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Theorem efabl 22913
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 ` fld )s  ran  F )
efabl.3  |-  ( ph  ->  A  e.  CC )
efabl.4  |-  ( ph  ->  X  e.  (SubGrp ` fld )
)
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.
StepHypRef Expression
1 eqid 2443 . 2  |-  ( Base `  (flds  X ) )  =  (
Base `  (flds  X ) )
2 eqid 2443 . 2  |-  ( Base `  G )  =  (
Base `  G )
3 eqid 2443 . 2  |-  ( +g  `  (flds  X ) )  =  ( +g  `  (flds  X ) )
4 eqid 2443 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
5 simp1 997 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  ph )
6 simp2 998 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  x  e.  ( Base `  (flds  X ) ) )
7 efabl.4 . . . . . 6  |-  ( ph  ->  X  e.  (SubGrp ` fld )
)
8 eqid 2443 . . . . . . 7  |-  (flds  X )  =  (flds  X )
98subgbas 16183 . . . . . 6  |-  ( X  e.  (SubGrp ` fld )  ->  X  =  ( Base `  (flds  X )
) )
107, 9syl 16 . . . . 5  |-  ( ph  ->  X  =  ( Base `  (flds  X ) ) )
11103ad2ant1 1018 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  X  =  ( Base `  (flds  X ) ) )
126, 11eleqtrrd 2534 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  x  e.  X )
13 simp3 999 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  -> 
y  e.  ( Base `  (flds  X ) ) )
1413, 11eleqtrrd 2534 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  -> 
y  e.  X )
15 efabl.3 . . . . . 6  |-  ( ph  ->  A  e.  CC )
1615, 7jca 532 . . . . 5  |-  ( ph  ->  ( A  e.  CC  /\  X  e.  (SubGrp ` fld )
) )
17 efabl.1 . . . . . 6  |-  F  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
1817efgh 22904 . . . . 5  |-  ( ( ( A  e.  CC  /\  X  e.  (SubGrp ` fld )
)  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  +  y ) )  =  ( ( F `  x
)  x.  ( F `
 y ) ) )
1916, 18syl3an1 1262 . . . 4  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  +  y ) )  =  ( ( F `  x
)  x.  ( F `
 y ) ) )
20 cnfldadd 18403 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
218, 20ressplusg 14720 . . . . . . . 8  |-  ( X  e.  (SubGrp ` fld )  ->  +  =  ( +g  `  (flds  X ) ) )
227, 21syl 16 . . . . . . 7  |-  ( ph  ->  +  =  ( +g  `  (flds  X ) ) )
23223ad2ant1 1018 . . . . . 6  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  +  =  ( +g  `  (flds  X ) ) )
2423oveqd 6298 . . . . 5  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( x  +  y )  =  ( x ( +g  `  (flds  X ) ) y ) )
2524fveq2d 5860 . . . 4  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  +  y ) )  =  ( F `  ( x ( +g  `  (flds  X )
) y ) ) )
26 mptexg 6127 . . . . . . . . 9  |-  ( X  e.  (SubGrp ` fld )  ->  ( x  e.  X  |->  ( exp `  ( A  x.  x
) ) )  e. 
_V )
2717, 26syl5eqel 2535 . . . . . . . 8  |-  ( X  e.  (SubGrp ` fld )  ->  F  e. 
