Theorem axtgcgrid 28485

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 28475	. . . . 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 4244	. . . . . 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 4244	. . . . . 6 ⊢ (TarskiGC ∩ TarskiGB) ⊆ TarskiGC
4	2, 3	sstri 4004	. . . . 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 4029	. . . 4 ⊢ TarskiG ⊆ TarskiGC
6		axtrkg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	5, 6	sselid 3992	. . 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 28476	. . . . 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 496	. . . 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 495	. . 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 7437	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 − 𝑦) = (𝑋 − 𝑦))
20	19	eqeq1d 2736	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑦) = (𝑧 − 𝑧)))
21		eqeq1 2738	. . . . 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 7438	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 − 𝑦) = (𝑋 − 𝑌))
24	23	eqeq1d 2736	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑋 − 𝑌) = (𝑧 − 𝑧)))
25		eqeq2 2746	. . . . 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 7448	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 − 𝑧) = (𝑍 − 𝑍))
29	28	eqeq2d 2745	. . . . 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 3637	. . 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 395   ∨ w3o 1085   = wceq 1536   ∈ wcel 2105  {cab 2711  ∀wral 3058  {crab 3432  Vcvv 3477  [wsbc 3790   ∖ cdif 3959   ∩ cin 3961  {csn 4630  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  TarskiG_Ccstrkgc 28450  TarskiG_Bcstrkgb 28451  TarskiG_CBcstrkgcb 28452  Itvcitv 28455  LineGclng 28456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-trkgc 28470  df-trkg 28475

This theorem is referenced by:  tgcgreqb  28503  tgcgrtriv  28506  tgsegconeq  28508  tgbtwntriv2  28509  tgbtwndiff  28528  tgifscgr  28530  tgbtwnxfr  28552  lnid  28592  tgbtwnconn1lem2  28595  tgbtwnconn1lem3  28596  legtri3  28612  legeq  28615  legbtwn  28616  mirreu3  28676  colmid  28710  krippenlem  28712  lmiisolem  28818  hypcgrlem1  28821  hypcgrlem2  28822  f1otrg  28893