Theorem axtgcgrid 28958

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
axtgcgrid.1	⊢ (𝜑 → 𝑋 ∈ 𝑃)
axtgcgrid.2	⊢ (𝜑 → 𝑌 ∈ 𝑃)
axtgcgrid.3	⊢ (𝜑 → 𝑍 ∈ 𝑃)
axtgcgrid.4	⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑍))

Assertion

Ref	Expression
axtgcgrid	⊢ (𝜑 → 𝑋 = 𝑌)

Proof of Theorem axtgcgrid

Dummy variables 𝑓 𝑖 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-trkg 28948	. . . . 5 ⊢ TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
2		inss1 4136	. . . . . 6 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ (TarskiGC ∩ TarskiGB)
3		inss1 4136	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 3922	. . . . 5 ⊢ ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) ⊆ TarskiGC
5	1, 4	eqsstri 3959	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3911	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiGC)
8		axtrkg.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
9		axtrkg.d	. . . . . 6 ⊢ − = (dist‘𝐺)
10		axtrkg.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
11	8, 9, 10	istrkgc 28949	. . . . 5 ⊢ (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))))
12	11	simprbi 503	. . . 4 ⊢ (𝐺 ∈ TarskiGC → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)))
13	12	simprd 501	. . 3 ⊢ (𝐺 ∈ TarskiGC → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))
14	7, 13	syl 18	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))
15		axtgcgrid.4	. 2 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑍))
16		axtgcgrid.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
17		axtgcgrid.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
18		axtgcgrid.3	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
19		oveq1 7453	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝑦) = (𝑋 − 𝑦))
20	19	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑦) = (𝑧 − 𝑧)))
21		eqeq1 2767	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
22	20, 21	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦) ↔ ((𝑋 − 𝑦) = (𝑧 − 𝑧) → 𝑋 = 𝑦)))
23		oveq2 7454	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 − 𝑦) = (𝑋 − 𝑌))
24	23	eqeq1d 2765	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑌) = (𝑧 − 𝑧)))
25		eqeq2 2775	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
26	24, 25	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝑌 → (((𝑋 − 𝑦) = (𝑧 − 𝑧) → 𝑋 = 𝑦) ↔ ((𝑋 − 𝑌) = (𝑧 − 𝑧) → 𝑋 = 𝑌)))
27		id 23	. . . . . . 7 ⊢ (𝑧 = 𝑍 → 𝑧 = 𝑍)
28	27, 27	oveq12d 7464	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 − 𝑧) = (𝑍 − 𝑍))
29	28	eqeq2d 2774	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑋 − 𝑌) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑌) = (𝑍 − 𝑍)))
30	29	imbi1d 344	. . . 4 ⊢ (𝑧 = 𝑍 → (((𝑋 − 𝑌) = (𝑧 − 𝑧) → 𝑋 = 𝑌) ↔ ((𝑋 − 𝑌) = (𝑍 − 𝑍) → 𝑋 = 𝑌)))
31	22, 26, 30	rspc3v 3595	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦) → ((𝑋 − 𝑌) = (𝑍 − 𝑍) → 𝑋 = 𝑌)))
32	16, 17, 18, 31	syl3anc 1399	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦) → ((𝑋 − 𝑌) = (𝑍 − 𝑍) → 𝑋 = 𝑌)))
33	14, 15, 32	mp2d 50	1 ⊢ (𝜑 → 𝑋 = 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  {crab 3416  Vcvv 3454  [wsbc 3742   ∖ cdif 4048   ∩ cin 4092  {csn 4588  ‘cfv 6561  (class class class)co 7446   ∈ cmpo 7448  Basecbs 17419  distcds 17469  TarskiGcstrkg 28922  TarskiG_Ccstrkgc 28923  TarskiG_Bcstrkgb 28924  TarskiG_CBcstrkgcb 28925  Itvcitv 28928  LineGclng 28929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5264

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-trkgc 28943  df-trkg 28948

This theorem is used by:  tgcgreqb  28976  tgcgrtriv  28979  tgsegconeq  28981  tgbtwntriv2  28983  tgbtwndiff  29002  tgifscgr  29004  tgbtwnxfr  29026  lnid  29066  tgbtwnconn1lem2  29069  tgbtwnconn1lem3  29070  legtri3  29086  legeq  29089  legbtwn  29090  mirreu3  29159  colmid  29193  krippenlem  29195  lmiisolem  29334  hypcgrlem1  29338  hypcgrlem2  29339  f1otrg  29481