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| Mirrors > Home > ILE Home > Th. List > mulreim | Unicode version | ||
| Description: Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.) |
| Ref | Expression |
|---|---|
| mulreim |
     ![/\](wedge.gif "/\"){kind=small}       
  |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpll 518 |
. . . 4
| |
| 2 | 1 | recnd 7794 |
. . 3
 |
| 3 | ax-icn 7715 |
. . . . 5
| |
| 4 | 3 | a1i 9 |
. . . 4
|
| 5 | simplr 519 |
. . . . 5
| |
| 6 | 5 | recnd 7794 |
. . . 4
|
| 7 | 4, 6 | mulcld 7786 |
. . 3
|
| 8 | simprl 520 |
. . . 4
| |
| 9 | 8 | recnd 7794 |
. . 3
|
| 10 | simprr 521 |
. . . . 5
| |
| 11 | 10 | recnd 7794 |
. . . 4
|
| 12 | 4, 11 | mulcld 7786 |
. . 3
|
| 13 | 2, 7, 9, 12 | muladdd 8178 |
. 2
|
| 14 | 4, 11, 4, 6 | mul4d 7917 |
. . . . . 6
|
| 15 | ixi 8345 |
. . . . . . 7
 | |
| 16 | 15 | oveq1i 5784 |
. . . . . 6
|
| 17 | 14, 16 | syl6eq 2188 |
. . . . 5
|
| 18 | 11, 6 | mulcld 7786 |
. . . . . 6
|
| 19 | 18 | mulm1d 8172 |
. . . . 5
|
| 20 | 11, 6 | mulcomd 7787 |
. . . . . 6
|
| 21 | 20 | negeqd 7957 |
. . . . 5
|
| 22 | 17, 19, 21 | 3eqtrd 2176 |
. . . 4
|
| 23 | 22 | oveq2d 5790 |
. . 3
|
| 24 | 11, 2 | mulcld 7786 |
. . . . . 6
|
| 25 | 4, 24 | mulcld 7786 |
. . . . 5
|
| 26 | 9, 6 | mulcld 7786 |
. . . . . 6
|
| 27 | 4, 26 | mulcld 7786 |
. . . . 5
|
| 28 | 25, 27 | addcomd 7913 |
. . . 4
|
| 29 | 2, 4, 11 | mul12d 7914 |
. . . . . 6
|
| 30 | 2, 11 | mulcomd 7787 |
. . . . . . 7
|
| 31 | 30 | oveq2d 5790 |
. . . . . 6
|
| 32 | 29, 31 | eqtrd 2172 |
. . . . 5
|
| 33 | 9, 4, 6 | mul12d 7914 |
. . . . 5
|
| 34 | 32, 33 | oveq12d 5792 |
. . . 4
|
| 35 | 4, 26, 24 | adddid 7790 |
. . . 4
|
| 36 | 28, 34, 35 | 3eqtr4d 2182 |
. . 3
|
| 37 | 23, 36 | oveq12d 5792 |
. 2
|
| 38 | 13, 37 | eqtrd 2172 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wceq 1331
wcel 1480
(class class class)co 5774 cc 7618 cr 7619 c1 7621 ci 7622 caddc 7623 cmul 7625 cneg 7934 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
| This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-neg 7936 |
| This theorem is referenced by: mulext1 8374 |
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