Theorem ercgrg 28540

Description: The shape congruence relation is an equivalence relation. Statement 4.4 of [Schwabhauser] p. 35. (Contributed by Thierry Arnoux, 9-Apr-2019.)

Hypothesis

Ref	Expression
ercgrg.p	⊢ 𝑃 = (Base‘𝐺)

Assertion

Ref	Expression
ercgrg	⊢ (𝐺 ∈ TarskiG → (cgrG‘𝐺) Er (𝑃 ↑pm ℝ))

Proof of Theorem ercgrg

Dummy variables 𝑎 𝑏 𝑔 𝑖 𝑗 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cgrg 28534	. . . 4 ⊢ cgrG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))})
2	1	relmptopab 7683	. . 3 ⊢ Rel (cgrG‘𝐺)
3	2	a1i 11	. 2 ⊢ (𝐺 ∈ TarskiG → Rel (cgrG‘𝐺))
4		ercgrg.p	. . . . . . 7 ⊢ 𝑃 = (Base‘𝐺)
5		eqid 2735	. . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺)
6		eqid 2735	. . . . . . 7 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
7	4, 5, 6	iscgrg 28535	. . . . . 6 ⊢ (𝐺 ∈ TarskiG → (𝑥(cgrG‘𝐺)𝑦 ↔ ((𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑦 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗))))))
8	7	biimpa 476	. . . . 5 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ((𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑦 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))))
9	8	simpld 494	. . . 4 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑦 ∈ (𝑃 ↑pm ℝ)))
10	9	ancomd 461	. . 3 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑦 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm ℝ)))
11	8	simprd 495	. . . . . 6 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗))))
12	11	simpld 494	. . . . 5 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑥 = dom 𝑦)
13	12	eqcomd 2741	. . . 4 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑦 = dom 𝑥)
14		simpl 482	. . . . . . 7 ⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → (𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦))
15		simprl 771	. . . . . . . 8 ⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → 𝑖 ∈ dom 𝑦)
16	12	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → dom 𝑥 = dom 𝑦)
17	15, 16	eleqtrrd 2842	. . . . . . 7 ⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → 𝑖 ∈ dom 𝑥)
18		simprr 773	. . . . . . . 8 ⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑦)
19	18, 16	eleqtrrd 2842	. . . . . . 7 ⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑥)
20	11	simprd 495	. . . . . . . . 9 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))
21	20	r19.21bi 3249	. . . . . . . 8 ⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ 𝑖 ∈ dom 𝑥) → ∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))
22	21	r19.21bi 3249	. . . . . . 7 ⊢ ((((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ 𝑖 ∈ dom 𝑥) ∧ 𝑗 ∈ dom 𝑥) → ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))
23	14, 17, 19, 22	syl21anc 838	. . . . . 6 ⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))
24	23	eqcomd 2741	. . . . 5 ⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)))
25	24	ralrimivva 3200	. . . 4 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑦∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)))
26	13, 25	jca 511	. . 3 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (dom 𝑦 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑦∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗))))
27	4, 5, 6	iscgrg 28535	. . . 4 ⊢ (𝐺 ∈ TarskiG → (𝑦(cgrG‘𝐺)𝑥 ↔ ((𝑦 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑦 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑦∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗))))))
28	27	adantr 480	. . 3 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑦(cgrG‘𝐺)𝑥 ↔ ((𝑦 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑦 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑦∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗))))))
29	10, 26, 28	mpbir2and 713	. 2 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑦(cgrG‘𝐺)𝑥)
30	9	simpld 494	. . . . 5 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑥 ∈ (𝑃 ↑pm ℝ))
31	30	adantrr 717	. . . 4 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → 𝑥 ∈ (𝑃 ↑pm ℝ))
32	4, 5, 6	iscgrg 28535	. . . . . . . 8 ⊢ (𝐺 ∈ TarskiG → (𝑦(cgrG‘𝐺)𝑧 ↔ ((𝑦 ∈ (𝑃 ↑pm ℝ) ∧ 𝑧 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑦 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑦∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗))))))
33	32	biimpa 476	. . . . . . 7 ⊢ ((𝐺 ∈ TarskiG ∧ 𝑦(cgrG‘𝐺)𝑧) → ((𝑦 ∈ (𝑃 ↑pm ℝ) ∧ 𝑧 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑦 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑦∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗)))))
34	33	adantrl 716	. . . . . 6 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → ((𝑦 ∈ (𝑃 ↑pm ℝ) ∧ 𝑧 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑦 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑦∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗)))))
35	34	simpld 494	. . . . 5 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → (𝑦 ∈ (𝑃 ↑pm ℝ) ∧ 𝑧 ∈ (𝑃 ↑pm ℝ)))
36	35	simprd 495	. . . 4 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → 𝑧 ∈ (𝑃 ↑pm ℝ))
