 25043  TarskiGBcstrkgb 25044  TarskiGCBcstrkgcb 25045  Itvcitv 25048  LineGclng 25049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-nul 4708
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-iota 5750  df-fv 5794  df-ov 6526  df-trkgc 25060  df-trkg 25065
This theorem is referenced by:  tgcgreqb  25089  tgcgrtriv  25092  tgsegconeq  25094  tgbtwntriv2  25095  tgbtwndiff  25114  tgifscgr  25117  tgbtwnxfr  25139  lnid  25179  tgbtwnconn1lem2  25182  tgbtwnconn1lem3  25183  legtri3  25199  legeq  25202  legbtwn  25203  mirreu3  25263  colmid  25297  krippenlem  25299  lmiisolem  25402  hypcgrlem1  25405  hypcgrlem2  25406  f1otrg  25465
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