mulreim - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > mulreim		Unicode version

Theorem mulreim 8015

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 496	. . . 4 $/\$ $/\$ $/\$
2	1	recnd 7453	. . 3 $/\$ $/\$ $/\$
3		ax-icn 7377	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 497	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 7453	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 7445	. . 3 $/\$ $/\$ $/\$
8		simprl 498	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 7453	. . 3 $/\$ $/\$ $/\$
10		simprr 499	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 7453	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 7445	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 7831	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 7574	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 7994	. . . . . . 7
16	15	oveq1i 5617	. . . . . 6
17	14, 16	syl6eq 2133	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 7445	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 7825	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 7446	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 7614	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2121	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5623	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 7445	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 7445	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 7445	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 7445	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 7570	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 7571	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 7446	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5623	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2117	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 7571	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5625	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 7449	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2127	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5625	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2117	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wceq 1287

wcel 1436 (class class class)co 5607

cc 7285

cr 7286

c1 7288

ci 7289

caddc 7290

cmul 7292

cneg 7591

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3931 ax-pow 3983 ax-pr 4009 ax-setind 4325 ax-resscn 7374 ax-1cn 7375 ax-icn 7377 ax-addcl 7378 ax-addrcl 7379 ax-mulcl 7380 ax-addcom 7382 ax-mulcom 7383 ax-addass 7384 ax-mulass 7385 ax-distr 7386 ax-i2m1 7387 ax-1rid 7389 ax-0id 7390 ax-rnegex 7391 ax-cnre 7393

This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2617 df-sbc 2830 df-dif 2990 df-un 2992 df-in 2994 df-ss 3001 df-pw 3417 df-sn 3437 df-pr 3438 df-op 3440 df-uni 3637 df-br 3821 df-opab 3875 df-id 4093 df-xp 4416 df-rel 4417 df-cnv 4418 df-co 4419 df-dm 4420 df-iota 4943 df-fun 4980 df-fv 4986 df-riota 5563 df-ov 5610 df-oprab 5611 df-mpt2 5612 df-sub 7592 df-neg 7593

This theorem is referenced by: mulext1 8023