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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 11080 | . . 3 ⊢ − :(ℂ × ℂ)⟶ℂ | |
2 | opelxpi 5588 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 〈𝐴, 𝐵〉 ∈ (ℂ × ℂ)) | |
3 | fvco3 6810 | . . 3 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ 〈𝐴, 𝐵〉 ∈ (ℂ × ℂ)) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘( − ‘〈𝐴, 𝐵〉))) | |
4 | 1, 2, 3 | sylancr 590 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs ∘ − )‘〈𝐴, 𝐵〉) = (abs‘( − ‘〈𝐴, 𝐵〉))) |
5 | df-ov 7216 | . . 3 ⊢ (𝐴𝐷𝐵) = (𝐷‘〈𝐴, 𝐵〉) | |
6 | cnmetdval.1 | . . . 4 ⊢ 𝐷 = (abs ∘ − ) | |
7 | 6 | fveq1i 6718 | . . 3 ⊢ (𝐷‘〈𝐴, 𝐵〉) = ((abs ∘ − )‘〈𝐴, 𝐵〉) |
8 | 5, 7 | eqtri 2765 | . 2 ⊢ (𝐴𝐷𝐵) = ((abs ∘ − )‘〈𝐴, 𝐵〉) |
9 | df-ov 7216 | . . 3 ⊢ (𝐴 − 𝐵) = ( − ‘〈𝐴, 𝐵〉) | |
10 | 9 | fveq2i 6720 | . 2 ⊢ (abs‘(𝐴 − 𝐵)) = (abs‘( − ‘〈𝐴, 𝐵〉)) |
11 | 4, 8, 10 | 3eqtr4g 2803 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 〈cop 4547 × cxp 5549 ∘ ccom 5555 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 − cmin 11062 abscabs 14797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-ltxr 10872 df-sub 11064 |
This theorem is referenced by: cnmet 23669 cnbl0 23671 cnblcld 23672 cnfldnm 23676 remetdval 23686 blcvx 23695 recld2 23711 zdis 23713 reperflem 23715 addcnlem 23761 divcn 23765 cncfmet 23806 cnheibor 23852 cnllycmp 23853 ipcn 24143 lmclim 24200 cncmet 24219 ovolfsval 24367 ellimc3 24776 lhop1lem 24910 ftc1lem6 24938 ulmdvlem1 25292 psercn 25318 pserdvlem2 25320 abelthlem2 25324 abelthlem3 25325 abelthlem5 25327 abelthlem7 25330 abelth 25333 dvlog2lem 25540 efopn 25546 logtayl 25548 logtayl2 25550 cxpcn3 25634 rlimcnp 25848 xrlimcnp 25851 efrlim 25852 lgamucov 25920 lgamcvg2 25937 ftalem3 25957 smcnlem 28778 hhcnf 29986 tpr2rico 31576 qqhcn 31653 qqhucn 31654 ftc1cnnc 35586 cntotbnd 35691 iccbnd 35735 cnmetcoval 42415 iooabslt 42712 limcrecl 42845 islpcn 42855 stirlinglem5 43294 ovolval2lem 43856 ovolval3 43860 |
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