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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 | β’ π· = (abs β β ) |
Ref | Expression |
---|---|
cnmetdval | β’ ((π΄ β β β§ π΅ β β) β (π΄π·π΅) = (absβ(π΄ β π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subf 11464 | . . 3 β’ β :(β Γ β)βΆβ | |
2 | opelxpi 5713 | . . 3 β’ ((π΄ β β β§ π΅ β β) β β¨π΄, π΅β© β (β Γ β)) | |
3 | fvco3 6990 | . . 3 β’ (( β :(β Γ β)βΆβ β§ β¨π΄, π΅β© β (β Γ β)) β ((abs β β )ββ¨π΄, π΅β©) = (absβ( β ββ¨π΄, π΅β©))) | |
4 | 1, 2, 3 | sylancr 587 | . 2 β’ ((π΄ β β β§ π΅ β β) β ((abs β β )ββ¨π΄, π΅β©) = (absβ( β ββ¨π΄, π΅β©))) |
5 | df-ov 7414 | . . 3 β’ (π΄π·π΅) = (π·ββ¨π΄, π΅β©) | |
6 | cnmetdval.1 | . . . 4 β’ π· = (abs β β ) | |
7 | 6 | fveq1i 6892 | . . 3 β’ (π·ββ¨π΄, π΅β©) = ((abs β β )ββ¨π΄, π΅β©) |
8 | 5, 7 | eqtri 2760 | . 2 β’ (π΄π·π΅) = ((abs β β )ββ¨π΄, π΅β©) |
9 | df-ov 7414 | . . 3 β’ (π΄ β π΅) = ( β ββ¨π΄, π΅β©) | |
10 | 9 | fveq2i 6894 | . 2 β’ (absβ(π΄ β π΅)) = (absβ( β ββ¨π΄, π΅β©)) |
11 | 4, 8, 10 | 3eqtr4g 2797 | 1 β’ ((π΄ β β β§ π΅ β β) β (π΄π·π΅) = (absβ(π΄ β π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β¨cop 4634 Γ cxp 5674 β ccom 5680 βΆwf 6539 βcfv 6543 (class class class)co 7411 βcc 11110 β cmin 11446 abscabs 15183 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-ltxr 11255 df-sub 11448 |
This theorem is referenced by: cnmet 24295 cnbl0 24297 cnblcld 24298 cnfldnm 24302 remetdval 24312 blcvx 24321 recld2 24337 zdis 24339 reperflem 24341 addcnlem 24387 divcn 24391 cncfmet 24432 cnheibor 24478 cnllycmp 24479 ipcn 24770 lmclim 24827 cncmet 24846 ovolfsval 24994 ellimc3 25403 lhop1lem 25537 ftc1lem6 25565 ulmdvlem1 25919 psercn 25945 pserdvlem2 25947 abelthlem2 25951 abelthlem3 25952 abelthlem5 25954 abelthlem7 25957 abelth 25960 dvlog2lem 26167 efopn 26173 logtayl 26175 logtayl2 26177 cxpcn3 26263 rlimcnp 26477 xrlimcnp 26480 efrlim 26481 lgamucov 26549 lgamcvg2 26566 ftalem3 26586 smcnlem 29988 hhcnf 31196 tpr2rico 32961 qqhcn 33040 qqhucn 33041 gg-divcn 35232 ftc1cnnc 36646 cntotbnd 36750 iccbnd 36794 cnmetcoval 43980 iooabslt 44291 limcrecl 44424 islpcn 44434 stirlinglem5 44873 ovolval2lem 45438 ovolval3 45442 |
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