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Theorem mulreim 8510

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 524	. . . 4 $/\$ $/\$ $/\$
2	1	recnd 7935	. . 3 $/\$ $/\$ $/\$
3		ax-icn 7856	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 525	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 7935	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 7927	. . 3 $/\$ $/\$ $/\$
8		simprl 526	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 7935	. . 3 $/\$ $/\$ $/\$
10		simprr 527	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 7935	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 7927	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8322	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8061	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8489	. . . . . . 7
16	15	oveq1i 5860	. . . . . 6
17	14, 16	eqtrdi 2219	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 7927	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8316	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 7928	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8101	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2207	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5866	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 7927	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 7927	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 7927	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 7927	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8057	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8058	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 7928	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5866	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2203	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8058	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5868	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 7931	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2213	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5868	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2203	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1348

wcel 2141 (class class class)co 5850

cc 7759

cr 7760

c1 7762

ci 7763

caddc 7764

cmul 7766

cneg 8078

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-resscn 7853 ax-1cn 7854 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-sub 8079 df-neg 8080

This theorem is referenced by: mulext1 8518