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Theorem efabl 23495
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 2423 . 2  |-  ( Base `  (flds  X ) )  =  (
Base `  (flds  X ) )
2 eqid 2423 . 2  |-  ( Base `  G )  =  (
Base `  G )
3 eqid 2423 . 2  |-  ( +g  `  (flds  X ) )  =  ( +g  `  (flds  X ) )
4 eqid 2423 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
5 simp1 1006 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  ph )
6 simp2 1007 . . . 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 2423 . . . . . . 7  |-  (flds  X )  =  (flds  X )
98subgbas 16818 . . . . . 6  |-  ( X  e.  (SubGrp ` fld )  ->  X  =  ( Base `  (flds  X )
) )
107, 9syl 17 . . . . 5  |-  ( ph  ->  X  =  ( Base `  (flds  X ) ) )
11103ad2ant1 1027 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  X  =  ( Base `  (flds  X ) ) )
126, 11eleqtrrd 2514 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  x  e.  X )
13 simp3 1008 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  -> 
y  e.  ( Base `  (flds  X ) ) )
1413, 11eleqtrrd 2514 . . 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 535 . . . . 5  |-  ( ph  ->  ( A  e.  CC  /\  X  e.  (SubGrp ` fld )
) )
17 efabl.1 . . . . . 6  |-  F  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
1817efgh 23486 . . . . 5  |-  ( ( ( A  e.  CC  /\  X  e.  (SubGrp ` fld )
)  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  +  y ) )  =  ( ( F `  x
)  x.  ( F `
 y ) ) )
1916, 18syl3an1 1298 . . . 4  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  +  y ) )  =  ( ( F `  x
)  x.  ( F `
 y ) ) )
20 cnfldadd 18972 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
218, 20ressplusg 15236 . . . . . . . 8  |-  ( X  e.  (SubGrp ` fld )  ->  +  =  ( +g  `  (flds  X ) ) )
227, 21syl 17 . . . . . . 7  |-  ( ph  ->  +  =  ( +g  `  (flds  X ) ) )
23223ad2ant1 1027 . . . . . 6  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  +  =  ( +g  `  (flds  X ) ) )
2423oveqd 6321 . . . . 5  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( x  +  y )  =  ( x ( +g  `  (flds  X ) ) y ) )
2524fveq2d 5884 . . . 4  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  +  y ) )  =  ( F `  ( x ( +g  `  (flds  X )
) y ) ) )
26 mptexg 6149 . . . . . . . . 9  |-  ( X  e.  (SubGrp ` fld )  ->  ( x  e.  X  |->  ( exp `  ( A  x.  x
) ) )  e. 
_V )
2717, 26syl5eqel 2515 . . . . . . . 8  |-  ( X  e.  (SubGrp ` fld )  ->  F  e. 
_V )
28 rnexg 6738 . . . . . . . 8  |-  ( F  e.  _V  ->  ran  F  e.  _V )
297, 27, 283syl 18 . . . . . . 7  |-  ( ph  ->  ran  F  e.  _V )
30 efabl.2 . . . . . . . 8  |-  G  =  ( (mulGrp ` fld )s  ran  F )
31 eqid 2423 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
32 cnfldmul 18973 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
3331, 32mgpplusg 17724 . . . . . . . 8  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3430, 33ressplusg 15236 . . . . . . 7  |-  ( ran 
F  e.  _V  ->  x.  =  ( +g  `  G
) )
3529, 34syl 17 . . . . . 6  |-  ( ph  ->  x.  =  ( +g  `  G ) )
36353ad2ant1 1027 . . . . 5  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  x.  =  ( +g  `  G ) )
3736oveqd 6321 . . . 4  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( ( F `  x )  x.  ( F `  y
) )  =  ( ( F `  x
) ( +g  `  G
) ( F `  y ) ) )
3819, 25, 373eqtr3d 2472 . . 3  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x ( +g  `  (flds  X ) ) y ) )  =  ( ( F `  x ) ( +g  `  G
) ( F `  y ) ) )
395, 12, 14, 38syl3anc 1265 . 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 5890 . . . . 5  |-  ( exp `  ( A  x.  x
) )  e.  _V
4140, 17fnmpti 5723 . . . 4  |-  F  Fn  X
42 dffn4 5815 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
4341, 42mpbi 212 . . 3  |-  F : X -onto-> ran  F
44 eqidd 2424 . . . 4  |-  ( ph  ->  F  =  F )
45 eff 14133 . . . . . . . 8  |-  exp : CC
--> CC
4645a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  exp : CC --> CC )
4715adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  CC )
48 cnfldbas 18971 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
4948subgss 16815 . . . . . . . . . 10  |-  ( X  e.  (SubGrp ` fld )  ->  X  C_  CC )
507, 49syl 17 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
5150sselda 3466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
5247, 51mulcld 9669 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A  x.  x )  e.  CC )
5346, 52ffvelrnd 6037 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( exp `  ( A  x.  x ) )  e.  CC )
5453ralrimiva 2840 . . . . 5  |-  ( ph  ->  A. x  e.  X  ( exp `  ( A  x.  x ) )  e.  CC )
5517rnmptss 6066 . . . . 5  |-  ( A. x  e.  X  ( exp `  ( A  x.  x ) )  e.  CC  ->  ran  F  C_  CC )
5631, 48mgpbas 17726 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
5730, 56ressbas2 15177 . . . . 5  |-  ( ran 
F  C_  CC  ->  ran 
F  =  ( Base `  G ) )
5854, 55, 573syl 18 . . . 4  |-  ( ph  ->  ran  F  =  (
Base `  G )
)
5944, 10, 58foeq123d 5826 . . 3  |-  ( ph  ->  ( F : X -onto-> ran  F  <->  F : ( Base `  (flds  X ) ) -onto-> ( Base `  G ) ) )
6043, 59mpbii 215 . 2  |-  ( ph  ->  F : ( Base `  (flds  X ) ) -onto-> ( Base `  G ) )
61 cnring 18987 . . . 4  |-fld  e.  Ring
62 ringabl 17807 . . . 4  |-  (fld  e.  Ring  ->fld  e.  Abel )
6361, 62ax-mp 5 . . 3  |-fld  e.  Abel
648subgabl 17473 . . 3  |-  ( (fld  e. 
