Theorem axtgcgrid 26824

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 26814	. . . . 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 4162	. . . . . 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 4162	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 3930	. . . . 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 3955	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3919	. . 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 26815	. . . . 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 7282	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝑦) = (𝑋 − 𝑦))
20	19	eqeq1d 2740	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑦) = (𝑧 − 𝑧)))
21		eqeq1 2742	. . . . 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 7283	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 − 𝑦) = (𝑋 − 𝑌))
24	23	eqeq1d 2740	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑌) = (𝑧 − 𝑧)))
25		eqeq2 2750	. . . . 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 7293	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 − 𝑧) = (𝑍 − 𝑍))
29	28	eqeq2d 2749	. . . . 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 3573	. . 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 1370	. 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 1085   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  {crab 3068  Vcvv 3432  [wsbc 3716   ∖ cdif 3884   ∩ cin 3886  {csn 4561  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  TarskiG_Ccstrkgc 26789  TarskiG_Bcstrkgb 26790  TarskiG_CBcstrkgcb 26791  Itvcitv 26794  LineGclng 26795

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-trkgc 26809  df-trkg 26814

This theorem is referenced by:  tgcgreqb  26842  tgcgrtriv  26845  tgsegconeq  26847  tgbtwntriv2  26848  tgbtwndiff  26867  tgifscgr  26869  tgbtwnxfr  26891  lnid  26931  tgbtwnconn1lem2  26934  tgbtwnconn1lem3  26935  legtri3  26951  legeq  26954  legbtwn  26955  mirreu3  27015  colmid  27049  krippenlem  27051  lmiisolem  27157  hypcgrlem1  27160  hypcgrlem2  27161  f1otrg  27232