Theorem axtgcgrid 27405

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 27395	. . . . 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 4188	. . . . . 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 4188	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 3953	. . . . 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 3978	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3942	. . 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 27396	. . . . 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 497	. . . 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 496	. . 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 7364	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝑦) = (𝑋 − 𝑦))
20	19	eqeq1d 2738	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑦) = (𝑧 − 𝑧)))
21		eqeq1 2740	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
22	20, 21	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑋 → (((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦) ↔ ((𝑋 − 𝑦) = (𝑧 − 𝑧) → 𝑋 = 𝑦)))
23		oveq2 7365	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 − 𝑦) = (𝑋 − 𝑌))
24	23	eqeq1d 2738	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑌) = (𝑧 − 𝑧)))
25		eqeq2 2748	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
26	24, 25	imbi12d 344	. . . 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 7375	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 − 𝑧) = (𝑍 − 𝑍))
29	28	eqeq2d 2747	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑋 − 𝑌) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑌) = (𝑍 − 𝑍)))
30	29	imbi1d 341	. . . 4 ⊢ (𝑧 = 𝑍 → (((𝑋 − 𝑌) = (𝑧 − 𝑧) → 𝑋 = 𝑌) ↔ ((𝑋 − 𝑌) = (𝑍 − 𝑍) → 𝑋 = 𝑌)))
31	22, 26, 30	rspc3v 3593	. . 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 1371	. 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 396   ∨ w3o 1086   = wceq 1541   ∈ wcel 2106  {cab 2713  ∀wral 3064  {crab 3407  Vcvv 3445  [wsbc 3739   ∖ cdif 3907   ∩ cin 3909  {csn 4586  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  TarskiG_Ccstrkgc 27370  TarskiG_Bcstrkgb 27371  TarskiG_CBcstrkgcb 27372  Itvcitv 27375  LineGclng 27376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360  df-trkgc 27390  df-trkg 27395

This theorem is referenced by:  tgcgreqb  27423  tgcgrtriv  27426  tgsegconeq  27428  tgbtwntriv2  27429  tgbtwndiff  27448  tgifscgr  27450  tgbtwnxfr  27472  lnid  27512  tgbtwnconn1lem2  27515  tgbtwnconn1lem3  27516  legtri3  27532  legeq  27535  legbtwn  27536  mirreu3  27596  colmid  27630  krippenlem  27632  lmiisolem  27738  hypcgrlem1  27741  hypcgrlem2  27742  f1otrg  27813