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Mirrors > Home > MPE Home > Th. List > 0cxp | Structured version Visualization version Unicode version |
Description: Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
0cxp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 9666 |
. . . 4
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2 | cxpval 23665 |
. . . 4
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3 | 1, 2 | mpan 681 |
. . 3
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4 | eqid 2462 |
. . . 4
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5 | 4 | iftruei 3900 |
. . 3
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6 | 3, 5 | syl6eq 2512 |
. 2
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7 | ifnefalse 3905 |
. 2
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8 | 6, 7 | sylan9eq 2516 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4541 ax-nul 4550 ax-pr 4656 ax-1cn 9628 ax-icn 9629 ax-addcl 9630 ax-mulcl 9632 ax-i2m1 9638 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4419 df-opab 4478 df-id 4771 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-iota 5569 df-fun 5607 df-fv 5613 df-ov 6323 df-oprab 6324 df-mpt2 6325 df-cxp 23563 |
This theorem is referenced by: cxpexp 23669 cxpeq0 23679 cxpge0 23684 mulcxplem 23685 cxpmul2 23690 cxple2 23698 cxpsqrt 23704 0cxpd 23711 abscxpbnd 23749 |
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