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Theorem mulreim 8625

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 8050	. . 3 $/\$ $/\$ $/\$
3		ax-icn 7969	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 528	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8050	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8042	. . 3 $/\$ $/\$ $/\$
8		simprl 529	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8050	. . 3 $/\$ $/\$ $/\$
10		simprr 531	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8050	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8042	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8437	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8176	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8604	. . . . . . 7
16	15	oveq1i 5929	. . . . . 6
17	14, 16	eqtrdi 2242	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8042	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8431	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8043	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8216	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2230	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5935	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8042	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8042	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8042	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8042	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8172	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8173	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8043	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5935	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2226	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8173	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5937	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8046	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2236	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5937	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2226	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1364

wcel 2164 (class class class)co 5919

cc 7872

cr 7873

c1 7875

ci 7876

caddc 7877

cmul 7879

cneg 8193

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-setind 4570 ax-resscn 7966 ax-1cn 7967 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-sub 8194 df-neg 8195

This theorem is referenced by: mulext1 8633