mulreim - Intuitionistic Logic Explorer

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Theorem mulreim 8390

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 7818	. . 3 $/\$ $/\$ $/\$
3		ax-icn 7739	. . . . 5
4	3	a1i 9	. . . 4 $/\$ $/\$ $/\$
5		simplr 520	. . . . 5 $/\$ $/\$ $/\$
6	5	recnd 7818	. . . 4 $/\$ $/\$ $/\$
7	4, 6	mulcld 7810	. . 3 $/\$ $/\$ $/\$
8		simprl 521	. . . 4 $/\$ $/\$ $/\$
9	8	recnd 7818	. . 3 $/\$ $/\$ $/\$
10		simprr 522	. . . . 5 $/\$ $/\$ $/\$
11	10	recnd 7818	. . . 4 $/\$ $/\$ $/\$
12	4, 11	mulcld 7810	. . 3 $/\$ $/\$ $/\$
13	2, 7, 9, 12	muladdd 8202	. 2 $/\$ $/\$ $/\$
14	4, 11, 4, 6	mul4d 7941	. . . . . 6 $/\$ $/\$ $/\$
15		ixi 8369	. . . . . . 7
16	15	oveq1i 5792	. . . . . 6
17	14, 16	eqtrdi 2189	. . . . 5 $/\$ $/\$ $/\$
18	11, 6	mulcld 7810	. . . . . 6 $/\$ $/\$ $/\$
19	18	mulm1d 8196	. . . . 5 $/\$ $/\$ $/\$
20	11, 6	mulcomd 7811	. . . . . 6 $/\$ $/\$ $/\$
21	20	negeqd 7981	. . . . 5 $/\$ $/\$ $/\$
22	17, 19, 21	3eqtrd 2177	. . . 4 $/\$ $/\$ $/\$
23	22	oveq2d 5798	. . 3 $/\$ $/\$ $/\$
24	11, 2	mulcld 7810	. . . . . 6 $/\$ $/\$ $/\$
25	4, 24	mulcld 7810	. . . . 5 $/\$ $/\$ $/\$
26	9, 6	mulcld 7810	. . . . . 6 $/\$ $/\$ $/\$
27	4, 26	mulcld 7810	. . . . 5 $/\$ $/\$ $/\$
28	25, 27	addcomd 7937	. . . 4 $/\$ $/\$ $/\$
29	2, 4, 11	mul12d 7938	. . . . . 6 $/\$ $/\$ $/\$
30	2, 11	mulcomd 7811	. . . . . . 7 $/\$ $/\$ $/\$
31	30	oveq2d 5798	. . . . . 6 $/\$ $/\$ $/\$
32	29, 31	eqtrd 2173	. . . . 5 $/\$ $/\$ $/\$
33	9, 4, 6	mul12d 7938	. . . . 5 $/\$ $/\$ $/\$
34	32, 33	oveq12d 5800	. . . 4 $/\$ $/\$ $/\$
35	4, 26, 24	adddid 7814	. . . 4 $/\$ $/\$ $/\$
36	28, 34, 35	3eqtr4d 2183	. . 3 $/\$ $/\$ $/\$
37	23, 36	oveq12d 5800	. 2 $/\$ $/\$ $/\$
38	13, 37	eqtrd 2173	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1332

wcel 1481 (class class class)co 5782

cc 7642

cr 7643

c1 7645

ci 7646

caddc 7647

cmul 7649

cneg 7958

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-neg 7960

This theorem is referenced by: mulext1 8398