Theorem istrkgc 28544

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 7377	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (𝑥𝑑𝑦) = (𝑥 − 𝑦))
6	4	oveqd 7377	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (𝑦𝑑𝑥) = (𝑦 − 𝑥))
7	5, 6	eqeq12d 2757	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → ((𝑥𝑑𝑦) = (𝑦𝑑𝑥) ↔ (𝑥 − 𝑦) = (𝑦 − 𝑥)))
8	3, 7	raleqbidv 3315	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ↔ ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥)))
9	3, 8	raleqbidv 3315	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ↔ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥)))
10	4	oveqd 7377	. . . . . . . . 9 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (𝑧𝑑𝑧) = (𝑧 − 𝑧))
11	5, 10	eqeq12d 2757	. . . . . . . 8 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) ↔ (𝑥 − 𝑦) = (𝑧 − 𝑧)))
12	11	imbi1d 343	. . . . . . 7 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦) ↔ ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)))
13	3, 12	raleqbidv 3315	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)))
14	3, 13	raleqbidv 3315	. . . . 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 3315	. . . 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 639	. . 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 17126	. 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 28538	. 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 3623	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 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433  [wsbc 3725  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  distcds 17224  TarskiG_Ccstrkgc 28518  Itvcitv 28523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5231

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363  df-trkgc 28538

This theorem is referenced by:  axtgcgrrflx  28552  axtgcgrid  28553  f1otrg  28961  xmstrkgc  28976  eengtrkg  29077