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Theorem ercgrg 26878
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 26872 . . . 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 7519 . . 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 26873 . . . . . 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 477 . . . . 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 495 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑦 ∈ (𝑃pm ℝ)))
109ancomd 462 . . 3 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑦 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)))
118simprd 496 . . . . . 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 495 . . . . 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 483 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → (𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦))
15 simprl 768 . . . . . . . 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 2842 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑖 ∈ dom 𝑥)
18 simprr 770 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑦)
1918, 16eleqtrrd 2842 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑥)
2011simprd 496 . . . . . . . . 9 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2120r19.21bi 3134 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ 𝑖 ∈ dom 𝑥) → ∀𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2221r19.21bi 3134 . . . . . . 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 835 . . . . . 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 3123 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
2613, 25jca 512 . . 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 26873 . . . 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 710 . 2 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑦(cgrG‘𝐺)𝑥)
309simpld 495 . . . . 5 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑥 ∈ (𝑃pm ℝ))
3130adantrr 714 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑥 ∈ (𝑃pm ℝ))
324, 5, 6iscgrg 26873 . . . . . . . 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 477 . . . . . . 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 713 . . . . . 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 495 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (𝑦 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)))
3635simprd 496 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑧 ∈ (𝑃pm ℝ))
3731, 36jca 512 . . 3 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)))
388adantrr 714 . . . . . . 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 496 . . . . . 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 495 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑦)
4134simprd 496 . . . . . 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 495 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑦 = dom 𝑧)
4340, 42eqtrd 2778 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑧)
4439simprd 496 . . . . . . . . 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 3134 . . . . . . . 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 3134 . . . . . . 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 467 . . . . . 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 483 . . . . . . 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 768 . . . . . . . 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 2841 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑖 ∈ dom 𝑦)
52 simprr 770 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑥)
5352, 50eleqtrd 2841 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑦)
5441simprd 496 . . . . . . . . 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 3134 . . . . . . . 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 3134 . . . . . . 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 835 . . . . . 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 2778 . . . . 5 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))
5958ralrimivva 3123 . . . 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 512 . . 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 26873 . . . 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 710 . 2 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑥(cgrG‘𝐺)𝑧)
64 pm4.24 564 . . . 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 3120 . . . . . 6 𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗))
6865, 67pm3.2i 471 . . . . 5 (dom 𝑥 = dom 𝑥 ∧ ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
6968biantru 530 . . . 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 274 . . 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 26873 . . 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 290 . 2 (𝐺 ∈ TarskiG → (𝑥 ∈ (𝑃pm ℝ) ↔ 𝑥(cgrG‘𝐺)𝑥))
733, 29, 63, 72iserd 8524 1 (𝐺 ∈ TarskiG → (cgrG‘𝐺) Er (𝑃pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432   class class class wbr 5074  dom cdm 5589  Rel wrel 5594  cfv 6433  (class class class)co 7275   Er wer 8495  pm cpm 8616  cr 10870  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  cgrGccgrg 26871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-er 8498  df-cgrg 26872
This theorem is referenced by: (None)
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