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Theorem mulreim 8895

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 8318	. . 3 $/\$ $/\$ $/\$
3		ax-icn 8238	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 529	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8318	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8310	. . 3 $/\$ $/\$ $/\$
8		simprl 531	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8318	. . 3 $/\$ $/\$ $/\$
10		simprr 533	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8318	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8310	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8706	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8444	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8874	. . . . . . 7
16	15	oveq1i 6068	. . . . . 6
17	14, 16	eqtrdi 2283	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8310	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8700	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8311	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8484	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2271	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 6074	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8310	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8310	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8310	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8310	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8440	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8441	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8311	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 6074	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2267	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8441	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 6076	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8314	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2277	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 6076	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2267	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1398

wcel 2205 (class class class)co 6058

cc 8141

cr 8142

c1 8144

ci 8145

caddc 8146

cmul 8148

cneg 8461

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 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-setind 4664 ax-resscn 8235 ax-1cn 8236 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-iota 5317 df-fun 5359 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-sub 8462 df-neg 8463

This theorem is referenced by: mulext1 8903