mulreim - Intuitionistic Logic Explorer

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Theorem mulreim 7595

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 481	. . . 4 $/\$ $/\$ $/\$
2	1	recnd 7054	. . 3 $/\$ $/\$ $/\$
3		ax-icn 6979	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 482	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 7054	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 7047	. . 3 $/\$ $/\$ $/\$
8		simprl 483	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 7054	. . 3 $/\$ $/\$ $/\$
10		simprr 484	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 7054	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 7047	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 7413	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 7168	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 7574	. . . . . . 7
16	15	oveq1i 5522	. . . . . 6
17	14, 16	syl6eq 2088	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 7047	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 7407	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 7048	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 7206	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2076	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5528	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 7047	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 7047	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 7047	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 7047	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 7164	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 7165	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 7048	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5528	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2072	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 7165	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5530	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 7051	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2082	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5530	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2072	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 97

wceq 1243

wcel 1393 (class class class)co 5512

cc 6887

cr 6888

c1 6890

ci 6891

caddc 6892

cmul 6894

cneg 7183

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-neg 7185

This theorem is referenced by: mulext1 7603