Theorem efsubm 24297

Description: The image of a subgroup of the group +, under the exponential function of a scaled complex number is a submonoid of the multiplicative group of ℂ_fld. (Contributed 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
efsubm	|- ( ph -> ran F e. (SubMnd ` (mulGrp ` ℂfld ) ) )

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

Proof of Theorem efsubm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eff 14812	. . . . . 6 |- exp : CC --> CC
2	1	a1i 11	. . . . 5 |- ( ( ph /\ x e. X ) -> exp : CC --> CC )
3		efabl.3	. . . . . . 7 |- ( ph -> A e. CC )
4	3	adantr 481	. . . . . 6 |- ( ( ph /\ x e. X ) -> A e. CC )
5		efabl.4	. . . . . . . 8 |- ( ph -> X e. (SubGrp ` ℂfld ) )
6		cnfldbas 19750	. . . . . . . . 9 |- CC = ( Base ` ℂfld )
7	6	subgss 17595	. . . . . . . 8 |- ( X e. (SubGrp ` ℂfld ) -> X C_ CC )
8	5, 7	syl 17	. . . . . . 7 |- ( ph -> X C_ CC )
9	8	sselda 3603	. . . . . 6 |- ( ( ph /\ x e. X ) -> x e. CC )
10	4, 9	mulcld 10060	. . . . 5 |- ( ( ph /\ x e. X ) -> ( A x. x ) e. CC )
11	2, 10	ffvelrnd 6360	. . . 4 |- ( ( ph /\ x e. X ) -> ( exp ` ( A x. x ) ) e. CC )
12	11	ralrimiva 2966	. . 3 |- ( ph -> A. x e. X ( exp ` ( A x. x ) ) e. CC )
13		efabl.1	. . . 4 |- F = ( x e. X |-> ( exp ` ( A x. x ) ) )
14	13	rnmptss 6392	. . 3 |- ( A. x e. X ( exp ` ( A x. x ) ) e. CC -> ran F C_ CC )
15	12, 14	syl 17	. 2 |- ( ph -> ran F C_ CC )
16	3	mul01d 10235	. . . . 5 |- ( ph -> ( A x. 0 ) = 0 )
17	16	fveq2d 6195	. . . 4 |- ( ph -> ( exp ` ( A x. 0 ) ) = ( exp ` 0 ) )
18		ef0 14821	. . . 4 |- ( exp ` 0 ) = 1
19	17, 18	syl6eq 2672	. . 3 |- ( ph -> ( exp ` ( A x. 0 ) ) = 1 )
20		cnfld0 19770	. . . . . 6 |- 0 = ( 0g ` ℂfld )
21	20	subg0cl 17602	. . . . 5 |- ( X e. (SubGrp ` ℂfld ) -> 0 e. X )
22	5, 21	syl 17	. . . 4 |- ( ph -> 0 e. X )
23		fvex 6201	. . . 4 |- ( exp ` ( A x. 0 ) ) e. _V
24		oveq2 6658	. . . . . 6 |- ( x = 0 -> ( A x. x ) = ( A x. 0 ) )
25	24	fveq2d 6195	. . . . 5 |- ( x = 0 -> ( exp ` ( A x. x ) ) = ( exp ` ( A x. 0 ) ) )
26	13, 25	elrnmpt1s 5373	. . . 4 |- ( ( 0 e. X /\ ( exp ` ( A x. 0 ) ) e. _V ) -> ( exp ` ( A x. 0 ) ) e. ran F )
27	22, 23, 26	sylancl 694	. . 3 |- ( ph -> ( exp ` ( A x. 0 ) ) e. ran F )
28	19, 27	eqeltrrd 2702	. 2 |- ( ph -> 1 e. ran F )
29		efabl.2	. . . . . . . . 9 |- G = ( (mulGrp ` ℂfld ) ↾s ran F )
30	13, 29, 3, 5	efabl 24296	. . . . . . . 8 |- ( ph -> G e. Abel )
31		ablgrp 18198	. . . . . . . 8 |- ( G e. Abel -> G e. Grp )
32	30, 31	syl 17	. . . . . . 7 |- ( ph -> G e. Grp )
33	32	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> G e. Grp )
34		simp2 1062	. . . . . . 7 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> x e. ran F )
35		eqid 2622	. . . . . . . . . . 11 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
36	35, 6	mgpbas 18495	. . . . . . . . . 10 |- CC = ( Base ` (mulGrp ` ℂfld ) )
37	29, 36	ressbas2 15931	. . . . . . . . 9 |- ( ran F C_ CC -> ran F = ( Base ` G ) )
