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Theorem mulreim 8932

Description: Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)

**Assertion**
Ref	Expression
mulreim	$/\$ $/\$ $/\$

**Proof of Theorem mulreim**
Step	Hyp	Ref	Expression
1		simpll 531	. . . 4 $/\$ $/\$ $/\$
2	1	recnd 8354	. . 3 $/\$ $/\$ $/\$
3		ax-icn 8274	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 533	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8354	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8346	. . 3 $/\$ $/\$ $/\$
8		simprl 535	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8354	. . 3 $/\$ $/\$ $/\$
10		simprr 537	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8354	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8346	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8743	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8481	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8911	. . . . . . 7
16	15	oveq1i 6095	. . . . . 6
17	14, 16	eqtrdi 2287	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8346	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8737	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8347	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8521	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2275	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 6101	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8346	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8346	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8346	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8346	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8477	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8478	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8347	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 6101	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2271	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8478	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 6103	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8350	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2281	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 6103	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2271	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wceq 1402

wcel 2209 (class class class)co 6085

cc 8177

cr 8178

c1 8180

ci 8181

caddc 8182

cmul 8184

cneg 8498

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-setind 4684 ax-resscn 8271 ax-1cn 8272 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-sub 8499 df-neg 8500

This theorem is used by: mulext1 8940