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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 7418 | . . 3 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
2 | remet.1 | . . . 4 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
3 | 2 | fveq1i 6893 | . . 3 ⊢ (𝐷‘〈𝐴, 𝐵〉) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
4 | 1, 3 | eqtri 2756 | . 2 ⊢ (𝐴𝐷𝐵) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
5 | opelxpi 5710 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 〈𝐴, 𝐵〉 ∈ (ℝ × ℝ)) | |
6 | 5 | fvresd 6912 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = ((abs ∘ − )‘〈𝐴, 𝐵〉)) |
7 | df-ov 7418 | . . . 4 ⊢ (𝐴(abs ∘ − )𝐵) = ((abs ∘ − )‘〈𝐴, 𝐵〉) | |
8 | recn 11223 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | recn 11223 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
10 | eqid 2728 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
11 | 10 | cnmetdval 24681 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
12 | 8, 9, 11 | syl2an 595 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
13 | 7, 12 | eqtr3id 2782 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
14 | 6, 13 | eqtrd 2768 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
15 | 4, 14 | eqtrid 2780 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 〈cop 4631 × cxp 5671 ↾ cres 5675 ∘ ccom 5677 ‘cfv 6543 (class class class)co 7415 ℂcc 11131 ℝcr 11132 − cmin 11469 abscabs 15208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7988 df-2nd 7989 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-ltxr 11278 df-sub 11471 |
This theorem is referenced by: bl2ioo 24702 xrsdsre 24720 reconnlem2 24737 rrxdstprj1 25331 dvlip2 25922 nmcvcn 30499 poimirlem29 37117 rrndstprj1 37298 rrndstprj2 37299 rrncmslem 37300 ismrer1 37306 rrnprjdstle 45680 |
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