Theorem istrkgc 27104

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 483	. . . . . 6 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → 𝑝 = 𝑃)
4	3	eqcomd 2742	. . . . 5 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → 𝑃 = 𝑝)
5	4	adantr 481	. . . . . 6 ⊢ (((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) → 𝑃 = 𝑝)
6		simpllr 773	. . . . . . . . 9 ⊢ ((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → 𝑑 = − )
7	6	eqcomd 2742	. . . . . . . 8 ⊢ ((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → − = 𝑑)
8	7	oveqd 7354	. . . . . . 7 ⊢ ((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → (𝑥 − 𝑦) = (𝑥𝑑𝑦))
9	7	oveqd 7354	. . . . . . 7 ⊢ ((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → (𝑦 − 𝑥) = (𝑦𝑑𝑥))
10	8, 9	eqeq12d 2752	. . . . . 6 ⊢ ((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → ((𝑥 − 𝑦) = (𝑦 − 𝑥) ↔ (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
11	5, 10	raleqbidva 3317	. . . . 5 ⊢ (((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) → (∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ↔ ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
12	4, 11	raleqbidva 3317	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ↔ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥)))
13	5	adantr 481	. . . . . . 7 ⊢ ((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → 𝑃 = 𝑝)
14	7	oveqdr 7365	. . . . . . . . 9 ⊢ (((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (𝑥 − 𝑦) = (𝑥𝑑𝑦))
15	7	oveqdr 7365	. . . . . . . . 9 ⊢ (((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (𝑧 − 𝑧) = (𝑧𝑑𝑧))
16	14, 15	eqeq12d 2752	. . . . . . . 8 ⊢ (((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → ((𝑥 − 𝑦) = (𝑧 − 𝑧) ↔ (𝑥𝑑𝑦) = (𝑧𝑑𝑧)))
17	16	imbi1d 341	. . . . . . 7 ⊢ (((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦) ↔ ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
18	13, 17	raleqbidva 3317	. . . . . 6 ⊢ ((((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) → (∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
19	5, 18	raleqbidva 3317	. . . . 5 ⊢ (((𝑝 = 𝑃 ∧ 𝑑 = − ) ∧ 𝑥 ∈ 𝑃) → (∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
20	4, 19	raleqbidva 3317	. . . 4 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → (∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦)))
21	12, 20	anbi12d 631	. . 3 ⊢ ((𝑝 = 𝑃 ∧ 𝑑 = − ) → ((∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 − 𝑦) = (𝑦 − 𝑥) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ((𝑥 − 𝑦) = (𝑧 − 𝑧) → 𝑥 = 𝑦)) ↔ (∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))))
22	1, 2, 21	sbcie2s 16959	. 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))))
23		df-trkgc 27098	. 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))}
24	22, 23	elab4g 3624	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 396   = wceq 1540   ∈ wcel 2105  ∀wral 3061  Vcvv 3441  [wsbc 3727  ‘cfv 6479  (class class class)co 7337  Basecbs 17009  distcds 17068  TarskiG_Ccstrkgc 27078  Itvcitv 27083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-nul 5250

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487  df-ov 7340  df-trkgc 27098

This theorem is referenced by:  axtgcgrrflx  27112  axtgcgrid  27113  f1otrg  27521  xmstrkgc  27542  eengtrkg  27643