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Theorem mulreim 8762

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 8186	. . 3 $/\$ $/\$ $/\$
3		ax-icn 8105	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 528	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8186	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8178	. . 3 $/\$ $/\$ $/\$
8		simprl 529	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8186	. . 3 $/\$ $/\$ $/\$
10		simprr 531	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8186	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8178	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8573	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8312	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8741	. . . . . . 7
16	15	oveq1i 6017	. . . . . 6
17	14, 16	eqtrdi 2278	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8178	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8567	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8179	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8352	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2266	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 6023	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8178	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8178	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8178	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8178	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8308	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8309	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8179	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 6023	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2262	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8309	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 6025	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8182	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2272	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 6025	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2262	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1395

wcel 2200 (class class class)co 6007

cc 8008

cr 8009

c1 8011

ci 8012

caddc 8013

cmul 8015

cneg 8329

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-setind 4629 ax-resscn 8102 ax-1cn 8103 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-cnre 8121

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-sub 8330 df-neg 8331

This theorem is referenced by: mulext1 8770