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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 24681 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 | β’ π = β¨β¨ + , Β· β©, absβ© |
cnims.7 | β’ π· = (abs β β ) |
Ref | Expression |
---|---|
cnims | β’ π· = (IndMetβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnims.7 | . 2 β’ π· = (abs β β ) | |
2 | cnims.6 | . . . 4 β’ π = β¨β¨ + , Β· β©, absβ© | |
3 | 2 | cnnv 30480 | . . 3 β’ π β NrmCVec |
4 | 2 | cnnvm 30485 | . . . 4 β’ β = ( βπ£ βπ) |
5 | 2 | cnnvnm 30484 | . . . 4 β’ abs = (normCVβπ) |
6 | eqid 2728 | . . . 4 β’ (IndMetβπ) = (IndMetβπ) | |
7 | 4, 5, 6 | imsval 30488 | . . 3 β’ (π β NrmCVec β (IndMetβπ) = (abs β β )) |
8 | 3, 7 | ax-mp 5 | . 2 β’ (IndMetβπ) = (abs β β ) |
9 | 1, 8 | eqtr4i 2759 | 1 β’ π· = (IndMetβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 β wcel 2099 β¨cop 4630 β ccom 5676 βcfv 6542 + caddc 11135 Β· cmul 11137 β cmin 11468 abscabs 15207 NrmCVeccnv 30387 IndMetcims 30394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 ax-mulf 11212 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-grpo 30296 df-gid 30297 df-ginv 30298 df-gdiv 30299 df-ablo 30348 df-vc 30362 df-nv 30395 df-va 30398 df-ba 30399 df-sm 30400 df-0v 30401 df-vs 30402 df-nmcv 30403 df-ims 30404 |
This theorem is referenced by: cnbn 30672 |
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