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Mirrors > Home > MPE Home > Th. List > mulnzcnopr | Structured version Visualization version Unicode version |
Description: Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.) |
Ref | Expression |
---|---|
mulnzcnopr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-mulf 9645 |
. . . . 5
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2 | ffnov 6427 |
. . . . 5
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3 | 1, 2 | mpbi 213 |
. . . 4
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4 | 3 | simpli 464 |
. . 3
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5 | difss 3572 |
. . . 4
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6 | xpss12 4959 |
. . . 4
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7 | 5, 5, 6 | mp2an 683 |
. . 3
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8 | fnssres 5711 |
. . 3
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9 | 4, 7, 8 | mp2an 683 |
. 2
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10 | ovres 6463 |
. . . 4
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11 | eldifsn 4110 |
. . . . . 6
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12 | eldifsn 4110 |
. . . . . 6
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13 | mulcl 9649 |
. . . . . . . 8
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14 | 13 | ad2ant2r 758 |
. . . . . . 7
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15 | mulne0 10282 |
. . . . . . 7
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16 | 14, 15 | jca 539 |
. . . . . 6
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17 | 11, 12, 16 | syl2anb 486 |
. . . . 5
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18 | eldifsn 4110 |
. . . . 5
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19 | 17, 18 | sylibr 217 |
. . . 4
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20 | 10, 19 | eqeltrd 2540 |
. . 3
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21 | 20 | rgen2a 2827 |
. 2
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22 | ffnov 6427 |
. 2
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23 | 9, 21, 22 | mpbir2an 936 |
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-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 ax-resscn 9622 ax-1cn 9623 ax-icn 9624 ax-addcl 9625 ax-addrcl 9626 ax-mulcl 9627 ax-mulrcl 9628 ax-mulcom 9629 ax-addass 9630 ax-mulass 9631 ax-distr 9632 ax-i2m1 9633 ax-1ne0 9634 ax-1rid 9635 ax-rnegex 9636 ax-rrecex 9637 ax-cnre 9638 ax-pre-lttri 9639 ax-pre-lttrn 9640 ax-pre-ltadd 9641 ax-pre-mulgt0 9642 ax-mulf 9645 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 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-nel 2636 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-iun 4294 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-po 4774 df-so 4775 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-riota 6277 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-er 7389 df-en 7596 df-dom 7597 df-sdom 7598 df-pnf 9703 df-mnf 9704 df-xr 9705 df-ltxr 9706 df-le 9707 df-sub 9888 df-neg 9889 |
This theorem is referenced by: ablomul 26132 mulid 26133 |
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