Theorem istrkgc 27705

Description: Property of being a Tarski geometry - congruence part. (Contributed by Thierry Arnoux, 14-Mar-2019.)

Hypotheses

Ref	Expression
istrkg.p	⊢ 𝑃 = (Base‘𝐺)
istrkg.d	⊢ − = (dist‘𝐺)
istrkg.i	⊢ 𝐼 = (Itv‘𝐺)

Assertion

Ref	Expression
istrkgc	⊢ (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐼   𝑥,𝑃,𝑦,𝑧   𝑥, − ,𝑦,𝑧

Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem istrkgc

Dummy variables 𝑓 𝑑 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		istrkg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		istrkg.d	. . 3 ⊢ − = (dist‘𝐺)
3		simpl 484	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → 𝑝 = 𝑃)
4		simpr 486	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → 𝑑 = − )
5	4	oveqd 7426	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (𝑥𝑑𝑦) = (𝑥 − 𝑦))
6	4	oveqd 7426	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (𝑦𝑑𝑥) = (𝑦 − 𝑥))
7	5, 6	eqeq12d 2749	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → ((𝑥𝑑𝑦) = (𝑦𝑑𝑥) ↔ (𝑥 − 𝑦) = (𝑦 − 𝑥)))
8	3, 7	raleqbidv 3343	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ↔ ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥)))
9	3, 8	raleqbidv 3343	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ↔ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥)))
10	4	oveqd 7426	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (𝑧𝑑𝑧) = (𝑧 − 𝑧))
11	5, 10	eqeq12d 2749	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) ↔ (𝑥 − 𝑦) = (𝑧 − 𝑧)))
12	11	imbi1d 342	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦) ↔ ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)))
13	3, 12	raleqbidv 3343	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)))
14	3, 13	raleqbidv 3343	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)))
15	3, 14	raleqbidv 3343	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)))
16	9, 15	anbi12d 632	. . 3 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → ((∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)) ↔ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))))
17	1, 2, 16	sbcie2s 17094	. 2 ⊢ (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)) ↔ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))))
18		df-trkgc 27699	. 2 ⊢ TarskiGC = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))}
19	17, 18	elab4g 3674	1 ⊢ (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  [wsbc 3778  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  distcds 17206  TarskiG_Ccstrkgc 27679  Itvcitv 27684

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 5307

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 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-trkgc 27699

This theorem is referenced by:  axtgcgrrflx  27713  axtgcgrid  27714  f1otrg  28122  xmstrkgc  28143  eengtrkg  28244