Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > isxmetd | GIF version |
Description: Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
isxmetd.0 | ⊢ (𝜑 → 𝑋 ∈ V) |
isxmetd.1 | ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
isxmetd.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)) |
isxmetd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) |
Ref | Expression |
---|---|
isxmetd | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isxmetd.1 | . 2 ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | isxmetd.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)) | |
3 | isxmetd.3 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) | |
4 | 3 | 3exp2 1203 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 → (𝑦 ∈ 𝑋 → (𝑧 ∈ 𝑋 → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) |
5 | 4 | imp32 255 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑧 ∈ 𝑋 → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))) |
6 | 5 | ralrimiv 2502 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) |
7 | 2, 6 | jca 304 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))) |
8 | 7 | ralrimivva 2512 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))) |
9 | isxmetd.0 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) | |
10 | isxmet 12503 | . . 3 ⊢ (𝑋 ∈ V → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) | |
11 | 9, 10 | syl 14 | . 2 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) |
12 | 1, 8, 11 | mpbir2and 928 | 1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∀wral 2414 Vcvv 2681 class class class wbr 3924 × cxp 4532 ⟶wf 5114 ‘cfv 5118 (class class class)co 5767 0cc0 7613 ℝ*cxr 7792 ≤ cle 7794 +𝑒 cxad 9550 ∞Metcxmet 12138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-map 6537 df-pnf 7795 df-mnf 7796 df-xr 7797 df-xmet 12146 |
This theorem is referenced by: isxmet2d 12506 xmetres2 12537 comet 12657 xmetxp 12665 |
Copyright terms: Public domain | W3C validator |