mulreim - Intuitionistic Logic Explorer

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Theorem mulreim 8774

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 8198	. . 3 $/\$ $/\$ $/\$
3		ax-icn 8117	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 528	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8198	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8190	. . 3 $/\$ $/\$ $/\$
8		simprl 529	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8198	. . 3 $/\$ $/\$ $/\$
10		simprr 531	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8198	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8190	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8585	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8324	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8753	. . . . . . 7
16	15	oveq1i 6023	. . . . . 6
17	14, 16	eqtrdi 2278	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8190	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8579	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8191	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8364	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2266	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 6029	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8190	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8190	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8190	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8190	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8320	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8321	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8191	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 6029	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2262	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8321	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 6031	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8194	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2272	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 6031	. 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 6013

cc 8020

cr 8021

c1 8023

ci 8024

caddc 8025

cmul 8027

cneg 8341

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 4205 ax-pow 4262 ax-pr 4297 ax-setind 4633 ax-resscn 8114 ax-1cn 8115 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-cnre 8133

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 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-iota 5284 df-fun 5326 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-sub 8342 df-neg 8343

This theorem is referenced by: mulext1 8782