Theorem efgh 24287

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.) (Revised by Mario Carneiro, 11-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)

Hypothesis

Ref	Expression
efgh.1	|- F = ( x e. X |-> ( exp ` ( A x. x ) ) )

Assertion

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

Distinct variable groups:   x, A   x, X

Allowed substitution hints:   B( x)   C( x)   F( x)

Proof of Theorem efgh

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1l 1085	. . . . 5 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> A e. CC )
2		simp1r 1086	. . . . . . 7 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> X e. (SubGrp ` ℂfld ) )
3		cnfldbas 19750	. . . . . . . 8 |- CC = ( Base ` ℂfld )
4	3	subgss 17595	. . . . . . 7 |- ( X e. (SubGrp ` ℂfld ) -> X C_ CC )
5	2, 4	syl 17	. . . . . 6 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> X C_ CC )
6		simp2 1062	. . . . . 6 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> B e. X )
7	5, 6	sseldd 3604	. . . . 5 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> B e. CC )
8		simp3 1063	. . . . . 6 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> C e. X )
9	5, 8	sseldd 3604	. . . . 5 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> C e. CC )
10	1, 7, 9	adddid 10064	. . . 4 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )
11	10	fveq2d 6195	. . 3 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( exp ` ( A x. ( B + C ) ) ) = ( exp ` ( ( A x. B ) + ( A x. C ) ) ) )
12	1, 7	mulcld 10060	. . . 4 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( A x. B ) e. CC )
13	1, 9	mulcld 10060	. . . 4 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( A x. C ) e. CC )
14		efadd 14824	. . . 4 |- ( ( ( 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 ) ) ) )
15	12, 13, 14	syl2anc 693	. . 3 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( exp ` ( ( A x. B ) + ( A x. C ) ) ) = ( ( exp ` ( A x. B ) ) x. ( exp ` ( A x. C ) ) ) )
16	11, 15	eqtrd 2656	. 2 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( exp ` ( A x. ( B + C ) ) ) = ( ( exp ` ( A x. B ) ) x. ( exp ` ( A x. C ) ) ) )
17		efgh.1	. . . . 5 |- F = ( x e. X |-> ( exp ` ( A x. x ) ) )
18		oveq2 6658	. . . . . . 7 |- ( x = y -> ( A x. x ) = ( A x. y ) )
19	18	fveq2d 6195	. . . . . 6 |- ( x = y -> ( exp ` ( A x. x ) ) = ( exp ` ( A x. y ) ) )
20	19	cbvmptv 4750	. . . . 5 |- ( x e. X |-> ( exp ` ( A x. x ) ) ) = ( y e. X |-> ( exp ` ( A x. y ) ) )
21	17, 20	eqtri 2644	. . . 4 |- F = ( y e. X |-> ( exp ` ( A x. y ) ) )
22	21	a1i 11	. . 3 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> F = ( y e. X |-> ( exp ` ( A x. y ) ) ) )
23		oveq2 6658	. . . . 5 |- ( y = ( B + C ) -> ( A x. y ) = ( A x. ( B + C ) ) )
24	23	fveq2d 6195	. . . 4 |- ( y = ( B + C ) -> ( exp ` ( A x. y ) ) = ( exp ` ( A x. ( B + C ) ) ) )
25	24	adantl 482	. . 3 |- ( ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) /\ y = ( B + C ) ) -> ( exp ` ( A x. y ) ) = ( exp ` ( A x. ( B + C ) ) ) )
26		cnfldadd 19751	. . . . 5 |- + = ( +g ` ℂfld )
27	26	subgcl 17604	. . . 4 |- ( ( X e. (SubGrp ` ℂfld ) /\ B e. X /\ C e. X ) -> ( B + C ) e. X )
28	27	3adant1l 1318	. . 3 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( B + C ) e. X )
29		fvexd 6203	. . 3 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( exp ` ( A x. ( B + C ) ) ) e. _V )
30	22, 25, 28, 29	fvmptd 6288	. 2 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( F ` ( B + C ) ) = ( exp ` ( A x. ( B + C ) ) ) )
31		oveq2 6658	. . . . . 6 |- ( y = B -> ( A x. y ) = ( A x. B ) )
32	31	fveq2d 6195	. . . . 5 |- ( y = B -> ( exp ` ( A x. y ) ) = ( exp ` ( A x. B ) ) )
33	32	adantl 482	. . . 4 |- ( ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) /\ y = B ) -> ( exp ` ( A x. y ) ) = ( exp ` ( A x. B ) ) )
34		fvexd 6203	. . . 4 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( exp ` ( A x. B ) ) e. _V )
35	22, 33, 6, 34	fvmptd 6288	. . 3 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( F ` B ) = ( exp ` ( A x. B ) ) )
36		oveq2 6658	. . . . . 6 |- ( y = C -> ( A x. y ) = ( A x. C ) )
37	36	fveq2d 6195	. . . . 5 |- ( y = C -> ( exp ` ( A x. y ) ) = ( exp ` ( A x. C ) ) )
38	37	adantl 482	. . . 4 |- ( ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) /\ y = C ) -> ( exp ` ( A x. y ) ) = ( exp ` ( A x. C ) ) )
39		fvexd 6203	. . . 4 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( exp ` ( A x. C ) ) e. _V )
40	22, 38, 8, 39	fvmptd 6288	. . 3 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( F ` C ) = ( exp ` ( A x. C ) ) )
41	35, 40	oveq12d 6668	. 2 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( ( F ` B ) x. ( F ` C ) ) = ( ( exp ` ( A x. B ) ) x. ( exp ` ( A x. C ) ) ) )
42	16, 30, 41	3eqtr4d 2666	1 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ B e. X /\ C e. X ) -> ( F ` ( B + C ) ) = ( ( F ` B ) x. ( F ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   + caddc 9939   x. cmul 9941   expce 14792  SubGrpcsubg 17588  ℂ_fldccnfld 19746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-subg 17591  df-cnfld 19747

This theorem is referenced by:  efabl  24296