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Mirrors > Home > MPE Home > Th. List > cnmetdval | Structured version Visualization version GIF version |
Description: Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
cnmetdval.1 | ⊢ 𝐷 = (abs ∘ − ) |
Ref | Expression |
---|---|
cnmetdval | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subf 10888 | . . 3 ⊢ − :(ℂ × ℂ)⟶ℂ | |
2 | opelxpi 5592 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 〈𝐴, 𝐵〉 ∈ (ℂ × ℂ)) | |
3 | fvco3 6760 | . . 3 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ 〈𝐴, 𝐵〉 ∈ (ℂ × ℂ)) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘( − ‘〈𝐴, 𝐵〉))) | |
4 | 1, 2, 3 | sylancr 589 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘( − ‘〈𝐴, 𝐵〉))) |
5 | df-ov 7159 | . . 3 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
6 | cnmetdval.1 | . . . 4 ⊢ 𝐷 = (abs ∘ − ) | |
7 | 6 | fveq1i 6671 | . . 3 ⊢ (𝐷‘〈𝐴, 𝐵〉) = ((abs ∘ − )‘〈𝐴, 𝐵〉) |
8 | 5, 7 | eqtri 2844 | . 2 ⊢ (𝐴𝐷𝐵) = ((abs ∘ − )‘〈𝐴, 𝐵〉) |
9 | df-ov 7159 | . . 3 ⊢ (𝐴 − 𝐵) = ( − ‘〈𝐴, 𝐵〉) | |
10 | 9 | fveq2i 6673 | . 2 ⊢ (abs‘(𝐴 − 𝐵)) = (abs‘( − ‘〈𝐴, 𝐵〉)) |
11 | 4, 8, 10 | 3eqtr4g 2881 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 × cxp 5553 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 − cmin 10870 abscabs 14593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 |
This theorem is referenced by: cnmet 23380 cnbl0 23382 cnblcld 23383 cnfldnm 23387 remetdval 23397 blcvx 23406 recld2 23422 zdis 23424 reperflem 23426 addcnlem 23472 divcn 23476 cncfmet 23516 cnheibor 23559 cnllycmp 23560 ipcn 23849 lmclim 23906 cncmet 23925 ovolfsval 24071 ellimc3 24477 lhop1lem 24610 ftc1lem6 24638 ulmdvlem1 24988 psercn 25014 pserdvlem2 25016 abelthlem2 25020 abelthlem3 25021 abelthlem5 25023 abelthlem7 25026 abelth 25029 dvlog2lem 25235 efopn 25241 logtayl 25243 logtayl2 25245 cxpcn3 25329 rlimcnp 25543 xrlimcnp 25546 efrlim 25547 lgamucov 25615 lgamcvg2 25632 ftalem3 25652 smcnlem 28474 hhcnf 29682 tpr2rico 31155 qqhcn 31232 qqhucn 31233 ftc1cnnc 34981 cntotbnd 35089 iccbnd 35133 cnmetcoval 41485 iooabslt 41794 limcrecl 41930 islpcn 41940 stirlinglem5 42383 ovolval2lem 42945 ovolval3 42949 |
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