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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 6552 | . . 3 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
2 | remet.1 | . . . 4 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
3 | 2 | fveq1i 6104 | . . 3 ⊢ (𝐷‘〈𝐴, 𝐵〉) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
4 | 1, 3 | eqtri 2632 | . 2 ⊢ (𝐴𝐷𝐵) = (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) |
5 | opelxpi 5072 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 〈𝐴, 𝐵〉 ∈ (ℝ × ℝ)) | |
6 | fvres 6117 | . . . 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 6552 | . . . 4 ⊢ (𝐴(abs ∘ − )𝐵) = ((abs ∘ − )‘〈𝐴, 𝐵〉) | |
9 | recn 9905 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
10 | recn 9905 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | eqid 2610 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
12 | 11 | cnmetdval 22384 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
13 | 9, 10, 12 | syl2an 493 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(abs ∘ − )𝐵) = (abs‘(𝐴 − 𝐵))) |
14 | 8, 13 | syl5eqr 2658 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
15 | 7, 14 | eqtrd 2644 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ))‘〈𝐴, 𝐵〉) = (abs‘(𝐴 − 𝐵))) |
16 | 4, 15 | syl5eq 2656 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 ↾ cres 5040 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 − cmin 10145 abscabs 13822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 |
This theorem is referenced by: bl2ioo 22403 xrsdsre 22421 reconnlem2 22438 rrxdstprj1 23000 dvlip2 23562 nmcvcn 26934 poimirlem29 32608 rrndstprj1 32799 rrndstprj2 32800 rrncmslem 32801 ismrer1 32807 rrnprjdstle 39197 |
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