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Mirrors > Home > MPE Home > Th. List > cnims | Structured version Visualization version GIF version |
Description: The metric induced on the complex numbers. cnmet 23073 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by NM, 15-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnims.6 | ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 |
cnims.7 | ⊢ 𝐷 = (abs ∘ − ) |
Ref | Expression |
---|---|
cnims | ⊢ 𝐷 = (IndMet‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnims.7 | . 2 ⊢ 𝐷 = (abs ∘ − ) | |
2 | cnims.6 | . . . 4 ⊢ 𝑈 = 〈〈 + , · 〉, abs〉 | |
3 | 2 | cnnv 28221 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
4 | 2 | cnnvm 28226 | . . . 4 ⊢ − = ( −𝑣 ‘𝑈) |
5 | 2 | cnnvnm 28225 | . . . 4 ⊢ abs = (normCV‘𝑈) |
6 | eqid 2772 | . . . 4 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
7 | 4, 5, 6 | imsval 28229 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) = (abs ∘ − )) |
8 | 3, 7 | ax-mp 5 | . 2 ⊢ (IndMet‘𝑈) = (abs ∘ − ) |
9 | 1, 8 | eqtr4i 2799 | 1 ⊢ 𝐷 = (IndMet‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2048 〈cop 4441 ∘ ccom 5404 ‘cfv 6182 + caddc 10330 · cmul 10332 − cmin 10662 abscabs 14444 NrmCVeccnv 28128 IndMetcims 28135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 ax-addf 10406 ax-mulf 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-z 11787 df-uz 12052 df-rp 12198 df-seq 13178 df-exp 13238 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-grpo 28037 df-gid 28038 df-ginv 28039 df-gdiv 28040 df-ablo 28089 df-vc 28103 df-nv 28136 df-va 28139 df-ba 28140 df-sm 28141 df-0v 28142 df-vs 28143 df-nmcv 28144 df-ims 28145 |
This theorem is referenced by: cnbn 28414 |
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