Theorem efabl 24296

Description: The image of a subgroup of the group +, under the exponential function of a scaled complex number, is an Abelian group. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)

Hypotheses

Ref	Expression
efabl.1	|- F = ( x e. X |-> ( exp ` ( A x. x ) ) )
efabl.2	|- G = ( (mulGrp ` ℂfld ) ↾s ran F )
efabl.3	|- ( ph -> A e. CC )
efabl.4	|- ( ph -> X e. (SubGrp ` ℂfld ) )

Assertion

Ref	Expression
efabl	|- ( ph -> G e. Abel )

Distinct variable groups:   x, A   x, F   x, G   x, X   ph, x

Proof of Theorem efabl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( Base ` (ℂfld ↾s X ) ) = ( Base ` (ℂfld ↾s X ) )
2		eqid 2622	. 2 |- ( Base ` G ) = ( Base ` G )
3		eqid 2622	. 2 |- ( +g ` (ℂfld ↾s X ) ) = ( +g ` (ℂfld ↾s X ) )
4		eqid 2622	. 2 |- ( +g ` G ) = ( +g ` G )
5		simp1 1061	. . 3 |- ( ( ph /\ x e. ( Base ` (ℂfld ↾s X ) ) /\ y e. ( Base ` (ℂfld ↾s X ) ) ) -> ph )
6		simp2 1062	. . . 4 |- ( ( ph /\ x e. ( Base ` (ℂfld ↾s X ) ) /\ y e. ( Base ` (ℂfld ↾s X ) ) ) -> x e. ( Base ` (ℂfld ↾s X ) ) )
7		efabl.4	. . . . . 6 |- ( ph -> X e. (SubGrp ` ℂfld ) )
8		eqid 2622	. . . . . . 7 |- (ℂfld ↾s X ) = (ℂfld ↾s X )
9	8	subgbas 17598	. . . . . 6 |- ( X e. (SubGrp ` ℂfld ) -> X = ( Base ` (ℂfld ↾s X ) ) )
10	7, 9	syl 17	. . . . 5 |- ( ph -> X = ( Base ` (ℂfld ↾s X ) ) )
11	10	3ad2ant1 1082	. . . 4 |- ( ( ph /\ x e. ( Base ` (ℂfld ↾s X ) ) /\ y e. ( Base ` (ℂfld ↾s X ) ) ) -> X = ( Base ` (ℂfld ↾s X ) ) )
12	6, 11	eleqtrrd 2704	. . 3 |- ( ( ph /\ x e. ( Base ` (ℂfld ↾s X ) ) /\ y e. ( Base ` (ℂfld ↾s X ) ) ) -> x e. X )
13		simp3 1063	. . . 4 |- ( ( ph /\ x e. ( Base ` (ℂfld ↾s X ) ) /\ y e. ( Base ` (ℂfld ↾s X ) ) ) -> y e. ( Base ` (ℂfld ↾s X ) ) )
14	13, 11	eleqtrrd 2704	. . 3 |- ( ( ph /\ x e. ( Base ` (ℂfld ↾s X ) ) /\ y e. ( Base ` (ℂfld ↾s X ) ) ) -> y e. X )
15		efabl.3	. . . . . 6 |- ( ph -> A e. CC )
16	15, 7	jca 554	. . . . 5 |- ( ph -> ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) )
17		efabl.1	. . . . . 6 |- F = ( x e. X |-> ( exp ` ( A x. x ) ) )
18	17	efgh 24287	. . . . 5 |- ( ( ( A e. CC /\ X e. (SubGrp ` ℂfld ) ) /\ x e. X /\ y e. X ) -> ( F ` ( x + y ) ) = ( ( F ` x ) x. ( F ` y ) ) )
19	16, 18	syl3an1 1359	. . . 4 |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x + y ) ) = ( ( F ` x ) x. ( F ` y ) ) )
20		cnfldadd 19751	. . . . . . . . 9 |- + = ( +g ` ℂfld )
21	8, 20	ressplusg 15993	. . . . . . . 8 |- ( X e. (SubGrp ` ℂfld ) -> + = ( +g ` (ℂfld ↾s X ) ) )
22	7, 21	syl 17	. . . . . . 7 |- ( ph -> + = ( +g ` (ℂfld ↾s X ) ) )
23	22	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ x e. X /\ y e. X ) -> + = ( +g ` (ℂfld ↾s X ) ) )
24	23	oveqd 6667	. . . . 5 |- ( ( ph /\ x e. X /\ y e. X ) -> ( x + y ) = ( x ( +g ` (ℂfld ↾s X ) ) y ) )
25	24	fveq2d 6195	. . . 4 |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x + y ) ) = ( F ` ( x ( +g ` (ℂfld ↾s X ) ) y ) ) )
26		mptexg 6484	. . . . . . . . 9 |- ( X e. (SubGrp ` ℂfld ) -> ( x e. X |-> ( exp ` ( A x. x ) ) ) e. _V )
27	17, 26	syl5eqel 2705	. . . . . . . 8 |- ( X e. (SubGrp ` ℂfld ) -> F e. _V )
28		rnexg 7098	. . . . . . . 8 |- ( F e. _V -> ran F e. _V )
29	7, 27, 28	3syl 18	. . . . . . 7 |- ( ph -> ran F e. _V )
30		efabl.2	. . . . . . . 8 |- G = ( (mulGrp ` ℂfld ) ↾s ran F )