37	31, 36	jca 511	. . 3 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → (𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑧 ∈ (𝑃 ↑pm ℝ)))
38	8	adantrr 717	. . . . . . 7 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → ((𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑦 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))))
39	38	simprd 495	. . . . . 6 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗))))
40	39	simpld 494	. . . . 5 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑦)
41	34	simprd 495	. . . . . 6 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → (dom 𝑦 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑦∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗))))
42	41	simpld 494	. . . . 5 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → dom 𝑦 = dom 𝑧)
43	40, 42	eqtrd 2775	. . . 4 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑧)
44	39	simprd 495	. . . . . . . . 9 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))
45	44	r19.21bi 3249	. . . . . . . 8 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ 𝑖 ∈ dom 𝑥) → ∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))
46	45	r19.21bi 3249	. . . . . . 7 ⊢ ((((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ 𝑖 ∈ dom 𝑥) ∧ 𝑗 ∈ dom 𝑥) → ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))
47	46	anasss 466	. . . . . 6 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)))
48		simpl 482	. . . . . . 7 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → (𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)))
49		simprl 771	. . . . . . . 8 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → 𝑖 ∈ dom 𝑥)
50	40	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → dom 𝑥 = dom 𝑦)
51	49, 50	eleqtrd 2841	. . . . . . 7 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → 𝑖 ∈ dom 𝑦)
52		simprr 773	. . . . . . . 8 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑥)
53	52, 50	eleqtrd 2841	. . . . . . 7 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑦)
54	41	simprd 495	. . . . . . . . 9 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → ∀𝑖 ∈ dom 𝑦∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗)))
55	54	r19.21bi 3249	. . . . . . . 8 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ 𝑖 ∈ dom 𝑦) → ∀𝑗 ∈ dom 𝑦((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗)))
56	55	r19.21bi 3249	. . . . . . 7 ⊢ ((((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ 𝑖 ∈ dom 𝑦) ∧ 𝑗 ∈ dom 𝑦) → ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗)))
57	48, 51, 53, 56	syl21anc 838	. . . . . 6 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → ((𝑦‘𝑖)(dist‘𝐺)(𝑦‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗)))
58	47, 57	eqtrd 2775	. . . . 5 ⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗)))
59	58	ralrimivva 3200	. . . 4 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗)))
60	43, 59	jca 511	. . 3 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → (dom 𝑥 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗))))
61	4, 5, 6	iscgrg 28535	. . . 4 ⊢ (𝐺 ∈ TarskiG → (𝑥(cgrG‘𝐺)𝑧 ↔ ((𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑧 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑥 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗))))))
62	61	adantr 480	. . 3 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → (𝑥(cgrG‘𝐺)𝑧 ↔ ((𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑧 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑥 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑧‘𝑖)(dist‘𝐺)(𝑧‘𝑗))))))
63	37, 60, 62	mpbir2and 713	. 2 ⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → 𝑥(cgrG‘𝐺)𝑧)
64		pm4.24 563	. . . 4 ⊢ (𝑥 ∈ (𝑃 ↑pm ℝ) ↔ (𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm ℝ)))
65		eqid 2735	. . . . . 6 ⊢ dom 𝑥 = dom 𝑥
66		eqidd 2736	. . . . . . 7 ⊢ ((𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥) → ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)))
67	66	rgen2 3197	. . . . . 6 ⊢ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗))
68	65, 67	pm3.2i 470	. . . . 5 ⊢ (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)))
69	68	biantru 529	. . . 4 ⊢ ((𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm ℝ)) ↔ ((𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)))))
70	64, 69	bitri 275	. . 3 ⊢ (𝑥 ∈ (𝑃 ↑pm ℝ) ↔ ((𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)))))
71	4, 5, 6	iscgrg 28535	. . 3 ⊢ (𝐺 ∈ TarskiG → (𝑥(cgrG‘𝐺)𝑥 ↔ ((𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm ℝ)) ∧ (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗))))))
72	70, 71	bitr4id 290	. 2 ⊢ (𝐺 ∈ TarskiG → (𝑥 ∈ (𝑃 ↑pm ℝ) ↔ 𝑥(cgrG‘𝐺)𝑥))
73	3, 29, 63, 72	iserd 8770	1 ⊢ (𝐺 ∈ TarskiG → (cgrG‘𝐺) Er (𝑃 ↑pm ℝ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   class class class wbr 5148  dom cdm 5689  Rel wrel 5694  ‘cfv 6563  (class class class)co 7431   Er wer 8741   ↑_pm cpm 8866  ℝcr 11152  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  cgrGccgrg 28533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-er 8744  df-cgrg 28534

This theorem is referenced by: (None)