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Mirrors > Home > MPE Home > Th. List > 0cxp | Structured version Visualization version GIF version |
Description: Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
0cxp | โข ((๐ด โ โ โง ๐ด โ 0) โ (0โ๐๐ด) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11208 | . . . 4 โข 0 โ โ | |
2 | cxpval 26179 | . . . 4 โข ((0 โ โ โง ๐ด โ โ) โ (0โ๐๐ด) = if(0 = 0, if(๐ด = 0, 1, 0), (expโ(๐ด ยท (logโ0))))) | |
3 | 1, 2 | mpan 688 | . . 3 โข (๐ด โ โ โ (0โ๐๐ด) = if(0 = 0, if(๐ด = 0, 1, 0), (expโ(๐ด ยท (logโ0))))) |
4 | eqid 2732 | . . . 4 โข 0 = 0 | |
5 | 4 | iftruei 4535 | . . 3 โข if(0 = 0, if(๐ด = 0, 1, 0), (expโ(๐ด ยท (logโ0)))) = if(๐ด = 0, 1, 0) |
6 | 3, 5 | eqtrdi 2788 | . 2 โข (๐ด โ โ โ (0โ๐๐ด) = if(๐ด = 0, 1, 0)) |
7 | ifnefalse 4540 | . 2 โข (๐ด โ 0 โ if(๐ด = 0, 1, 0) = 0) | |
8 | 6, 7 | sylan9eq 2792 | 1 โข ((๐ด โ โ โง ๐ด โ 0) โ (0โ๐๐ด) = 0) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 396 = wceq 1541 โ wcel 2106 โ wne 2940 ifcif 4528 โcfv 6543 (class class class)co 7411 โcc 11110 0cc0 11112 1c1 11113 ยท cmul 11117 expce 16007 logclog 26070 โ๐ccxp 26071 |
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-pr 5427 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-mulcl 11174 ax-i2m1 11180 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-cxp 26073 |
This theorem is referenced by: cxpexp 26183 cxpeq0 26193 cxpge0 26198 mulcxplem 26199 cxpmul2 26204 cxple2 26212 cxpsqrt 26218 0cxpd 26225 cxpsqrtth 26245 abscxpbnd 26268 |
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