Step | Hyp | Ref
| 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 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))}) |
2 | 1 | relmptopab 7411 |
. . 3
⊢ Rel
(cgrG‘𝐺) |
3 | 2 | a1i 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) |
7 | 4, 5, 6 | iscgrg 26458 |
. . . . . 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 480 |
. . . . 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 498 |
. . . 4
⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑦 ∈ (𝑃 ↑pm
ℝ))) |
10 | 9 | ancomd 465 |
. . 3
⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → (𝑦 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm
ℝ))) |
11 | 8 | simprd 499 |
. . . . . 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 498 |
. . . . 5
⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑥 = dom 𝑦) |
13 | 12 | eqcomd 2744 |
. . . 4
⊢ ((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) → dom 𝑦 = dom 𝑥) |
14 | | simpl 486 |
. . . . . . 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 484 |
. . . . . . . 8
⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → dom 𝑥 = dom 𝑦) |
17 | 15, 16 | eleqtrrd 2836 |
. . . . . . 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 2836 |
. . . . . . 7
⊢ (((𝐺 ∈ TarskiG ∧ 𝑥(cgrG‘𝐺)𝑦) ∧ (𝑖 ∈ dom 𝑦 ∧ 𝑗 ∈ dom 𝑦)) → 𝑗 ∈ dom 𝑥) |
20 | 11 | simprd 499 |
. . . . . . . . 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 3121 |
. . . . . . . 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 3121 |
. . . . . . 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 837 |
. . . . . 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 2744 |
. . . . 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 3103 |
. . . 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 515 |
. . 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 26458 |
. . . 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 484 |
. . 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 498 |
. . . . 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 26458 |
. . . . . . . 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 480 |
. . . . . . 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 498 |
. . . . 5
⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → (𝑦 ∈ (𝑃 ↑pm ℝ) ∧ 𝑧 ∈ (𝑃 ↑pm
ℝ))) |
36 | 35 | simprd 499 |
. . . 4
⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → 𝑧 ∈ (𝑃 ↑pm
ℝ)) |
37 | 31, 36 | jca 515 |
. . 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 499 |
. . . . . 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 498 |
. . . . 5
⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑦) |
41 | 34 | simprd 499 |
. . . . . 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 498 |
. . . . 5
⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → dom 𝑦 = dom 𝑧) |
43 | 40, 42 | eqtrd 2773 |
. . . 4
⊢ ((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) → dom 𝑥 = dom 𝑧) |
44 | 39 | simprd 499 |
. . . . . . . . 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 3121 |
. . . . . . . 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 3121 |
. . . . . . 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 470 |
. . . . . 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 486 |
. . . . . . 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 484 |
. . . . . . . 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 2835 |
. . . . . . 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 2835 |
. . . . . . 7
⊢ (((𝐺 ∈ TarskiG ∧ (𝑥(cgrG‘𝐺)𝑦 ∧ 𝑦(cgrG‘𝐺)𝑧)) ∧ (𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥)) → 𝑗 ∈ dom 𝑦) |
54 | 41 | simprd 499 |
. . . . . . . . 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 3121 |
. . . . . . . 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 3121 |
. . . . . . 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 837 |
. . . . . 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 2773 |
. . . . 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 3103 |
. . . 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 515 |
. . 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 26458 |
. . . 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 484 |
. . 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 | | pm4.24 567 |
. . . 4
⊢ (𝑥 ∈ (𝑃 ↑pm ℝ) ↔ (𝑥 ∈ (𝑃 ↑pm ℝ) ∧ 𝑥 ∈ (𝑃 ↑pm
ℝ))) |
65 | | eqid 2738 |
. . . . . 6
⊢ dom 𝑥 = dom 𝑥 |
66 | | eqidd 2739 |
. . . . . . 7
⊢ ((𝑖 ∈ dom 𝑥 ∧ 𝑗 ∈ dom 𝑥) → ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗))) |
67 | 66 | rgen2 3115 |
. . . . . 6
⊢
∀𝑖 ∈ dom
𝑥∀𝑗 ∈ dom 𝑥((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) = ((𝑥‘𝑖)(dist‘𝐺)(𝑥‘𝑗)) |
68 | 65, 67 | pm3.2i 474 |
. . . . 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 533 |
. . . 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 278 |
. . 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 26458 |
. . 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 293 |
. 2
⊢ (𝐺 ∈ TarskiG → (𝑥 ∈ (𝑃 ↑pm ℝ) ↔ 𝑥(cgrG‘𝐺)𝑥)) |
73 | 3, 29, 63, 72 | iserd 8346 |
1
⊢ (𝐺 ∈ TarskiG →
(cgrG‘𝐺) Er (𝑃 ↑pm
ℝ)) |