mulreim - Intuitionistic Logic Explorer

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Theorem mulreim 8591

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 8016	. . 3 $/\$ $/\$ $/\$
3		ax-icn 7936	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 528	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 8016	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 8008	. . 3 $/\$ $/\$ $/\$
8		simprl 529	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 8016	. . 3 $/\$ $/\$ $/\$
10		simprr 531	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 8016	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 8008	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8403	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8142	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8570	. . . . . . 7
16	15	oveq1i 5906	. . . . . 6
17	14, 16	eqtrdi 2238	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 8008	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8397	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 8009	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8182	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2226	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5912	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 8008	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 8008	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 8008	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 8008	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8138	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8139	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 8009	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5912	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2222	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8139	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5914	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 8012	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2232	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5914	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2222	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1364

wcel 2160 (class class class)co 5896

cc 7839

cr 7840

c1 7842

ci 7843

caddc 7844

cmul 7846

cneg 8159

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 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-resscn 7933 ax-1cn 7934 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952

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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-sub 8160 df-neg 8161

This theorem is referenced by: mulext1 8599