| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧
∣ ∃𝑎 ∈
𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧
∣ ∃𝑎 ∈
𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) |
| 2 | | vex 3484 |
. . . . . . 7
⊢ 𝑥 ∈ V |
| 3 | | vex 3484 |
. . . . . . 7
⊢ 𝑦 ∈ V |
| 4 | 2, 3 | ab2rexex 8004 |
. . . . . 6
⊢ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V |
| 5 | 4 | uniex 7761 |
. . . . 5
⊢ ∪ {𝑧
∣ ∃𝑎 ∈
𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V |
| 6 | 1, 5 | fnmpoi 8095 |
. . . 4
⊢ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧
∣ ∃𝑎 ∈
𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ )) |
| 7 | | pstmval.1 |
. . . . . 6
⊢ ∼ =
(~Met‘𝐷) |
| 8 | 7 | pstmval 33894 |
. . . . 5
⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧
∣ ∃𝑎 ∈
𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})) |
| 9 | 8 | fneq1d 6661 |
. . . 4
⊢ (𝐷 ∈ (PsMet‘𝑋) → ((pstoMet‘𝐷) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ )) ↔ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧
∣ ∃𝑎 ∈
𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ∼ ) × (𝑋 / ∼
)))) |
| 10 | 6, 9 | mpbiri 258 |
. . 3
⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) Fn ((𝑋 / ∼ ) × (𝑋 / ∼
))) |
| 11 | | simpllr 776 |
. . . . . . . . 9
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑥 = [𝑎] ∼ ) |
| 12 | | simpr 484 |
. . . . . . . . 9
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑦 = [𝑏] ∼ ) |
| 13 | 11, 12 | oveq12d 7449 |
. . . . . . . 8
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = ([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ )) |
| 14 | | simp-5l 785 |
. . . . . . . . 9
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝐷 ∈ (PsMet‘𝑋)) |
| 15 | | simp-4r 784 |
. . . . . . . . 9
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑎 ∈ 𝑋) |
| 16 | | simplr 769 |
. . . . . . . . 9
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑏 ∈ 𝑋) |
| 17 | 7 | pstmfval 33895 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏)) |
| 18 | 14, 15, 16, 17 | syl3anc 1373 |
. . . . . . . 8
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → ([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏)) |
| 19 | 13, 18 | eqtrd 2777 |
. . . . . . 7
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏)) |
| 20 | | psmetf 24316 |
. . . . . . . . 9
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
| 21 | 14, 20 | syl 17 |
. . . . . . . 8
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
| 22 | 21, 15, 16 | fovcdmd 7605 |
. . . . . . 7
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑎𝐷𝑏) ∈
ℝ*) |
| 23 | 19, 22 | eqeltrd 2841 |
. . . . . 6
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈
ℝ*) |
| 24 | | elqsi 8810 |
. . . . . . . 8
⊢ (𝑦 ∈ (𝑋 / ∼ ) →
∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ ) |
| 25 | 24 | ad2antll 729 |
. . . . . . 7
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) →
∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ ) |
| 26 | 25 | ad2antrr 726 |
. . . . . 6
⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) →
∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ ) |
| 27 | 23, 26 | r19.29a 3162 |
. . . . 5
⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈
ℝ*) |
| 28 | | elqsi 8810 |
. . . . . 6
⊢ (𝑥 ∈ (𝑋 / ∼ ) →
∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ ) |
| 29 | 28 | ad2antrl 728 |
. . . . 5
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) →
∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ ) |
| 30 | 27, 29 | r19.29a 3162 |
. . . 4
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → (𝑥(pstoMet‘𝐷)𝑦) ∈
ℝ*) |
| 31 | 30 | ralrimivva 3202 |
. . 3
⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ∈
ℝ*) |
| 32 | | ffnov 7559 |
. . 3
⊢
((pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼
))⟶ℝ* ↔ ((pstoMet‘𝐷) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ )) ∧
∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ∈
ℝ*)) |
| 33 | 10, 31, 32 | sylanbrc 583 |
. 2
⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼
))⟶ℝ*) |
| 34 | 17 | 3expa 1119 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏)) |
| 35 | 34 | eqeq1d 2739 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ (𝑎𝐷𝑏) = 0)) |
| 36 | 7 | breqi 5149 |
. . . . . . . . . . . 12
⊢ (𝑎 ∼ 𝑏 ↔ 𝑎(~Met‘𝐷)𝑏) |
| 37 | | metidv 33891 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(~Met‘𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0)) |
| 38 | 37 | anassrs 467 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (𝑎(~Met‘𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0)) |
| 39 | 36, 38 | bitrid 283 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (𝑎 ∼ 𝑏 ↔ (𝑎𝐷𝑏) = 0)) |
| 40 | | metider 33893 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ∈ (PsMet‘𝑋) →
(~Met‘𝐷)
Er 𝑋) |
| 41 | 40 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (~Met‘𝐷) Er 𝑋) |
| 42 | | ereq1 8752 |
. . . . . . . . . . . . . 14
⊢ ( ∼ =
(~Met‘𝐷)
→ ( ∼ Er 𝑋 ↔
(~Met‘𝐷)
Er 𝑋)) |
| 43 | 7, 42 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ( ∼ Er
𝑋 ↔
(~Met‘𝐷)
Er 𝑋) |
| 44 | 41, 43 | sylibr 234 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ∼ Er 𝑋) |
| 45 | | simplr 769 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → 𝑎 ∈ 𝑋) |
| 46 | 44, 45 | erth 8796 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (𝑎 ∼ 𝑏 ↔ [𝑎] ∼ = [𝑏] ∼ )) |
| 47 | 35, 39, 46 | 3bitr2d 307 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ )) |
| 48 | 47 | adantllr 719 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ )) |
| 49 | 48 | adantlr 715 |
. . . . . . . 8
