mulreim - Intuitionistic Logic Explorer

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Theorem mulreim 8502

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 519	. . . 4 $/\$ $/\$ $/\$
2	1	recnd 7927	. . 3 $/\$ $/\$ $/\$
3		ax-icn 7848	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 520	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 7927	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 7919	. . 3 $/\$ $/\$ $/\$
8		simprl 521	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 7927	. . 3 $/\$ $/\$ $/\$
10		simprr 522	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 7927	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 7919	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8314	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8053	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8481	. . . . . . 7
16	15	oveq1i 5852	. . . . . 6
17	14, 16	eqtrdi 2215	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 7919	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8308	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 7920	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8093	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2202	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5858	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 7919	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 7919	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 7919	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 7919	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8049	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8050	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 7920	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5858	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2198	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8050	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5860	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 7923	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2208	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5860	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2198	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1343

wcel 2136 (class class class)co 5842

cc 7751

cr 7752

c1 7754

ci 7755

caddc 7756

cmul 7758

cneg 8070

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-setind 4514 ax-resscn 7845 ax-1cn 7846 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-sub 8071 df-neg 8072

This theorem is referenced by: mulext1 8510