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Theorem mulreim 8878

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 527	. . . 4 $/\$ $/\$ $/\$
2	1	recnd 8302	. . 3 $/\$ $/\$ $/\$
3		ax-icn 8222	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 529	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8302	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8294	. . 3 $/\$ $/\$ $/\$
8		simprl 531	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8302	. . 3 $/\$ $/\$ $/\$
10		simprr 533	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8302	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8294	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8689	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8428	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8857	. . . . . . 7
16	15	oveq1i 6060	. . . . . 6
17	14, 16	eqtrdi 2281	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8294	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8683	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8295	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8468	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2269	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 6066	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8294	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8294	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8294	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8294	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8424	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8425	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8295	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 6066	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2265	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8425	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 6068	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8298	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2275	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 6068	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2265	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1398

wcel 2203 (class class class)co 6050

cc 8125

cr 8126

c1 8128

ci 8129

caddc 8130

cmul 8132

cneg 8445

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-setind 4659 ax-resscn 8219 ax-1cn 8220 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-cnre 8238

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-iota 5312 df-fun 5354 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-sub 8446 df-neg 8447

This theorem is referenced by: mulext1 8886