Theorem axtgcgrid 27694

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 27684	. . . . 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 4227	. . . . . 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 4227	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 3990	. . . . 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 4015	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3979	. . 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 27685	. . . . 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 498	. . . 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 497	. . 3 ⊢ (𝐺 ∈ TarskiGC → ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))
14	7, 13	syl 17	. 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 7411	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝑦) = (𝑋 − 𝑦))
20	19	eqeq1d 2735	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑦) = (𝑧 − 𝑧)))
21		eqeq1 2737	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
22	20, 21	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦) ↔ ((𝑋 − 𝑦) = (𝑧 − 𝑧) → 𝑋 = 𝑦)))
23		oveq2 7412	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 − 𝑦) = (𝑋 − 𝑌))
24	23	eqeq1d 2735	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑌) = (𝑧 − 𝑧)))
25		eqeq2 2745	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
26	24, 25	imbi12d 345	. . . 4 ⊢ (𝑦 = 𝑌 → (((𝑋 − 𝑦) = (𝑧 − 𝑧) → 𝑋 = 𝑦) ↔ ((𝑋 − 𝑌) = (𝑧 − 𝑧) → 𝑋 = 𝑌)))
27		id 22	. . . . . . 7 ⊢ (𝑧 = 𝑍 → 𝑧 = 𝑍)
28	27, 27	oveq12d 7422	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 − 𝑧) = (𝑍 − 𝑍))
29	28	eqeq2d 2744	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑋 − 𝑌) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑌) = (𝑍 − 𝑍)))
30	29	imbi1d 342	. . . 4 ⊢ (𝑧 = 𝑍 → (((𝑋 − 𝑌) = (𝑧 − 𝑧) → 𝑋 = 𝑌) ↔ ((𝑋 − 𝑌) = (𝑍 − 𝑍) → 𝑋 = 𝑌)))
31	22, 26, 30	rspc3v 3626	. . 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 1372	. 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 49	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Syntax hints:   → wi 4   ∧ wa 397   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  {crab 3433  Vcvv 3475  [wsbc 3776   ∖ cdif 3944   ∩ cin 3946  {csn 4627  ‘cfv 6540  (class class class)co 7404   ∈ cmpo 7406  Basecbs 17140  distcds 17202  TarskiGcstrkg 27658  TarskiG_Ccstrkgc 27659  TarskiG_Bcstrkgb 27660  TarskiG_CBcstrkgcb 27661  Itvcitv 27664  LineGclng 27665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7407  df-trkgc 27679  df-trkg 27684

This theorem is referenced by:  tgcgreqb  27712  tgcgrtriv  27715  tgsegconeq  27717  tgbtwntriv2  27718  tgbtwndiff  27737  tgifscgr  27739  tgbtwnxfr  27761  lnid  27801  tgbtwnconn1lem2  27804  tgbtwnconn1lem3  27805  legtri3  27821  legeq  27824  legbtwn  27825  mirreu3  27885  colmid  27919  krippenlem  27921  lmiisolem  28027  hypcgrlem1  28030  hypcgrlem2  28031  f1otrg  28102