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Theorem mulreim 8712

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 8136	. . 3 $/\$ $/\$ $/\$
3		ax-icn 8055	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 528	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8136	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8128	. . 3 $/\$ $/\$ $/\$
8		simprl 529	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8136	. . 3 $/\$ $/\$ $/\$
10		simprr 531	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8136	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8128	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8523	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8262	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8691	. . . . . . 7
16	15	oveq1i 5977	. . . . . 6
17	14, 16	eqtrdi 2256	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8128	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8517	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8129	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8302	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2244	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5983	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8128	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8128	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8128	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8128	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8258	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8259	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8129	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5983	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2240	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8259	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5985	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8132	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2250	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5985	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2240	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1373

wcel 2178 (class class class)co 5967

cc 7958

cr 7959

c1 7961

ci 7962

caddc 7963

cmul 7965

cneg 8279

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-setind 4603 ax-resscn 8052 ax-1cn 8053 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-cnre 8071

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-id 4358 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-iota 5251 df-fun 5292 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-sub 8280 df-neg 8281

This theorem is referenced by: mulext1 8720