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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 11467 | . . 3 β’ β :(β Γ β)βΆβ | |
2 | opelxpi 5714 | . . 3 β’ ((π΄ β β β§ π΅ β β) β β¨π΄, π΅β© β (β Γ β)) | |
3 | fvco3 6991 | . . 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 586 | . 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 7415 | . . 3 β’ (π΄π·π΅) = (π·ββ¨π΄, π΅β©) | |
6 | cnmetdval.1 | . . . 4 β’ π· = (abs β β ) | |
7 | 6 | fveq1i 6893 | . . 3 β’ (π·ββ¨π΄, π΅β©) = ((abs β β )ββ¨π΄, π΅β©) |
8 | 5, 7 | eqtri 2759 | . 2 β’ (π΄π·π΅) = ((abs β β )ββ¨π΄, π΅β©) |
9 | df-ov 7415 | . . 3 β’ (π΄ β π΅) = ( β ββ¨π΄, π΅β©) | |
10 | 9 | fveq2i 6895 | . 2 β’ (absβ(π΄ β π΅)) = (absβ( β ββ¨π΄, π΅β©)) |
11 | 4, 8, 10 | 3eqtr4g 2796 | 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 395 = wceq 1540 β wcel 2105 β¨cop 4635 Γ cxp 5675 β ccom 5681 βΆwf 6540 βcfv 6544 (class class class)co 7412 βcc 11111 β cmin 11449 abscabs 15186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-ltxr 11258 df-sub 11451 |
This theorem is referenced by: cnmet 24509 cnbl0 24511 cnblcld 24512 cnfldnm 24516 remetdval 24526 blcvx 24535 recld2 24551 zdis 24553 reperflem 24555 addcnlem 24601 divcnOLD 24605 divcn 24607 cncfmet 24650 cnheibor 24702 cnllycmp 24703 ipcn 24995 lmclim 25052 cncmet 25071 ovolfsval 25220 ellimc3 25629 lhop1lem 25763 ftc1lem6 25791 ulmdvlem1 26145 psercn 26171 pserdvlem2 26173 abelthlem2 26177 abelthlem3 26178 abelthlem5 26180 abelthlem7 26183 abelth 26186 dvlog2lem 26393 efopn 26399 logtayl 26401 logtayl2 26403 cxpcn3 26489 rlimcnp 26703 xrlimcnp 26706 efrlim 26707 lgamucov 26775 lgamcvg2 26792 ftalem3 26812 smcnlem 30214 hhcnf 31422 tpr2rico 33187 qqhcn 33266 qqhucn 33267 ftc1cnnc 36864 cntotbnd 36968 iccbnd 37012 cnmetcoval 44201 iooabslt 44512 limcrecl 44645 islpcn 44655 stirlinglem5 45094 ovolval2lem 45659 ovolval3 45663 |
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