31		eqid 2622	. . . . . . . . 9 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
32		cnfldmul 19752	. . . . . . . . 9 |- x. = ( .r ` ℂfld )
33	31, 32	mgpplusg 18493	. . . . . . . 8 |- x. = ( +g ` (mulGrp ` ℂfld ) )
34	30, 33	ressplusg 15993	. . . . . . 7 |- ( ran F e. _V -> x. = ( +g ` G ) )
35	29, 34	syl 17	. . . . . 6 |- ( ph -> x. = ( +g ` G ) )
36	35	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ x e. X /\ y e. X ) -> x. = ( +g ` G ) )
37	36	oveqd 6667	. . . 4 |- ( ( ph /\ x e. X /\ y e. X ) -> ( ( F ` x ) x. ( F ` y ) ) = ( ( F ` x ) ( +g ` G ) ( F ` y ) ) )
38	19, 25, 37	3eqtr3d 2664	. . 3 |- ( ( ph /\ x e. X /\ y e. X ) -> ( F ` ( x ( +g ` (ℂfld ↾s X ) ) y ) ) = ( ( F ` x ) ( +g ` G ) ( F ` y ) ) )
39	5, 12, 14, 38	syl3anc 1326	. 2 |- ( ( ph /\ x e. ( Base ` (ℂfld ↾s X ) ) /\ y e. ( Base ` (ℂfld ↾s X ) ) ) -> ( F ` ( x ( +g ` (ℂfld ↾s X ) ) y ) ) = ( ( F ` x ) ( +g ` G ) ( F ` y ) ) )
40		fvex 6201	. . . . 5 |- ( exp ` ( A x. x ) ) e. _V
41	40, 17	fnmpti 6022	. . . 4 |- F Fn X
42		dffn4 6121	. . . 4 |- ( F Fn X <-> F : X -onto-> ran F )
43	41, 42	mpbi 220	. . 3 |- F : X -onto-> ran F
44		eqidd 2623	. . . 4 |- ( ph -> F = F )
45		eff 14812	. . . . . . . 8 |- exp : CC --> CC
46	45	a1i 11	. . . . . . 7 |- ( ( ph /\ x e. X ) -> exp : CC --> CC )
47	15	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> A e. CC )
48		cnfldbas 19750	. . . . . . . . . . 11 |- CC = ( Base ` ℂfld )
49	48	subgss 17595	. . . . . . . . . 10 |- ( X e. (SubGrp ` ℂfld ) -> X C_ CC )
50	7, 49	syl 17	. . . . . . . . 9 |- ( ph -> X C_ CC )
51	50	sselda 3603	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x e. CC )
52	47, 51	mulcld 10060	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A x. x ) e. CC )
53	46, 52	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( exp ` ( A x. x ) ) e. CC )
54	53	ralrimiva 2966	. . . . 5 |- ( ph -> A. x e. X ( exp ` ( A x. x ) ) e. CC )
55	17	rnmptss 6392	. . . . 5 |- ( A. x e. X ( exp ` ( A x. x ) ) e. CC -> ran F C_ CC )
56	31, 48	mgpbas 18495	. . . . . 6 |- CC = ( Base ` (mulGrp ` ℂfld ) )
57	30, 56	ressbas2 15931	. . . . 5 |- ( ran F C_ CC -> ran F = ( Base ` G ) )
58	54, 55, 57	3syl 18	. . . 4 |- ( ph -> ran F = ( Base ` G ) )
59	44, 10, 58	foeq123d 6132	. . 3 |- ( ph -> ( F : X -onto-> ran F <-> F : ( Base ` (ℂfld ↾s X ) ) -onto-> ( Base ` G ) ) )
60	43, 59	mpbii 223	. 2 |- ( ph -> F : ( Base ` (ℂfld ↾s X ) ) -onto-> ( Base ` G ) )
61		cnring 19768	. . . 4 |- ℂfld e. Ring
62		ringabl 18580	. . . 4 |- (ℂfld e. Ring -> ℂfld e. Abel )
63	61, 62	ax-mp 5	. . 3 |- ℂfld e. Abel
64	8	subgabl 18241	. . 3 |- ( (ℂfld e. Abel /\ X e. (SubGrp ` ℂfld ) ) -> (ℂfld ↾s X ) e. Abel )
65	63, 7, 64	sylancr 695	. 2 |- ( ph -> (ℂfld ↾s X ) e. Abel )
66	1, 2, 3, 4, 39, 60, 65	ghmabl 18238	1 |- ( ph -> G e. Abel )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   CCcc 9934   + caddc 9939   x. cmul 9941   expce 14792   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941  SubGrpcsubg 17588   Abelcabl 18194  mulGrpcmgp 18489   Ringcrg 18547  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-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-cnfld 19747

This theorem is referenced by:  efsubm  24297  circgrp  24298