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Theorem mulreim 8232

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 499	. . . 4 $/\$ $/\$ $/\$
2	1	recnd 7666	. . 3 $/\$ $/\$ $/\$
3		ax-icn 7590	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 500	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 7666	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 7658	. . 3 $/\$ $/\$ $/\$
8		simprl 501	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 7666	. . 3 $/\$ $/\$ $/\$
10		simprr 502	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 7666	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 7658	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8045	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 7788	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8211	. . . . . . 7
16	15	oveq1i 5716	. . . . . 6
17	14, 16	syl6eq 2148	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 7658	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8039	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 7659	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 7828	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2136	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5722	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 7658	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 7658	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 7658	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 7658	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 7784	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 7785	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 7659	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5722	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2132	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 7785	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5724	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 7662	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2142	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5724	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2132	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1299

wcel 1448 (class class class)co 5706

cc 7498

cr 7499

c1 7501

ci 7502

caddc 7503

cmul 7505

cneg 7805

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-setind 4390 ax-resscn 7587 ax-1cn 7588 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606

This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-sub 7806 df-neg 7807

This theorem is referenced by: mulext1 8240