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Theorem mulreim 8922

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 531	. . . 4 $/\$ $/\$ $/\$
2	1	recnd 8344	. . 3 $/\$ $/\$ $/\$
3		ax-icn 8264	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 533	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8344	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8336	. . 3 $/\$ $/\$ $/\$
8		simprl 535	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8344	. . 3 $/\$ $/\$ $/\$
10		simprr 537	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8344	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8336	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8733	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8471	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8901	. . . . . . 7
16	15	oveq1i 6085	. . . . . 6
17	14, 16	eqtrdi 2287	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8336	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8727	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8337	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8511	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2275	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 6091	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8336	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8336	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8336	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8336	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8467	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8468	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8337	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 6091	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2271	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8468	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 6093	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8340	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2281	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 6093	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2271	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1402

wcel 2209 (class class class)co 6075

cc 8167

cr 8168

c1 8170

ci 8171

caddc 8172

cmul 8174

cneg 8488

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-setind 4679 ax-resscn 8261 ax-1cn 8262 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-cnre 8280

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-sub 8489 df-neg 8490

This theorem is referenced by: mulext1 8930