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Mirrors > Home > MPE Home > Th. List > adddir | Unicode version |
Description: Distributive law for complex numbers. (Contributed by NM, 10-Oct-2004.) |
Ref | Expression |
---|---|
adddir |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adddi 9043 |
. . 3
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2 | 1 | 3coml 1160 |
. 2
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3 | addcl 9036 |
. . . 4
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4 | mulcom 9040 |
. . . 4
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5 | 3, 4 | sylan 458 |
. . 3
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6 | 5 | 3impa 1148 |
. 2
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7 | mulcom 9040 |
. . . 4
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8 | 7 | 3adant2 976 |
. . 3
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9 | mulcom 9040 |
. . . 4
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10 | 9 | 3adant1 975 |
. . 3
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11 | 8, 10 | oveq12d 6066 |
. 2
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12 | 2, 6, 11 | 3eqtr4d 2454 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: mulid1 9052 adddiri 9065 adddird 9077 muladd11 9200 00id 9205 cnegex2 9212 muladd 9430 ser1const 11342 hashxplem 11659 demoivreALT 12765 dvds2ln 12843 dvds2add 12844 odd2np1lem 12870 cncrng 16685 icccvx 18936 sincosq1eq 20381 abssinper 20387 sineq0 20390 bposlem9 21037 cnrngo 21952 cncvc 22023 ipasslem1 22293 ipasslem11 22302 cdj3i 23905 mblfinlem2 26152 expgrowth 27428 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-addcl 9014 ax-mulcom 9018 ax-distr 9021 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-rex 2680 df-rab 2683 df-v 2926 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-nul 3597 df-if 3708 df-sn 3788 df-pr 3789 df-op 3791 df-uni 3984 df-br 4181 df-iota 5385 df-fv 5429 df-ov 6051 |
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