Abel  /\  X  e.  (SubGrp ` fld ) )  ->  (flds  X )  e.  Abel )
6563, 7, 64sylancr 668 . 2  |-  ( ph  ->  (flds  X )  e.  Abel )
661, 2, 3, 4, 39, 60, 65ghmabl 17470 1  |-  ( ph  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776   _Vcvv 3082    C_ wss 3438    |-> cmpt 4481   ran crn 4853    Fn wfn 5595   -->wf 5596   -onto->wfo 5598   ` cfv 5600  (class class class)co 6304   CCcc 9543    + caddc 9548    x. cmul 9550   expce 14111   Basecbs 15118   ↾s cress 15119   +g cplusg 15187  SubGrpcsubg 16808   Abelcabl 17428  mulGrpcmgp 17720   Ringcrg 17777  ℂfldccnfld 18967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4535  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596  ax-inf2 8154  ax-cnex 9601  ax-resscn 9602  ax-1cn 9603  ax-icn 9604  ax-addcl 9605  ax-addrcl 9606  ax-mulcl 9607  ax-mulrcl 9608  ax-mulcom 9609  ax-addass 9610  ax-mulass 9611  ax-distr 9612  ax-i2m1 9613  ax-1ne0 9614  ax-1rid 9615  ax-rnegex 9616  ax-rrecex 9617  ax-cnre 9618  ax-pre-lttri 9619  ax-pre-lttrn 9620  ax-pre-ltadd 9621  ax-pre-mulgt0 9622  ax-pre-sup 9623  ax-addf 9624  ax-mulf 9625
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-pss 3454  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4219  df-int 4255  df-iun 4300  df-br 4423  df-opab 4482  df-mpt 4483  df-tr 4518  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6266  df-ov 6307  df-oprab 6308  df-mpt2 6309  df-om 6706  df-1st 6806  df-2nd 6807  df-wrecs 7038  df-recs 7100  df-rdg 7138  df-1o 7192  df-oadd 7196  df-er 7373  df-pm 7485  df-en 7580  df-dom 7581  df-sdom 7582  df-fin 7583  df-sup 7964  df-inf 7965  df-oi 8033  df-card 8380  df-pnf 9683  df-mnf 9684  df-xr 9685  df-ltxr 9686  df-le 9687  df-sub 9868  df-neg 9869  df-div 10276  df-nn 10616  df-2 10674  df-3 10675  df-4 10676  df-5 10677  df-6 10678  df-7 10679  df-8 10680  df-9 10681  df-10 10682  df-n0 10876  df-z 10944  df-dec 11058  df-uz 11166  df-rp 11309  df-ico 11647  df-fz 11791  df-fzo 11922  df-fl 12033  df-seq 12219  df-exp 12278  df-fac 12465  df-bc 12493  df-hash 12521  df-shft 13128  df-cj 13160  df-re 13161  df-im 13162  df-sqrt 13296  df-abs 13297  df-limsup 13523  df-clim 13549  df-rlim 13550  df-sum 13750  df-ef 14118  df-struct 15120  df-ndx 15121  df-slot 15122  df-base 15123  df-sets 15124  df-ress 15125  df-plusg 15200  df-mulr 15201  df-starv 15202  df-tset 15206  df-ple 15207  df-ds 15209  df-unif 15210  df-0g 15337  df-mgm 16485  df-sgrp 16524  df-mnd 16534  df-grp 16670  df-minusg 16671  df-subg 16811  df-cmn 17429  df-abl 17430  df-mgp 17721  df-ur 17733  df-ring 17779  df-cring 17780  df-cnfld 18968
This theorem is referenced by:  efsubm  23496  circgrp  23497
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