Theorem efgh 10072

Description: The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.)

Hypothesis

Ref	Expression
efgh.1	|- F = {<.x, y>. | (x e. CC /\ y = (exp` (A x. x)))}

Assertion

Ref	Expression
efgh	|- ((A e. CC /\ B e. CC /\ C e. CC) -> (F` (B + C)) = ((F` B) x. (F` C)))

Distinct variable groups:   x,A,y   x,B,y   x,C,y

Proof of Theorem efgh

Step	Hyp	Ref	Expression
1		adddi 6462	. . . 4 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B + C)) = ((A x. B) + (A x. C)))
2	1	fveq2d 4685	. . 3 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (exp` (A x. (B + C))) = (exp` ((A x. B) + (A x. C))))
3		efadd 8629	. . . . 5 |- (((A x. B) e. CC /\ (A x. C) e. CC) -> (exp` ((A x. B) + (A x. C))) = ((exp` (A x. B)) x. (exp` (A x. C))))
4		mulcl 6456	. . . . 5 |- ((A e. CC /\ B e. CC) -> (A x. B) e. CC)
5		mulcl 6456	. . . . 5 |- ((A e. CC /\ C e. CC) -> (A x. C) e. CC)
6	3, 4, 5	syl2an 503	. . . 4 |- (((A e. CC /\ B e. CC) /\ (A e. CC /\ C e. CC)) -> (exp` ((A x. B) + (A x. C))) = ((exp` (A x. B)) x. (exp` (A x. C))))
7	6	3impdi 1152	. . 3 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (exp` ((A x. B) + (A x. C))) = ((exp` (A x. B)) x. (exp` (A x. C))))
8	2, 7	eqtrd 1925	. 2 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (exp` (A x. (B + C))) = ((exp` (A x. B)) x. (exp` (A x. C))))
9		addcl 6454	. . . 4 |- ((B e. CC /\ C e. CC) -> (B + C) e. CC)
10		opreq2 4890	. . . . . 6 |- (x = (B + C) -> (A x. x) = (A x. (B + C)))
11	10	fveq2d 4685	. . . . 5 |- (x = (B + C) -> (exp` (A x. x)) = (exp` (A x. (B + C))))
12		efgh.1	. . . . 5 |- F = {<.x, y>. | (x e. CC /\ y = (exp` (A x. x)))}
13		fvex 4689	. . . . 5 |- (exp` (A x. (B + C))) e. _V
14	11, 12, 13	fvopab4 4743	. . . 4 |- ((B + C) e. CC -> (F` (B + C)) = (exp` (A x. (B + C))))
15	9, 14	syl 12	. . 3 |- ((B e. CC /\ C e. CC) -> (F` (B + C)) = (exp` (A x. (B + C))))
16	15	3adant1 894	. 2 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (F` (B + C)) = (exp` (A x. (B + C))))
17		opreq2 4890	. . . . . 6 |- (x = B -> (A x. x) = (A x. B))
18	17	fveq2d 4685	. . . . 5 |- (x = B -> (exp` (A x. x)) = (exp` (A x. B)))
19		fvex 4689	. . . . 5 |- (exp` (A x. B)) e. _V
20	18, 12, 19	fvopab4 4743	. . . 4 |- (B e. CC -> (F` B) = (exp` (A x. B)))
21		opreq2 4890	. . . . . 6 |- (x = C -> (A x. x) = (A x. C))
22	21	fveq2d 4685	. . . . 5 |- (x = C -> (exp` (A x. x)) = (exp` (A x. C)))
23		fvex 4689	. . . . 5 |- (exp` (A x. C)) e. _V
24	22, 12, 23	fvopab4 4743	. . . 4 |- (C e. CC -> (F` C) = (exp` (A x. C)))
25	20, 24	opreqan12d 4902	. . 3 |- ((B e. CC /\ C e. CC) -> ((F` B) x. (F` C)) = ((exp` (A x. B)) x. (exp` (A x. C))))
26	25	3adant1 894	. 2 |- ((A e. CC /\ B e. CC /\ C e. CC) -> ((F` B) x. (F` C)) = ((exp` (A x. B)) x. (exp` (A x. C))))
27	8, 16, 26	3eqtr4d 1937	1 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (F` (B + C)) = ((F` B) x. (F` C)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384   + caddc 6389   x. cmul 6391  expce 8555

This theorem is referenced by:  efghgrpilem 10073

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-bc 8209  df-clim 8235  df-sum 8240  df-ef 8560

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