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Theorem ercgrg 25307
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 (𝑃pm ℝ))

Proof of Theorem ercgrg
Dummy variables 𝑎 𝑏 𝑔 𝑖 𝑗 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cgrg 25301 . . . 4 cgrG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎𝑗 ∈ dom 𝑎((𝑎𝑖)(dist‘𝑔)(𝑎𝑗)) = ((𝑏𝑖)(dist‘𝑔)(𝑏𝑗))))})
21relmptopab 6837 . . 3 Rel (cgrG‘𝐺)
32a1i 11 . 2 (𝐺 ∈ TarskiG → Rel (cgrG‘𝐺))
4 ercgrg.p . . . . . . 7 𝑃 = (Base‘𝐺)
5 eqid 2626 . . . . . . 7 (dist‘𝐺) = (dist‘𝐺)
6 eqid 2626 . . . . . . 7 (cgrG‘𝐺) = (cgrG‘𝐺)
74, 5, 6iscgrg 25302 . . . . . 6 (𝐺 ∈ TarskiG → (𝑥(cgrG‘𝐺)𝑦 ↔ ((𝑥 ∈ (𝑃pm ℝ) ∧ 𝑦 ∈ (𝑃pm ℝ)) ∧ (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗))))))
87biimpa 501 . . . . 5 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ((𝑥 ∈ (𝑃pm ℝ) ∧ 𝑦 ∈ (𝑃pm ℝ)) ∧ (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))))
98simpld 475 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑦 ∈ (𝑃pm ℝ)))
109ancomd 467 . . 3 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑦 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)))
118simprd 479 . . . . . 6 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗))))
1211simpld 475 . . . . 5 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑥 = dom 𝑦)
1312eqcomd 2632 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑦 = dom 𝑥)
14 simpl 473 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → (𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦))
15 simprl 793 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑖 ∈ dom 𝑦)
1612adantr 481 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → dom 𝑥 = dom 𝑦)
1715, 16eleqtrrd 2707 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑖 ∈ dom 𝑥)
18 simprr 795 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑦)
1918, 16eleqtrrd 2707 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑥)
2011simprd 479 . . . . . . . . 9 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2120r19.21bi 2932 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ 𝑖 ∈ dom 𝑥) → ∀𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2221r19.21bi 2932 . . . . . . 7 ((((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ 𝑖 ∈ dom 𝑥) ∧ 𝑗 ∈ dom 𝑥) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2314, 17, 19, 22syl21anc 1322 . . . . . 6 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2423eqcomd 2632 . . . . 5 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
2524ralrimivva 2970 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
2613, 25jca 554 . . 3 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (dom 𝑦 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗))))
274, 5, 6iscgrg 25302 . . . 4 (𝐺 ∈ TarskiG → (𝑦(cgrG‘𝐺)𝑥 ↔ ((𝑦 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)) ∧ (dom 𝑦 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗))))))
2827adantr 481 . . 3 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑦(cgrG‘𝐺)𝑥 ↔ ((𝑦 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)) ∧ (dom 𝑦 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗))))))
2910, 26, 28mpbir2and 956 . 2 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑦(cgrG‘𝐺)𝑥)
309simpld 475 . . . . 5 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑥 ∈ (𝑃pm ℝ))
3130adantrr 752 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑥 ∈ (𝑃pm ℝ))
324, 5, 6iscgrg 25302 . . . . . . . 8 (𝐺 ∈ TarskiG → (𝑦(cgrG‘𝐺)𝑧 ↔ ((𝑦 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)) ∧ (dom 𝑦 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗))))))
3332biimpa 501 . . . . . . 7 ((𝐺 ∈ TarskiG ∧ 𝑦(cgrG‘𝐺)𝑧) → ((𝑦 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)) ∧ (dom 𝑦 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))))
3433adantrl 751 . . . . . 6 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → ((𝑦 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)) ∧ (dom 𝑦 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))))
3534simpld 475 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (𝑦 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)))
3635simprd 479 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑧 ∈ (𝑃pm ℝ))
3731, 36jca 554 . . 3 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)))
388adantrr 752 . . . . . . 7 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → ((𝑥 ∈ (𝑃pm ℝ) ∧ 𝑦 ∈ (𝑃pm ℝ)) ∧ (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))))
