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Theorem efabl 23022
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 2382 . 2  |-  ( Base `  (flds  X ) )  =  (
Base `  (flds  X ) )
2 eqid 2382 . 2  |-  ( Base `  G )  =  (
Base `  G )
3 eqid 2382 . 2  |-  ( +g  `  (flds  X ) )  =  ( +g  `  (flds  X ) )
4 eqid 2382 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
5 simp1 994 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  ph )
6 simp2 995 . . . 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 2382 . . . . . . 7  |-  (flds  X )  =  (flds  X )
98subgbas 16322 . . . . . 6  |-  ( X  e.  (SubGrp ` fld )  ->  X  =  ( Base `  (flds  X )
) )
107, 9syl 16 . . . . 5  |-  ( ph  ->  X  =  ( Base `  (flds  X ) ) )
11103ad2ant1 1015 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  X  =  ( Base `  (flds  X ) ) )
126, 11eleqtrrd 2473 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  ->  x  e.  X )
13 simp3 996 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  (flds  X ) )  /\  y  e.  ( Base `  (flds  X ) ) )  -> 
y  e.  ( Base `  (flds  X ) ) )
1413, 11eleqtrrd 2473 . . 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 530 . . . . 5  |-  ( ph  ->  ( A  e.  CC  /\  X  e.  (SubGrp ` fld )
) )
17 efabl.1 . . . . . 6  |-  F  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
1817efgh 23013 . . . . 5  |-  ( ( ( A  e.  CC  /\  X  e.  (SubGrp ` fld )
)  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  +  y ) )  =  ( ( F `  x
)  x.  ( F `
 y ) ) )
1916, 18syl3an1 1259 . . . 4  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  +  y ) )  =  ( ( F `  x
)  x.  ( F `
 y ) ) )
20 cnfldadd 18538 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
218, 20ressplusg 14748 . . . . . . . 8  |-  ( X  e.  (SubGrp ` fld )  ->  +  =  ( +g  `  (flds  X ) ) )
227, 21syl 16 . . . . . . 7  |-  ( ph  ->  +  =  ( +g  `  (flds  X ) ) )
23223ad2ant1 1015 . . . . . 6  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  +  =  ( +g  `  (flds  X ) ) )
2423oveqd 6213 . . . . 5  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( x  +  y )  =  ( x ( +g  `  (flds  X ) ) y ) )
2524fveq2d 5778 . . . 4  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x  +  y ) )  =  ( F `  ( x ( +g  `  (flds  X )
) y ) ) )
26 mptexg 6043 . . . . . . . . 9  |-  ( X  e.  (SubGrp ` fld )  ->  ( x  e.  X  |->  ( exp `  ( A  x.  x
) ) )  e. 
_V )
2717, 26syl5eqel 2474 . . . . . . . 8  |-  ( X  e.  (SubGrp ` fld )  ->  F  e. 
_V )
28 rnexg 6631 . . . . . . . 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 2382 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
32 cnfldmul 18539 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
3331, 32mgpplusg 17258 . . . . . . . 8  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3430, 33ressplusg 14748 . . . . . . 7  |-  ( ran 
F  e.  _V  ->  x.  =  ( +g  `  G
) )
3529, 34syl 16 . . . . . 6  |-  ( ph  ->  x.  =  ( +g  `  G ) )
36353ad2ant1 1015 . . . . 5  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  x.  =  ( +g  `  G ) )
3736oveqd 6213 . . . 4  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( ( F `  x )  x.  ( F `  y
) )  =  ( ( F `  x
) ( +g  `  G
) ( F `  y ) ) )
3819, 25, 373eqtr3d 2431 . . 3  |-  ( (
ph  /\  x  e.  X  /\  y  e.  X
)  ->  ( F `  ( x ( +g  `  (flds  X ) ) y ) )  =  ( ( F `  x ) ( +g  `  G
) ( F `  y ) ) )
395, 12, 14, 38syl3anc 1226 . 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 5784 . . . . 5  |-  ( exp `  ( A  x.  x
) )  e.  _V
4140, 17fnmpti 5617 . . . 4  |-  F  Fn  X
42 dffn4 5709 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
4341, 42mpbi 208 . . 3  |-  F : X -onto-> ran  F
44 eqidd 2383 . . . 4  |-  ( ph  ->  F  =  F )
45 eff 13819 . . . . . . . 8  |-  exp : CC
--> CC
4645a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  exp : CC --> CC )
4715adantr 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  CC )
48 cnfldbas 18537 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
4948subgss 16319 . . . . . . . . . 10  |-  ( X  e.  (SubGrp ` fld )  ->  X  C_  CC )
507, 49syl 16 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
5150sselda 3417 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  CC )
5247, 51mulcld 9527 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A  x.  x )  e.  CC )
5346, 52ffvelrnd 5934 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( exp `  ( A  x.  x ) )  e.  CC )
5453ralrimiva 2796 . . . . 5  |-  ( ph  ->  A. x  e.  X  ( exp `  ( A  x.  x ) )  e.  CC )
5517rnmptss 5962 . . . . 5  |-  ( A. x  e.  X  ( exp `  ( A  x.  x ) )  e.  CC  ->  ran  F  C_  CC )
5631, 48mgpbas 17260 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
5730, 56ressbas2 14692 . . . . 5  |-  ( ran 
F  C_  CC  ->  ran 
F  =  ( Base `  G ) )
5854, 55, 573syl 20 . . . 4  |-  ( ph  ->  ran  F  =  (
Base `  G )
)
5944, 10, 58foeq123d 5720 . . 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 18553 . . . 4  |-fld  e.  Ring
62 ringabl 17341 . . . 4  |-  (fld  e.  Ring  ->fld  e.  Abel )
6361, 62ax-mp 5 . . 3  |-fld  e.  Abel
648subgabl 16961 . . 3  |-  ( (fld  e. 
Abel  /\  X  e.  (SubGrp ` fld ) )  ->  (flds  X )  e.  Abel )
6563, 7, 64sylancr 661 . 2  |-  ( ph  ->  (flds  X )  e.  Abel )
661, 2, 3, 4, 39, 60, 65ghmabl 16958 1  |-  ( ph  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034    C_ wss 3389    |-> cmpt 4425   ran crn 4914    Fn wfn 5491   -->wf 5492   -onto->wfo 5494   ` cfv 5496  (class class class)co 6196   CCcc 9401    + caddc 9406    x. cmul 9408   expce 13799   Basecbs 14634   ↾s cress 14635   +g cplusg 14702  SubGrpcsubg 16312   Abelcabl 16916  mulGrpcmgp 17254   Ringcrg 17311  ℂfldccnfld 18533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-rp 11140  df-ico 11456  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-fac 12256  df-bc 12283  df-hash 12308  df-shft 12902  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-limsup 13296  df-clim 13313  df-rlim 13314  df-sum 13511  df-ef 13805  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-subg 16315  df-cmn 16917  df-abl 16918  df-mgp 17255  df-ur 17267  df-ring 17313  df-cring 17314  df-cnfld 18534
This theorem is referenced by:  efsubm  23023  circgrp  23024
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