| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-cgrg 28519 | . . . 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 2737 | . . . . . . 7
⊢
(dist‘𝐺) =
(dist‘𝐺) | 
| 6 |  | eqid 2737 | . . . . . . 7
⊢
(cgrG‘𝐺) =
(cgrG‘𝐺) | 
| 7 | 4, 5, 6 | iscgrg 28520 | . . . . . 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 2743 | . . . 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 2844 | . . . . . . 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 2844 | . . . . . . 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 3251 | . . . . . . . 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 3251 | . . . . . . 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 2743 | . . . . 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 3202 | . . . 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 28520 | . . . 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 28520 | . . . . . . . 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 2777 | . . . 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 3251 | . . . . . . . 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 3251 | . . . . . . 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 2843 | . . . . . . 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 2843 | . . . . . . 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 3251 | . . . . . . . 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 3251 | . . . . . . 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 2777 | . . . . 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 3202 | . . . 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 28520 | . . . 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 2737 | . . . . . 6
⊢ dom 𝑥 = dom 𝑥 | 
| 66 |  | eqidd 2738 | . . . . . . 7
⊢ ((𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥) → ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗))) | 
| 67 | 66 | rgen2 3199 | . . . . . 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 28520 | . . 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 8771 | 1
⊢ (𝐺 ∈ TarskiG →
(cgrG‘𝐺) Er (𝑃 ↑pm
ℝ)) |