3938simprd 479 . . . . . 6 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (dom 𝑥 = dom 𝑦 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗))))
4039simpld 475 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑦)
4134simprd 479 . . . . . 6 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (dom 𝑦 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗))))
4241simpld 475 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑦 = dom 𝑧)
4340, 42eqtrd 2660 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑧)
4439simprd 479 . . . . . . . . 9 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
4544r19.21bi 2932 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ 𝑖 ∈ dom 𝑥) → ∀𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
4645r19.21bi 2932 . . . . . . 7 ((((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ 𝑖 ∈ dom 𝑥) ∧ 𝑗 ∈ dom 𝑥) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
4746anasss 678 . . . . . 6 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
48 simpl 473 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → (𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)))
49 simprl 793 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑖 ∈ dom 𝑥)
5040adantr 481 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → dom 𝑥 = dom 𝑦)
5149, 50eleqtrd 2706 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑖 ∈ dom 𝑦)
52 simprr 795 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑥)
5352, 50eleqtrd 2706 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑦)
5441simprd 479 . . . . . . . . 9 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))
5554r19.21bi 2932 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ 𝑖 ∈ dom 𝑦) → ∀𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))
5655r19.21bi 2932 . . . . . . 7 ((((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ 𝑖 ∈ dom 𝑦) ∧ 𝑗 ∈ dom 𝑦) → ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))
5748, 51, 53, 56syl21anc 1322 . . . . . 6 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))
5847, 57eqtrd 2660 . . . . 5 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))
5958ralrimivva 2970 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))
6043, 59jca 554 . . 3 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (dom 𝑥 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗))))
614, 5, 6iscgrg 25302 . . . 4 (𝐺 ∈ TarskiG → (𝑥(cgrG‘𝐺)𝑧 ↔ ((𝑥 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)) ∧ (dom 𝑥 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗))))))
6261adantr 481 . . 3 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (𝑥(cgrG‘𝐺)𝑧 ↔ ((𝑥 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)) ∧ (dom 𝑥 = dom 𝑧 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗))))))
6337, 60, 62mpbir2and 956 . 2 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑥(cgrG‘𝐺)𝑧)
644, 5, 6iscgrg 25302 . . 3 (𝐺 ∈ TarskiG → (𝑥(cgrG‘𝐺)𝑥 ↔ ((𝑥 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)) ∧ (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗))))))
65 pm4.24 674 . . . 4 (𝑥 ∈ (𝑃pm ℝ) ↔ (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)))
66 eqid 2626 . . . . . 6 dom 𝑥 = dom 𝑥
67 eqidd 2627 . . . . . . 7 ((𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
6867rgen2a 2976 . . . . . 6 𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗))
6966, 68pm3.2i 471 . . . . 5 (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
7069biantru 526 . . . 4 ((𝑥 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)) ↔ ((𝑥 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)) ∧ (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))))
7165, 70bitri 264 . . 3 (𝑥 ∈ (𝑃pm ℝ) ↔ ((𝑥 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)) ∧ (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))))
7264, 71syl6rbbr 279 . 2 (𝐺 ∈ TarskiG → (𝑥 ∈ (𝑃pm ℝ) ↔ 𝑥(cgrG‘𝐺)𝑥))
733, 29, 63, 72iserd 7714 1 (𝐺 ∈ TarskiG → (cgrG‘𝐺) Er (𝑃pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191   class class class wbr 4618  dom cdm 5079  Rel wrel 5084  cfv 5850  (class class class)co 6605   Er wer 7685  pm cpm 7804  cr 9880  Basecbs 15776  distcds 15866  TarskiGcstrkg 25224  cgrGccgrg 25300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-er 7688  df-cgrg 25301
This theorem is referenced by: (None)
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