Theorem crmul 6740

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 5270	. . . . 5 |- i e. CC
4	2, 3	mulcl 5321	. . . 4 |- (B x. i) e. CC
5		crmul.3	. . . . 5 |- C e. CC
6		crmul.4	. . . . . 6 |- D e. CC
7	6, 3	mulcl 5321	. . . . 5 |- (D x. i) e. CC
8	5, 7	addcl 5320	. . . 4 |- (C + (D x. i)) e. CC
9	1, 4, 8	adddir 5327	. . 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 5321	. . . . 5 |- (A x. C) e. CC
11	1, 6	mulcl 5321	. . . . . 6 |- (A x. D) e. CC
12	11, 3	mulcl 5321	. . . . 5 |- ((A x. D) x. i) e. CC
13	2, 6	mulcl 5321	. . . . . 6 |- (B x. D) e. CC
14	13	negcl 5369	. . . . 5 |- -u(B x. D) e. CC
15	2, 5	mulcl 5321	. . . . . 6 |- (B x. C) e. CC
16	15, 3	mulcl 5321	. . . . 5 |- ((B x. C) x. i) e. CC
17	10, 12, 14, 16	add4 5342	. . . 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 5326	. . . . . 6 |- (A x. (C + (D x. i))) = ((A x. C) + (A x. (D x. i)))
19	1, 6, 3	mulass 5325	. . . . . . 7 |- ((A x. D) x. i) = (A x. (D x. i))
20	19	opreq2i 3972	. . . . . 6 |- ((A x. C) + ((A x. D) x. i)) = ((A x. C) + (A x. (D x. i)))
21	18, 20	eqtr4 1498	. . . . 5 |- (A x. (C + (D x. i))) = ((A x. C) + ((A x. D) x. i))
22	4, 5, 7	adddi 5326	. . . . . 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 5424	. . . . . . 7 |- ((B x. i) x. C) = ((B x. C) x. i)
24	2, 3, 6, 3	mul4 5425	. . . . . . . 8 |- ((B x. i) x. (D x. i)) = ((B x. D) x. (i x. i))
25		ixi 5681	. . . . . . . . 9 |- (i x. i) = -u1
26	25	opreq2i 3972	. . . . . . . 8 |- ((B x. D) x. (i x. i)) = ((B x. D) x. -u1)
27		ax1cn 5269	. . . . . . . . . . 11 |- 1 e. CC
28	27	negcl 5369	. . . . . . . . . 10 |- -u1 e. CC
29	13, 28	mulcom 5323	. . . . . . . . 9 |- ((B x. D) x. -u1) = (-u1 x. (B x. D))
30	27, 13	mulneg1 5445	. . . . . . . . 9 |- (-u1 x. (B x. D)) = -u(1 x. (B x. D))
31	13	mulid2 5333	. . . . . . . . . 10 |- (1 x. (B x. D)) = (B x. D)
32	31	negeqi 5360	. . . . . . . . 9 |- -u(1 x. (B x. D)) = -u(B x. D)
33	29, 30, 32	3eqtr 1499	. . . . . . . 8 |- ((B x. D) x. -u1) = -u(B x. D)
34	24, 26, 33	3eqtr 1499	. . . . . . 7 |- ((B x. i) x. (D x. i)) = -u(B x. D)
35	23, 34	opreq12i 3973	. . . . . 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 5322	. . . . . 6 |- (((B x. C) x. i) + -u(B x. D)) = (-u(B x. D) + ((B x. C) x. i))
37	22, 35, 36	3eqtr 1499	. . . . 5 |- ((B x. i) x. (C + (D x. i))) = (-u(B x. D) + ((B x. C) x. i))
38	21, 37	opreq12i 3973	. . . 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 5327	. . . . 5 |- (((A x. D) + (B x. C)) x. i) = (((A x. D) x. i) + ((B x. C) x. i))
40	39	opreq2i 3972	. . . 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 1505	. . 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 5381	. . . 4 |- ((A x. C) + -u(B x. D)) = ((A x. C) - (B x. D))
43	42	opreq1i 3971	. . 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 1499	. 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 5323	. . . 4 |- (B x. i) = (i x. B)
46	45	opreq2i 3972	. . 3 |- (A + (B x. i)) = (A + (i x. B))
47	6, 3	mulcom 5323	. . . 4 |- (D x. i) = (i x. D)
48	47	opreq2i 3972	. . 3 |- (C + (D x. i)) = (C + (i x. D))
49	46, 48	opreq12i 3973	. 2 |- ((A + (B x. i)) x. (C + (D x. i))) = ((A + (i x. B)) x. (C + (i x. D)))
50	11, 15	addcl 5320	. . . 4 |- ((A x. D) + (B x. C)) e. CC
51	50, 3	mulcom 5323	. . 3 |- (((A x. D) + (B x. C)) x. i) = (i x. ((A x. D) + (B x. C)))
52	51	opreq2i 3972	. 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 1503	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 956   e. wcel 958  (class class class)co 3963  CCcc 5232  1c1 5235  ici 5236   + caddc 5237   x. cmul 5239   - cmin 5292  -ucneg 5293

This theorem is referenced by:  remul 6786  immul 6787  cjmul 6789

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-sub 5356  df-neg 5358

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