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Theorem ercgrg 26463
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 26457 . . . 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 7411 . . 3 Rel (cgrG‘𝐺)
32a1i 11 . 2 (𝐺 ∈ TarskiG → Rel (cgrG‘𝐺))
4 ercgrg.p . . . . . . 7 𝑃 = (Base‘𝐺)
5 eqid 2738 . . . . . . 7 (dist‘𝐺) = (dist‘𝐺)
6 eqid 2738 . . . . . . 7 (cgrG‘𝐺) = (cgrG‘𝐺)
74, 5, 6iscgrg 26458 . . . . . 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 480 . . . . 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 498 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑦 ∈ (𝑃pm ℝ)))
109ancomd 465 . . 3 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑦 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)))
118simprd 499 . . . . . 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 498 . . . . 5 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑥 = dom 𝑦)
1312eqcomd 2744 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑦 = dom 𝑥)
14 simpl 486 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → (𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦))
15 simprl 771 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑖 ∈ dom 𝑦)
1612adantr 484 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → dom 𝑥 = dom 𝑦)
1715, 16eleqtrrd 2836 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑖 ∈ dom 𝑥)
18 simprr 773 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑦)
1918, 16eleqtrrd 2836 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑥)
2011simprd 499 . . . . . . . . 9 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2120r19.21bi 3121 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ 𝑖 ∈ dom 𝑥) → ∀𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2221r19.21bi 3121 . . . . . . 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 837 . . . . . 6 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2423eqcomd 2744 . . . . 5 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
2524ralrimivva 3103 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
2613, 25jca 515 . . 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 26458 . . . 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 484 . . 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 713 . 2 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑦(cgrG‘𝐺)𝑥)
309simpld 498 . . . . 5 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑥 ∈ (𝑃pm ℝ))
3130adantrr 717 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑥 ∈ (𝑃pm ℝ))
324, 5, 6iscgrg 26458 . . . . . . . 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 480 . . . . . . 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 716 . . . . . 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 498 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (𝑦 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)))
3635simprd 499 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑧 ∈ (𝑃pm ℝ))
3731, 36jca 515 . . 3 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)))
388adantrr 717 . . . . . . 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 499 . . . . . 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 498 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑦)
4134simprd 499 . . . . . 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 498 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑦 = dom 𝑧)
4340, 42eqtrd 2773 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑧)
4439simprd 499 . . . . . . . . 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 3121 . . . . . . . 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 3121 . . . . . . 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 470 . . . . . 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 486 . . . . . . 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 771 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑖 ∈ dom 𝑥)
5040adantr 484 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → dom 𝑥 = dom 𝑦)
5149, 50eleqtrd 2835 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑖 ∈ dom 𝑦)
52 simprr 773 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑥)
5352, 50eleqtrd 2835 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑦)
5441simprd 499 . . . . . . . . 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 3121 . . . . . . . 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 3121 . . . . . . 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 837 . . . . . 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 2773 . . . . 5 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))
5958ralrimivva 3103 . . . 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 515 . . 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 26458 . . . 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 484 . . 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 713 . 2 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑥(cgrG‘𝐺)𝑧)
64 pm4.24 567 . . . 4 (𝑥 ∈ (𝑃pm ℝ) ↔ (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)))
65 eqid 2738 . . . . . 6 dom 𝑥 = dom 𝑥
66 eqidd 2739 . . . . . . 7 ((𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
6766rgen2 3115 . . . . . 6 𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗))
6865, 67pm3.2i 474 . . . . 5 (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
6968biantru 533 . . . 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)))))
7064, 69bitri 278 . . 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)))))
714, 5, 6iscgrg 26458 . . 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))))))
7270, 71bitr4id 293 . 2 (𝐺 ∈ TarskiG → (𝑥 ∈ (𝑃pm ℝ) ↔ 𝑥(cgrG‘𝐺)𝑥))
733, 29, 63, 72iserd 8346 1 (𝐺 ∈ TarskiG → (cgrG‘𝐺) Er (𝑃pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wral 3053  Vcvv 3398   class class class wbr 5030  dom cdm 5525  Rel wrel 5530  cfv 6339  (class class class)co 7170   Er wer 8317  pm cpm 8438  cr 10614  Basecbs 16586  distcds 16677  TarskiGcstrkg 26376  cgrGccgrg 26456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-er 8320  df-cgrg 26457
This theorem is referenced by: (None)
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