_V )
28 rnexg 6717 . . . . . . . 8  |-  ( F  e.  _V  ->  ran  F  e.  _V )
297, 27, 283syl 20 . . . . . . 7  |-  ( ph  ->  ran  F  e.  _V )
30 efabl.2 . . . . . . . 8  |-  G  =  ( (mulGrp ` fld )s  ran  F )
31 eqid 2443 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
32 cnfldmul 18404 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
3331, 32mgpplusg 17123 . . . . . . . 8  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3430, 33ressplusg 14720 . . . . . . 7  |-  ( ran 
F  e.  _V  ->  x.  =  ( +g  `  G
) )
3529, 34syl 16 . . . . . 6  |-  ( ph  ->  x.  =  ( +g  `  G ) )
36353ad2ant1 1018 . . . . 5  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  x.  =  ( +g  `  G ) )
3736oveqd 6298 . . . 4  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( ( F `  x )  x.  ( F `  y
) )  =  ( ( F `  x
) ( +g  `  G
) ( F `  y ) ) )
3819, 25, 373eqtr3d 2492 . . 3  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x ( +g  `  (flds  X ) ) y ) )  =  ( ( F `  x ) ( +g  `  G
) ( F `  y ) ) )
395, 12, 14, 38syl3anc 1229 . 2  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  -> 
( F `  (
x ( +g  `  (flds  X )
) y ) )  =  ( ( F `
 x ) ( +g  `  G ) ( F `  y
) ) )
40 fvex 5866 . . . . 5  |-  ( exp `  ( A  x.  x
) )  e.  _V
4140, 17fnmpti 5699 . . . 4  |-  F  Fn  X
42 dffn4 5791 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
4341, 42mpbi 208 . . 3  |-  F : X -onto-> ran  F
44 eqidd 2444 . . . 4  |-  ( ph  ->  F  =  F )
45 eff 13798 . . . . . . . 8  |-  exp : CC
--> CC
4645a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  exp : CC --> CC )
4715adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  CC )
48 cnfldbas 18402 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
4948subgss 16180 . . . . . . . . . 10  |-  ( X  e.  (SubGrp ` fld )  ->  X  C_  CC )
507, 49syl 16 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
5150sselda 3489 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
5247, 51mulcld 9619 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A  x.  x )  e.  CC )
5346, 52ffvelrnd 6017 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( exp `  ( A  x.  x ) )  e.  CC )
5453ralrimiva 2857 . . . . 5  |-  ( ph  ->  A. x  e.  X  ( exp `  ( A  x.  x ) )  e.  CC )
5517rnmptss 6045 . . . . 5  |-  ( A. x  e.  X  ( exp `  ( A  x.  x ) )  e.  CC  ->  ran  F  C_  CC )
5631, 48mgpbas 17125 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
5730, 56ressbas2 14669 . . . . 5  |-  ( ran 
F  C_  CC  ->  ran 
F  =  ( Base `  G ) )
5854, 55, 573syl 20 . . . 4  |-  ( ph  ->  ran  F  =  (
Base `  G )
)
5944, 10, 58foeq123d 5802 . . 3  |-  ( ph  ->  ( F : X -onto-> ran  F  <->  F : ( Base `  (flds  X ) ) -onto-> ( Base `  G ) ) )
6043, 59mpbii 211 . 2  |-  ( ph  ->  F : ( Base `  (flds  X ) ) -onto-> ( Base `  G ) )
61 cnring 18418 . . . 4  |-fld  e.  Ring
62 ringabl 17206 . . . 4  |-  (fld  e.  Ring  ->fld  e.  Abel )
6361, 62ax-mp 5 . . 3  |-fld  e.  Abel
648subgabl 16822 . . 3  |-  ( (fld  e. 
Abel  /\  X  e.  (SubGrp ` fld ) )  ->  (flds  X )  e.  Abel )
6563, 7, 64sylancr 663 . 2  |-  ( ph  ->  (flds  X )  e.  Abel )
661, 2, 3, 4, 39, 60, 65ghmabl 16819 1  |-  ( ph  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    C_ wss 3461    |-> cmpt 4495   ran crn 4990    Fn wfn 5573   -->wf 5574   -onto->wfo 5576   ` cfv 5578  (class class class)co 6281   CCcc 9493    + caddc 9498    x. cmul 9500   expce 13778   Basecbs 14613   ↾s cress 14614   +g cplusg 14678  SubGrpcsubg 16173   Abelcabl 16777  mulGrpcmgp 17119   Ringcrg 17176  ℂfldccnfld 18398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-rp 11231  df-ico 11545  df-fz 11683  df-fzo 11806  df-fl 11910  df-seq 12089  df-exp 12148  df-fac 12335  df-bc 12362  df-hash 12387  df-shft 12881  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-limsup 13275  df-clim 13292  df-rlim 13293  df-sum 13490  df-ef 13784  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-starv 14693  df-tset 14697  df-ple 14698  df-ds 14700  df-unif 14701  df-0g 14820  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-grp 16035  df-minusg 16036  df-subg 16176  df-cmn 16778  df-abl 16779  df-mgp 17120  df-ur 17132  df-ring 17178  df-cring 17179  df-cnfld 18399
This theorem is referenced by:  efsubm  22914  circgrp  22915
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