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Theorem axtgcgrid 25667
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 25657 . . . . 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 3994 . . . . . 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 3994 . . . . . 6 (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
42, 3sstri 3772 . . . . 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 3797 . . . 4 TarskiG ⊆ TarskiGC
6 axtrkg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
75, 6sseldi 3761 . . 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 25658 . . . . 5 (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦))))
1211simprbi 490 . . . 4 (𝐺 ∈ TarskiGC → (∀𝑥𝑃𝑦𝑃 (𝑥 𝑦) = (𝑦 𝑥) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦)))
1312simprd 489 . . 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 6853 . . . . . 6 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
2019eqeq1d 2767 . . . . 5 (𝑥 = 𝑋 → ((𝑥 𝑦) = (𝑧 𝑧) ↔ (𝑋 𝑦) = (𝑧 𝑧)))
21 eqeq1 2769 . . . . 5 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
2220, 21imbi12d 335 . . . 4 (𝑥 = 𝑋 → (((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) ↔ ((𝑋 𝑦) = (𝑧 𝑧) → 𝑋 = 𝑦)))
23 oveq2 6854 . . . . . 6 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
2423eqeq1d 2767 . . . . 5 (𝑦 = 𝑌 → ((𝑋 𝑦) = (𝑧 𝑧) ↔ (𝑋 𝑌) = (𝑧 𝑧)))
25 eqeq2 2776 . . . . 5 (𝑦 = 𝑌 → (𝑋 = 𝑦𝑋 = 𝑌))
2624, 25imbi12d 335 . . . 4 (𝑦 = 𝑌 → (((𝑋 𝑦) = (𝑧 𝑧) → 𝑋 = 𝑦) ↔ ((𝑋 𝑌) = (𝑧 𝑧) → 𝑋 = 𝑌)))
27 id 22 . . . . . . 7 (𝑧 = 𝑍𝑧 = 𝑍)
2827, 27oveq12d 6864 . . . . . 6 (𝑧 = 𝑍 → (𝑧 𝑧) = (𝑍 𝑍))
2928eqeq2d 2775 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌) = (𝑧 𝑧) ↔ (𝑋 𝑌) = (𝑍 𝑍)))
3029imbi1d 332 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑌) = (𝑧 𝑧) → 𝑋 = 𝑌) ↔ ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3122, 26, 30rspc3v 3478 . . 3 ((𝑋𝑃𝑌𝑃𝑍𝑃) → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) → ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3216, 17, 18, 31syl3anc 1490 . 2 (𝜑 → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ((𝑥 𝑦) = (𝑧 𝑧) → 𝑥 = 𝑦) → ((𝑋 𝑌) = (𝑍 𝑍) → 𝑋 = 𝑌)))
3314, 15, 32mp2d 49 1 (𝜑𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3o 1106   = wceq 1652  wcel 2155  {cab 2751  wral 3055  {crab 3059  Vcvv 3350  [wsbc 3598  cdif 3731  cin 3733  {csn 4336  cfv 6070  (class class class)co 6846  cmpt2 6848  Basecbs 16144  distcds 16237  TarskiGcstrkg 25634  TarskiGCcstrkgc 25635  TarskiGBcstrkgb 25636  TarskiGCBcstrkgcb 25637  Itvcitv 25640  LineGclng 25641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-nul 4951
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-iota 6033  df-fv 6078  df-ov 6849  df-trkgc 25652  df-trkg 25657
This theorem is referenced by:  tgcgreqb  25681  tgcgrtriv  25684  tgsegconeq  25686  tgbtwntriv2  25687  tgbtwndiff  25706  tgifscgr  25708  tgbtwnxfr  25730  lnid  25770  tgbtwnconn1lem2  25773  tgbtwnconn1lem3  25774  legtri3  25790  legeq  25793  legbtwn  25794  mirreu3  25854  colmid  25888  krippenlem  25890  lmiisolem  25993  hypcgrlem1  25996  hypcgrlem2  25997  f1otrg  26056
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