Theorem crmul 6679

Description: Multiplication rule for complex number representation. Remark in [Apostol] p. 361. In normal use, the arguments are the real components of two complex numbers, but the theorem works for complex components as well.

Hypotheses

Ref	Expression
crmul.1	|- A e. CC
crmul.2	|- B e. CC
crmul.3	|- C e. CC
crmul.4	|- D e. CC

Assertion

Ref	Expression
crmul	|- ((A + (i x. B)) x. (C + (i x. D))) = (((A x. C) - (B x. D)) + (i x. ((A x. D) + (B x. C))))

Proof of Theorem crmul

Step	Hyp	Ref	Expression
1		crmul.1	. . . 4 |- A e. CC
2		crmul.2	. . . . 5 |- B e. CC
3		axicn 5250	. . . . 5 |- i e. CC
4	2, 3	mulcl 5301	. . . 4 |- (B x. i) e. CC
5		crmul.3	. . . . 5 |- C e. CC
6		crmul.4	. . . . . 6 |- D e. CC
7	6, 3	mulcl 5301	. . . . 5 |- (D x. i) e. CC
8	5, 7	addcl 5300	. . . 4 |- (C + (D x. i)) e. CC
9	1, 4, 8	adddir 5307	. . 3 |- ((A + (B x. i)) x. (C + (D x. i))) = ((A x. (C + (D x. i))) + ((B x. i) x. (C + (D x. i))))
10	1, 5	mulcl 5301	. . . . 5 |- (A x. C) e. CC
11	1, 6	mulcl 5301	. . . . . 6 |- (A x. D) e. CC
12	11, 3	mulcl 5301	. . . . 5 |- ((A x. D) x. i) e. CC
13	2, 6	mulcl 5301	. . . . . 6 |- (B x. D) e. CC
14	13	negcl 5349	. . . . 5 |- -u(B x. D) e. CC
15	2, 5	mulcl 5301	. . . . . 6 |- (B x. C) e. CC
16	15, 3	mulcl 5301	. . . . 5 |- ((B x. C) x. i) e. CC
17	10, 12, 14, 16	add4 5322	. . . 4 |- (((A x. C) + ((A x. D) x. i)) + (-u(B x. D) + ((B x. C) x. i))) = (((A x. C) + -u(B x. D)) + (((A x. D) x. i) + ((B x. C) x. i)))
18	1, 5, 7	adddi 5306	. . . . . 6 |- (A x. (C + (D x. i))) = ((A x. C) + (A x. (D x. i)))
19	1, 6, 3	mulass 5305	. . . . . . 7 |- ((A x. D) x. i) = (A x. (D x. i))
20	19	opreq2i 3963	. . . . . 6 |- ((A x. C) + ((A x. D) x. i)) = ((A x. C) + (A x. (D x. i)))
21	18, 20	eqtr4 1495	. . . . 5 |- (A x. (C + (D x. i))) = ((A x. C) + ((A x. D) x. i))
22	4, 5, 7	adddi 5306	. . . . . 6 |- ((B x. i) x. (C + (D x. i))) = (((B x. i) x. C) + ((B x. i) x. (D x. i)))
23	2, 3, 5	mul23 5404	. . . . . . 7 |- ((B x. i) x. C) = ((B x. C) x. i)
24	2, 3, 6, 3	mul4 5405	. . . . . . . 8 |- ((B x. i) x. (D x. i)) = ((B x. D) x. (i x. i))
25		ixi 5662	. . . . . . . . 9 |- (i x. i) = -u1
26	25	opreq2i 3963	. . . . . . . 8 |- ((B x. D) x. (i x. i)) = ((B x. D) x. -u1)
27		ax1cn 5249	. . . . . . . . . . 11 |- 1 e. CC
28	27	negcl 5349	. . . . . . . . . 10 |- -u1 e. CC
29	13, 28	mulcom 5303	. . . . . . . . 9 |- ((B x. D) x. -u1) = (-u1 x. (B x. D))
30	27, 13	mulneg1 5425	. . . . . . . . 9 |- (-u1 x. (B x. D)) = -u(1 x. (B x. D))
31	13	mulid2 5313	. . . . . . . . . 10 |- (1 x. (B x. D)) = (B x. D)