38	15, 37	syl 17	. . . . . . . 8 |- ( ph -> ran F = ( Base ` G ) )
39	38	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> ran F = ( Base ` G ) )
40	34, 39	eleqtrd 2703	. . . . . 6 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> x e. ( Base ` G ) )
41		simp3 1063	. . . . . . 7 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> y e. ran F )
42	41, 39	eleqtrd 2703	. . . . . 6 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> y e. ( Base ` G ) )
43		eqid 2622	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
44		eqid 2622	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
45	43, 44	grpcl 17430	. . . . . 6 |- ( ( G e. Grp /\ x e. ( Base ` G ) /\ y e. ( Base ` G ) ) -> ( x ( +g ` G ) y ) e. ( Base ` G ) )
46	33, 40, 42, 45	syl3anc 1326	. . . . 5 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> ( x ( +g ` G ) y ) e. ( Base ` G ) )
47		mptexg 6484	. . . . . . . . . 10 |- ( X e. (SubGrp ` ℂfld ) -> ( x e. X |-> ( exp ` ( A x. x ) ) ) e. _V )
48	5, 47	syl 17	. . . . . . . . 9 |- ( ph -> ( x e. X |-> ( exp ` ( A x. x ) ) ) e. _V )
49	13, 48	syl5eqel 2705	. . . . . . . 8 |- ( ph -> F e. _V )
50		rnexg 7098	. . . . . . . 8 |- ( F e. _V -> ran F e. _V )
51		cnfldmul 19752	. . . . . . . . . 10 |- x. = ( .r ` ℂfld )
52	35, 51	mgpplusg 18493	. . . . . . . . 9 |- x. = ( +g ` (mulGrp ` ℂfld ) )
53	29, 52	ressplusg 15993	. . . . . . . 8 |- ( ran F e. _V -> x. = ( +g ` G ) )
54	49, 50, 53	3syl 18	. . . . . . 7 |- ( ph -> x. = ( +g ` G ) )
55	54	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> x. = ( +g ` G ) )
56	55	oveqd 6667	. . . . 5 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> ( x x. y ) = ( x ( +g ` G ) y ) )
57	46, 56, 39	3eltr4d 2716	. . . 4 |- ( ( ph /\ x e. ran F /\ y e. ran F ) -> ( x x. y ) e. ran F )
58	57	3expb 1266	. . 3 |- ( ( ph /\ ( x e. ran F /\ y e. ran F ) ) -> ( x x. y ) e. ran F )
59	58	ralrimivva 2971	. 2 |- ( ph -> A. x e. ran F A. y e. ran F ( x x. y ) e. ran F )
60		cnring 19768	. . 3 |- ℂfld e. Ring
61	35	ringmgp 18553	. . 3 |- (ℂfld e. Ring -> (mulGrp ` ℂfld ) e. Mnd )
62		cnfld1 19771	. . . . 5 |- 1 = ( 1r ` ℂfld )
63	35, 62	ringidval 18503	. . . 4 |- 1 = ( 0g ` (mulGrp ` ℂfld ) )
64	36, 63, 52	issubm 17347	. . 3 |- ( (mulGrp ` ℂfld ) e. Mnd -> ( ran F e. (SubMnd ` (mulGrp ` ℂfld ) ) <-> ( ran F C_ CC /\ 1 e. ran F /\ A. x e. ran F A. y e. ran F ( x x. y ) e. ran F ) ) )
65	60, 61, 64	mp2b 10	. 2 |- ( ran F e. (SubMnd ` (mulGrp ` ℂfld ) ) <-> ( ran F C_ CC /\ 1 e. ran F /\ A. x e. ran F A. y e. ran F ( x x. y ) e. ran F ) )
66	15, 28, 59, 65	syl3anbrc 1246	1 |- ( ph -> ran F e. (SubMnd ` (mulGrp ` ℂfld ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   expce 14792   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   Mndcmnd 17294  SubMndcsubmnd 17334   Grpcgrp 17422  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-submnd 17336  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:  circsubm  24299