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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 11361 | . . 3 ⊢ − :(ℂ × ℂ)⟶ℂ | |
2 | opelxpi 5668 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 〈𝐴, 𝐵〉 ∈ (ℂ × ℂ)) | |
3 | fvco3 6937 | . . 3 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ 〈𝐴, 𝐵〉 ∈ (ℂ × ℂ)) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘( − ‘〈𝐴, 𝐵〉))) | |
4 | 1, 2, 3 | sylancr 587 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘( − ‘〈𝐴, 𝐵〉))) |
5 | df-ov 7354 | . . 3 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
6 | cnmetdval.1 | . . . 4 ⊢ 𝐷 = (abs ∘ − ) | |
7 | 6 | fveq1i 6840 | . . 3 ⊢ (𝐷‘〈𝐴, 𝐵〉) = ((abs ∘ − )‘〈𝐴, 𝐵〉) |
8 | 5, 7 | eqtri 2765 | . 2 ⊢ (𝐴𝐷𝐵) = ((abs ∘ − )‘〈𝐴, 𝐵〉) |
9 | df-ov 7354 | . . 3 ⊢ (𝐴 − 𝐵) = ( − ‘〈𝐴, 𝐵〉) | |
10 | 9 | fveq2i 6842 | . 2 ⊢ (abs‘(𝐴 − 𝐵)) = (abs‘( − ‘〈𝐴, 𝐵〉)) |
11 | 4, 8, 10 | 3eqtr4g 2802 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 〈cop 4590 × cxp 5629 ∘ ccom 5635 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 − cmin 11343 abscabs 15079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-sub 11345 |
This theorem is referenced by: cnmet 24087 cnbl0 24089 cnblcld 24090 cnfldnm 24094 remetdval 24104 blcvx 24113 recld2 24129 zdis 24131 reperflem 24133 addcnlem 24179 divcn 24183 cncfmet 24224 cnheibor 24270 cnllycmp 24271 ipcn 24562 lmclim 24619 cncmet 24638 ovolfsval 24786 ellimc3 25195 lhop1lem 25329 ftc1lem6 25357 ulmdvlem1 25711 psercn 25737 pserdvlem2 25739 abelthlem2 25743 abelthlem3 25744 abelthlem5 25746 abelthlem7 25749 abelth 25752 dvlog2lem 25959 efopn 25965 logtayl 25967 logtayl2 25969 cxpcn3 26053 rlimcnp 26267 xrlimcnp 26270 efrlim 26271 lgamucov 26339 lgamcvg2 26356 ftalem3 26376 smcnlem 29468 hhcnf 30676 tpr2rico 32305 qqhcn 32384 qqhucn 32385 ftc1cnnc 36088 cntotbnd 36193 iccbnd 36237 cnmetcoval 43328 iooabslt 43638 limcrecl 43771 islpcn 43781 stirlinglem5 44220 ovolval2lem 44785 ovolval3 44789 |
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