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Mirrors > Home > MPE Home > Th. List > stdbdmetval | Structured version Visualization version GIF version |
Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
stdbdmet.1 | ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅)) |
Ref | Expression |
---|---|
stdbdmetval | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7178 | . . . 4 ⊢ (𝐴𝐶𝐵) ∈ V | |
2 | ifexg 4510 | . . . 4 ⊢ (((𝐴𝐶𝐵) ∈ V ∧ 𝑅 ∈ 𝑉) → if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V) | |
3 | 1, 2 | mpan 686 | . . 3 ⊢ (𝑅 ∈ 𝑉 → if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V) |
4 | oveq12 7154 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐶𝑦) = (𝐴𝐶𝐵)) | |
5 | 4 | breq1d 5067 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥𝐶𝑦) ≤ 𝑅 ↔ (𝐴𝐶𝐵) ≤ 𝑅)) |
6 | 5, 4 | ifbieq1d 4486 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅)) |
7 | stdbdmet.1 | . . . 4 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅)) | |
8 | 6, 7 | ovmpoga 7293 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅)) |
9 | 3, 8 | syl3an3 1157 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑅 ∈ 𝑉) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅)) |
10 | 9 | 3comr 1117 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ifcif 4463 class class class wbr 5057 (class class class)co 7145 ∈ cmpo 7147 ≤ cle 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 |
This theorem is referenced by: stdbdbl 23054 |
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