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Mirrors > Home > MPE Home > Th. List > remetdval | Structured version Visualization version 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 6925 | . . 3 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
2 | remet.1 | . . . 4 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
3 | 2 | fveq1i 6447 | . . 3 ⊢ (𝐷‘〈𝐴, 𝐵〉) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
4 | 1, 3 | eqtri 2802 | . 2 ⊢ (𝐴𝐷𝐵) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
5 | opelxpi 5392 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 〈𝐴, 𝐵〉 ∈ (ℝ × ℝ)) | |
6 | fvres 6465 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (ℝ × ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = ((abs ∘ − )‘〈𝐴, 𝐵〉)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = ((abs ∘ − )‘〈𝐴, 𝐵〉)) |
8 | df-ov 6925 | . . . 4 ⊢ (𝐴(abs ∘ − )𝐵) = ((abs ∘ − )‘〈𝐴, 𝐵〉) | |
9 | recn 10362 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
10 | recn 10362 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | eqid 2778 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
12 | 11 | cnmetdval 22982 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
13 | 9, 10, 12 | syl2an 589 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
14 | 8, 13 | syl5eqr 2828 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
15 | 7, 14 | eqtrd 2814 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
16 | 4, 15 | syl5eq 2826 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 〈cop 4404 × cxp 5353 ↾ cres 5357 ∘ ccom 5359 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ℝcr 10271 − cmin 10606 abscabs 14381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 |
This theorem is referenced by: bl2ioo 23003 xrsdsre 23021 reconnlem2 23038 rrxdstprj1 23615 dvlip2 24195 nmcvcn 28122 poimirlem29 34064 rrndstprj1 34253 rrndstprj2 34254 rrncmslem 34255 ismrer1 34261 rrnprjdstle 41445 |
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