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ )) |
| 50 | 49 | adantr 480 |
. . . . . . 7
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ )) |
| 51 | 13 | eqeq1d 2739 |
. . . . . . 7
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ ([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) =
0)) |
| 52 | 11, 12 | eqeq12d 2753 |
. . . . . . 7
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥 = 𝑦 ↔ [𝑎] ∼ = [𝑏] ∼ )) |
| 53 | 50, 51, 52 | 3bitr4d 311 |
. . . . . 6
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧
(𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦)) |
| 54 | 53, 26 | r19.29a 3162 |
. . . . 5
⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦)) |
| 55 | 54, 29 | r19.29a 3162 |
. . . 4
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦)) |
| 56 | | simp-6l 787 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝐷 ∈ (PsMet‘𝑋)) |
| 57 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑐 ∈ 𝑋) |
| 58 | | simp-6r 788 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑎 ∈ 𝑋) |
| 59 | | simp-4r 784 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑏 ∈ 𝑋) |
| 60 | | psmettri2 24319 |
. . . . . . . . . . . . . 14
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) |
| 61 | 56, 57, 58, 59, 60 | syl13anc 1374 |
. . . . . . . . . . . . 13
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) |
| 62 | | simp-5r 786 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑥 = [𝑎] ∼ ) |
| 63 | | simpllr 776 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑦 = [𝑏] ∼ ) |
| 64 | 62, 63 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = ([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ )) |
| 65 | 56, 58, 59, 17 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → ([𝑎] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏)) |
| 66 | 64, 65 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏)) |
| 67 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑧 = [𝑐] ∼ ) |
| 68 | 67, 62 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑧(pstoMet‘𝐷)𝑥) = ([𝑐] ∼
(pstoMet‘𝐷)[𝑎] ∼ )) |
| 69 | 7 | pstmfval 33895 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → ([𝑐] ∼
(pstoMet‘𝐷)[𝑎] ∼ ) = (𝑐𝐷𝑎)) |
| 70 | 56, 57, 58, 69 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → ([𝑐] ∼
(pstoMet‘𝐷)[𝑎] ∼ ) = (𝑐𝐷𝑎)) |
| 71 | 68, 70 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑧(pstoMet‘𝐷)𝑥) = (𝑐𝐷𝑎)) |
| 72 | 67, 63 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑧(pstoMet‘𝐷)𝑦) = ([𝑐] ∼
(pstoMet‘𝐷)[𝑏] ∼ )) |
| 73 | 7 | pstmfval 33895 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑐] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = (𝑐𝐷𝑏)) |
| 74 | 56, 57, 59, 73 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → ([𝑐] ∼
(pstoMet‘𝐷)[𝑏] ∼ ) = (𝑐𝐷𝑏)) |
| 75 | 72, 74 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑧(pstoMet‘𝐷)𝑦) = (𝑐𝐷𝑏)) |
| 76 | 71, 75 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)) = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))) |
| 77 | 61, 66, 76 | 3brtr4d 5175 |
. . . . . . . . . . . 12
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))) |
| 78 | 77 | adantl6r 764 |
. . . . . . . . . . 11
⊢
((((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))) |
| 79 | | elqsi 8810 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ (𝑋 / ∼ ) →
∃𝑐 ∈ 𝑋 𝑧 = [𝑐] ∼ ) |
| 80 | 79 | ad5antlr 735 |
. . . . . . . . . . 11
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) →
∃𝑐 ∈ 𝑋 𝑧 = [𝑐] ∼ ) |
| 81 | 78, 80 | r19.29a 3162 |
. . . . . . . . . 10
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))) |
| 82 | 81 | adantl5r 763 |
. . . . . . . . 9
⊢
(((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))) |
| 83 | 24 | ad4antlr 733 |
. . . . . . . . 9
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) →
∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ ) |
| 84 | 82, 83 | r19.29a 3162 |
. . . . . . . 8
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))) |
| 85 | 84 | adantl4r 755 |
. . . . . . 7
⊢
((((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))) |
| 86 | 28 | ad3antlr 731 |
. . . . . . 7
⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) →
∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ ) |
| 87 | 85, 86 | r19.29a 3162 |
. . . . . 6
⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))) |
| 88 | 87 | ralrimiva 3146 |
. . . . 5
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) →
∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))) |
| 89 | 88 | anasss 466 |
. . . 4
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) →
∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))) |
| 90 | 55, 89 | jca 511 |
. . 3
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → (((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))) |
| 91 | 90 | ralrimivva 3202 |
. 2
⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))) |
| 92 | | elfvex 6944 |
. . 3
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V) |
| 93 | | qsexg 8815 |
. . 3
⊢ (𝑋 ∈ V → (𝑋 / ∼ ) ∈
V) |
| 94 | | isxmet 24334 |
. . 3
⊢ ((𝑋 / ∼ ) ∈ V →
((pstoMet‘𝐷) ∈
(∞Met‘(𝑋
/ ∼ )) ↔
((pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼
))⟶ℝ* ∧ ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))))) |
| 95 | 92, 93, 94 | 3syl 18 |
. 2
⊢ (𝐷 ∈ (PsMet‘𝑋) → ((pstoMet‘𝐷) ∈
(∞Met‘(𝑋
/ ∼ )) ↔
((pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼
))⟶ℝ* ∧ ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))))) |
| 96 | 33, 91, 95 | mpbir2and 713 |
1
⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈
(∞Met‘(𝑋
/ ∼
))) |