MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ercgrg Structured version   Visualization version   GIF version

Theorem ercgrg 25768
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 25762 . . . 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 7117 . . 3 Rel (cgrG‘𝐺)
32a1i 11 . 2 (𝐺 ∈ TarskiG → Rel (cgrG‘𝐺))
4 ercgrg.p . . . . . . 7 𝑃 = (Base‘𝐺)
5 eqid 2799 . . . . . . 7 (dist‘𝐺) = (dist‘𝐺)
6 eqid 2799 . . . . . . 7 (cgrG‘𝐺) = (cgrG‘𝐺)
74, 5, 6iscgrg 25763 . . . . . 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 469 . . . . 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 489 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑦 ∈ (𝑃pm ℝ)))
109ancomd 454 . . 3 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑦 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)))
118simprd 490 . . . . . 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 489 . . . . 5 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑥 = dom 𝑦)
1312eqcomd 2805 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑦 = dom 𝑥)
14 simpl 475 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → (𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦))
15 simprl 788 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑖 ∈ dom 𝑦)
1612adantr 473 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → dom 𝑥 = dom 𝑦)
1715, 16eleqtrrd 2881 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑖 ∈ dom 𝑥)
18 simprr 790 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑦)
1918, 16eleqtrrd 2881 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑥)
2011simprd 490 . . . . . . . . 9 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2120r19.21bi 3113 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ 𝑖 ∈ dom 𝑥) → ∀𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2221r19.21bi 3113 . . . . . . 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 867 . . . . . 6 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)))
2423eqcomd 2805 . . . . 5 (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦)) → ((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
2524ralrimivva 3152 . . . 4 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → ∀𝑖 ∈ dom 𝑦𝑗 ∈ dom 𝑦((𝑦𝑖)(dist‘𝐺)(𝑦𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
2613, 25jca 508 . . 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 25763 . . . 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 473 . . 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 705 . 2 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑦(cgrG‘𝐺)𝑥)
309simpld 489 . . . . 5 ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → 𝑥 ∈ (𝑃pm ℝ))
3130adantrr 709 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑥 ∈ (𝑃pm ℝ))
324, 5, 6iscgrg 25763 . . . . . . . 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 469 . . . . . . 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 708 . . . . . 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 489 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (𝑦 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)))
3635simprd 490 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑧 ∈ (𝑃pm ℝ))
3731, 36jca 508 . . 3 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑧 ∈ (𝑃pm ℝ)))
388adantrr 709 . . . . . . 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 490 . . . . . 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 489 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑦)
4134simprd 490 . . . . . 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 489 . . . . 5 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑦 = dom 𝑧)
4340, 42eqtrd 2833 . . . 4 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑧)
4439simprd 490 . . . . . . . . 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 3113 . . . . . . . 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 3113 . . . . . . 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 459 . . . . . 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 475 . . . . . . 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 788 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑖 ∈ dom 𝑥)
5040adantr 473 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → dom 𝑥 = dom 𝑦)
5149, 50eleqtrd 2880 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑖 ∈ dom 𝑦)
52 simprr 790 . . . . . . . 8 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑥)
5352, 50eleqtrd 2880 . . . . . . 7 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑦)
5441simprd 490 . . . . . . . . 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 3113 . . . . . . . 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 3113 . . . . . . 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 867 . . . . . 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 2833 . . . . 5 (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥)) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑧𝑖)(dist‘𝐺)(𝑧𝑗)))
5958ralrimivva 3152 . . . 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 508 . . 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 25763 . . . 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 473 . . 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 705 . 2 ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦𝑦(cgrG‘𝐺)𝑧)) → 𝑥(cgrG‘𝐺)𝑧)
644, 5, 6iscgrg 25763 . . 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 pm4.24 560 . . . 4 (𝑥 ∈ (𝑃pm ℝ) ↔ (𝑥 ∈ (𝑃pm ℝ) ∧ 𝑥 ∈ (𝑃pm ℝ)))
66 eqid 2799 . . . . . 6 dom 𝑥 = dom 𝑥
67 eqidd 2800 . . . . . . 7 ((𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥) → ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)))
6867rgen2a 3158 . . . . . 6 𝑖 ∈ dom 𝑥𝑗 ∈ dom 𝑥((𝑥𝑖)(dist‘𝐺)(𝑥𝑗)) = ((𝑥𝑖)(dist‘𝐺)(𝑥𝑗))
6966, 68pm3.2i 463 . . . . 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 267 . . 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 282 . 2 (𝐺 ∈ TarskiG → (𝑥 ∈ (𝑃pm ℝ) ↔ 𝑥(cgrG‘𝐺)𝑥))
733, 29, 63, 72iserd 8008 1 (𝐺 ∈ TarskiG → (cgrG‘𝐺) Er (𝑃pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385   class class class wbr 4843  dom cdm 5312  Rel wrel 5317  cfv 6101  (class class class)co 6878   Er wer 7979  pm cpm 8096  cr 10223  Basecbs 16184  distcds 16276  TarskiGcstrkg 25681  cgrGccgrg 25761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-er 7982  df-cgrg 25762
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator