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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 6817 | . . 3 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
2 | remet.1 | . . . 4 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
3 | 2 | fveq1i 6354 | . . 3 ⊢ (𝐷‘〈𝐴, 𝐵〉) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
4 | 1, 3 | eqtri 2782 | . 2 ⊢ (𝐴𝐷𝐵) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
5 | opelxpi 5305 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 〈𝐴, 𝐵〉 ∈ (ℝ × ℝ)) | |
6 | fvres 6369 | . . . 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 6817 | . . . 4 ⊢ (𝐴(abs ∘ − )𝐵) = ((abs ∘ − )‘〈𝐴, 𝐵〉) | |
9 | recn 10238 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
10 | recn 10238 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | eqid 2760 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
12 | 11 | cnmetdval 22795 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
13 | 9, 10, 12 | syl2an 495 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
14 | 8, 13 | syl5eqr 2808 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
15 | 7, 14 | eqtrd 2794 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
16 | 4, 15 | syl5eq 2806 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 〈cop 4327 × cxp 5264 ↾ cres 5268 ∘ ccom 5270 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 ℝcr 10147 − cmin 10478 abscabs 14193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-1st 7334 df-2nd 7335 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-ltxr 10291 df-sub 10480 |
This theorem is referenced by: bl2ioo 22816 xrsdsre 22834 reconnlem2 22851 rrxdstprj1 23412 dvlip2 23977 nmcvcn 27880 poimirlem29 33769 rrndstprj1 33960 rrndstprj2 33961 rrncmslem 33962 ismrer1 33968 rrnprjdstle 41042 |
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