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Theorem mulreim 8677

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 8101	. . 3 $/\$ $/\$ $/\$
3		ax-icn 8020	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 528	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8101	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8093	. . 3 $/\$ $/\$ $/\$
8		simprl 529	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8101	. . 3 $/\$ $/\$ $/\$
10		simprr 531	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8101	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8093	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8488	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8227	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8656	. . . . . . 7
16	15	oveq1i 5954	. . . . . 6
17	14, 16	eqtrdi 2254	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8093	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8482	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8094	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8267	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2242	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5960	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8093	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8093	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8093	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8093	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8223	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8224	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8094	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5960	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2238	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8224	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5962	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8097	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2248	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5962	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2238	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1373

wcel 2176 (class class class)co 5944

cc 7923

cr 7924

c1 7926

ci 7927

caddc 7928

cmul 7930

cneg 8244

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-setind 4585 ax-resscn 8017 ax-1cn 8018 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-cnre 8036

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-sub 8245 df-neg 8246

This theorem is referenced by: mulext1 8685