32	31	negeqi 5340	. . . . . . . . 9 |- -u(1 x. (B x. D)) = -u(B x. D)
33	29, 30, 32	3eqtr 1496	. . . . . . . 8 |- ((B x. D) x. -u1) = -u(B x. D)
34	24, 26, 33	3eqtr 1496	. . . . . . 7 |- ((B x. i) x. (D x. i)) = -u(B x. D)
35	23, 34	opreq12i 3964	. . . . . 6 |- (((B x. i) x. C) + ((B x. i) x. (D x. i))) = (((B x. C) x. i) + -u(B x. D))
36	16, 14	addcom 5302	. . . . . 6 |- (((B x. C) x. i) + -u(B x. D)) = (-u(B x. D) + ((B x. C) x. i))
37	22, 35, 36	3eqtr 1496	. . . . 5 |- ((B x. i) x. (C + (D x. i))) = (-u(B x. D) + ((B x. C) x. i))
38	21, 37	opreq12i 3964	. . . 4 |- ((A x. (C + (D x. i))) + ((B x. i) x. (C + (D x. i)))) = (((A x. C) + ((A x. D) x. i)) + (-u(B x. D) + ((B x. C) x. i)))
39	11, 15, 3	adddir 5307	. . . . 5 |- (((A x. D) + (B x. C)) x. i) = (((A x. D) x. i) + ((B x. C) x. i))
40	39	opreq2i 3963	. . . 4 |- (((A x. C) + -u(B x. D)) + (((A x. D) + (B x. C)) x. i)) = (((A x. C) + -u(B x. D)) + (((A x. D) x. i) + ((B x. C) x. i)))
41	17, 38, 40	3eqtr4 1502	. . 3 |- ((A x. (C + (D x. i))) + ((B x. i) x. (C + (D x. i)))) = (((A x. C) + -u(B x. D)) + (((A x. D) + (B x. C)) x. i))
42	10, 13	negsub 5361	. . . 4 |- ((A x. C) + -u(B x. D)) = ((A x. C) - (B x. D))
43	42	opreq1i 3962	. . 3 |- (((A x. C) + -u(B x. D)) + (((A x. D) + (B x. C)) x. i)) = (((A x. C) - (B x. D)) + (((A x. D) + (B x. C)) x. i))
44	9, 41, 43	3eqtr 1496	. 2 |- ((A + (B x. i)) x. (C + (D x. i))) = (((A x. C) - (B x. D)) + (((A x. D) + (B x. C)) x. i))
45	2, 3	mulcom 5303	. . . 4 |- (B x. i) = (i x. B)
46	45	opreq2i 3963	. . 3 |- (A + (B x. i)) = (A + (i x. B))
47	6, 3	mulcom 5303	. . . 4 |- (D x. i) = (i x. D)
48	47	opreq2i 3963	. . 3 |- (C + (D x. i)) = (C + (i x. D))
49	46, 48	opreq12i 3964	. 2 |- ((A + (B x. i)) x. (C + (D x. i))) = ((A + (i x. B)) x. (C + (i x. D)))
50	11, 15	addcl 5300	. . . 4 |- ((A x. D) + (B x. C)) e. CC
51	50, 3	mulcom 5303	. . 3 |- (((A x. D) + (B x. C)) x. i) = (i x. ((A x. D) + (B x. C)))
52	51	opreq2i 3963	. 2 |- (((A x. C) - (B x. D)) + (((A x. D) + (B x. C)) x. i)) = (((A x. C) - (B x. D)) + (i x. ((A x. D) + (B x. C))))
53	44, 49, 52	3eqtr3 1500	1 |- ((A + (i x. B)) x. (C + (i x. D))) = (((A x. C) - (B x. D)) + (i x. ((A x. D) + (B x. C))))

Syntax hints:   = wceq 954   e. wcel 956  (class class class)co 3954  CCcc 5212  1c1 5215  ici 5216   + caddc 5217   x. cmul 5219   - cmin 5272  -ucneg 5273

This theorem is referenced by:  remul 6729  immul 6730  cjmul 6732

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-sub 5336  df-neg 5338

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