mulreim - Intuitionistic Logic Explorer

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Theorem mulreim 8563

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 7988	. . 3 $/\$ $/\$ $/\$
3		ax-icn 7908	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 528	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 7988	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 7980	. . 3 $/\$ $/\$ $/\$
8		simprl 529	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 7988	. . 3 $/\$ $/\$ $/\$
10		simprr 531	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 7988	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 7980	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8375	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 8114	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8542	. . . . . . 7
16	15	oveq1i 5887	. . . . . 6
17	14, 16	eqtrdi 2226	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 7980	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8369	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 7981	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 8154	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2214	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5893	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 7980	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 7980	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 7980	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 7980	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 8110	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 8111	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 7981	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5893	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2210	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 8111	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5895	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 7984	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2220	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5895	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2210	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1353

wcel 2148 (class class class)co 5877

cc 7811

cr 7812

c1 7814

ci 7815

caddc 7816

cmul 7818

cneg 8131

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-setind 4538 ax-resscn 7905 ax-1cn 7906 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-sub 8132 df-neg 8133

This theorem is referenced by: mulext1 8571