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Theorem mulreim 7704

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 495	. . . 4 $/\$ $/\$ $/\$
2	1	recnd 7147	. . 3 $/\$ $/\$ $/\$
3		ax-icn 7071	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 496	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 7147	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 7139	. . 3 $/\$ $/\$ $/\$
8		simprl 497	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 7147	. . 3 $/\$ $/\$ $/\$
10		simprr 498	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 7147	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 7139	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 7520	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 7263	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 7683	. . . . . . 7
16	15	oveq1i 5542	. . . . . 6
17	14, 16	syl6eq 2129	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 7139	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 7514	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 7140	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 7303	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2117	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5548	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 7139	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 7139	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 7139	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 7139	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 7259	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 7260	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 7140	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5548	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2113	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 7260	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5550	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 7143	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2123	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5550	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2113	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wceq 1284

wcel 1433 (class class class)co 5532

cc 6979

cr 6980

c1 6982

ci 6983

caddc 6984

cmul 6986

cneg 7280

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282

This theorem is referenced by: mulext1 7712