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Mirrors > Home > ILE Home > Th. List > bdmetval | GIF version |
Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.) |
Ref | Expression |
---|---|
stdbdmet.1 | ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < )) |
Ref | Expression |
---|---|
bdmetval | ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 520 | . 2 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
2 | simprr 521 | . 2 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋) | |
3 | simpll 518 | . . . 4 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐶:(𝑋 × 𝑋)⟶ℝ*) | |
4 | 3, 1, 2 | fovrnd 5915 | . . 3 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐶𝐵) ∈ ℝ*) |
5 | simplr 519 | . . 3 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝑅 ∈ ℝ*) | |
6 | xrmincl 11035 | . . 3 ⊢ (((𝐴𝐶𝐵) ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < ) ∈ ℝ*) | |
7 | 4, 5, 6 | syl2anc 408 | . 2 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < ) ∈ ℝ*) |
8 | oveq12 5783 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐶𝑦) = (𝐴𝐶𝐵)) | |
9 | 8 | preq1d 3606 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {(𝑥𝐶𝑦), 𝑅} = {(𝐴𝐶𝐵), 𝑅}) |
10 | 9 | infeq1d 6899 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < )) |
11 | stdbdmet.1 | . . 3 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < )) | |
12 | 10, 11 | ovmpoga 5900 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < ) ∈ ℝ*) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < )) |
13 | 1, 2, 7, 12 | syl3anc 1216 | 1 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < )) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {cpr 3528 × cxp 4537 ⟶wf 5119 (class class class)co 5774 ∈ cmpo 5776 infcinf 6870 ℝ*cxr 7799 < clt 7800 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-xneg 9559 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 |
This theorem is referenced by: bdbl 12672 |
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