mulreim - Intuitionistic Logic Explorer

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Theorem mulreim 8783

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 8207	. . 3 $/\$ $/\$ $/\$
3		ax-icn 8126	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 529	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8207	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8199	. . 3 $/\$ $/\$ $/\$
8		simprl 531	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8207	. . 3 $/\$ $/\$ $/\$
10		simprr 533	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8207	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8199	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8594	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8333	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8762	. . . . . . 7
16	15	oveq1i 6027	. . . . . 6
17	14, 16	eqtrdi 2280	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8199	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8588	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8200	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8373	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2268	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 6033	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8199	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8199	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8199	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8199	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8329	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8330	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8200	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 6033	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8330	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 6035	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8203	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2274	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 6035	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2264	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1397

wcel 2202 (class class class)co 6017

cc 8029

cr 8030

c1 8032

ci 8033

caddc 8034

cmul 8036

cneg 8350

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-setind 4635 ax-resscn 8123 ax-1cn 8124 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-cnre 8142

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-sub 8351 df-neg 8352

This theorem is referenced by: mulext1 8791