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Mirrors > Home > ILE Home > Th. List > remetdval | GIF version |
Description: Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.) |
Ref | Expression |
---|---|
remet.1 | ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) |
Ref | Expression |
---|---|
remetdval | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5877 | . . 3 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
2 | remet.1 | . . . 4 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
3 | 2 | fveq1i 5516 | . . 3 ⊢ (𝐷‘〈𝐴, 𝐵〉) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
4 | 1, 3 | eqtri 2198 | . 2 ⊢ (𝐴𝐷𝐵) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
5 | opelxpi 4658 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 〈𝐴, 𝐵〉 ∈ (ℝ × ℝ)) | |
6 | fvres 5539 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (ℝ × ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = ((abs ∘ − )‘〈𝐴, 𝐵〉)) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = ((abs ∘ − )‘〈𝐴, 𝐵〉)) |
8 | df-ov 5877 | . . . 4 ⊢ (𝐴(abs ∘ − )𝐵) = ((abs ∘ − )‘〈𝐴, 𝐵〉) | |
9 | recn 7943 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
10 | recn 7943 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | eqid 2177 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
12 | 11 | cnmetdval 13922 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
13 | 9, 10, 12 | syl2an 289 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
14 | 8, 13 | eqtr3id 2224 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
15 | 7, 14 | eqtrd 2210 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
16 | 4, 15 | eqtrid 2222 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 〈cop 3595 × cxp 4624 ↾ cres 4628 ∘ ccom 4630 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 ℝcr 7809 − cmin 8126 abscabs 11001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-sub 8128 |
This theorem is referenced by: bl2ioo 13935 |
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