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Theorem efghgrp 21809
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.) (New usage is discouraged.)
Hypotheses
Ref Expression
efghgrp.1  |-  S  =  { y  |  E. x  e.  X  y  =  ( exp `  ( A  x.  x )
) }
efghgrp.2  |-  G  =  (  x.  |`  ( S  X.  S ) )
efghgrp.3  |-  ( ph  ->  A  e.  CC )
efghgrp.4  |-  ( ph  ->  X  C_  CC )
efghgrp.5  |-  (  +  |`  ( X  X.  X
) )  e.  (
SubGrpOp `  +  )
Assertion
Ref Expression
efghgrp  |-  ( ph  ->  G  e.  AbelOp )
Distinct variable groups:    x, y, A    ph, x, y    x, X, y
Allowed substitution hints:    S( x, y)    G( x, y)

Proof of Theorem efghgrp
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 efghgrp.2 . . 3  |-  G  =  (  x.  |`  ( S  X.  S ) )
2 efghgrp.1 . . . . . . 7  |-  S  =  { y  |  E. x  e.  X  y  =  ( exp `  ( A  x.  x )
) }
3 eqid 2387 . . . . . . . 8  |-  ( x  e.  X  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
43rnmpt 5056 . . . . . . 7  |-  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) )  =  { y  |  E. x  e.  X  y  =  ( exp `  ( A  x.  x ) ) }
52, 4eqtr4i 2410 . . . . . 6  |-  S  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
6 df-ima 4831 . . . . . . . 8  |-  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " X )  =  ran  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )  |`  X )
7 efghgrp.4 . . . . . . . . . 10  |-  ( ph  ->  X  C_  CC )
8 resmpt 5131 . . . . . . . . . 10  |-  ( X 
C_  CC  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )  |`  X )  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) ) )
97, 8syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) )  |`  X )  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) ) )
109rneqd 5037 . . . . . . . 8  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  |`  X )  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) ) )
116, 10syl5eq 2431 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) ) )
12 ax-addf 9002 . . . . . . . . . . 11  |-  +  :
( CC  X.  CC )
--> CC
1312fdmi 5536 . . . . . . . . . 10  |-  dom  +  =  ( CC  X.  CC )
1413a1i 11 . . . . . . . . 9  |-  ( ph  ->  dom  +  =  ( CC  X.  CC ) )
15 cnaddablo 21786 . . . . . . . . . . . 12  |-  +  e.  AbelOp
16 efghgrp.5 . . . . . . . . . . . 12  |-  (  +  |`  ( X  X.  X
) )  e.  (
SubGrpOp `  +  )
17 subgoablo 21747 . . . . . . . . . . . 12  |-  ( (  +  e.  AbelOp  /\  (  +  |`  ( X  X.  X ) )  e.  ( SubGrpOp `  +  )
)  ->  (  +  |`  ( X  X.  X
) )  e.  AbelOp )
1815, 16, 17mp2an 654 . . . . . . . . . . 11  |-  (  +  |`  ( X  X.  X
) )  e.  AbelOp
1918a1i 11 . . . . . . . . . 10  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  AbelOp )
20 ablogrpo 21720 . . . . . . . . . 10  |-  ( (  +  |`  ( X  X.  X ) )  e. 
AbelOp  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
2119, 20syl 16 . . . . . . . . 9  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
22 eqid 2387 . . . . . . . . . 10  |-  (  +  |`  ( X  X.  X
) )  =  (  +  |`  ( X  X.  X ) )
2322resgrprn 21716 . . . . . . . . 9  |-  ( ( dom  +  =  ( CC  X.  CC )  /\  (  +  |`  ( X  X.  X ) )  e.  GrpOp  /\  X  C_  CC )  ->  X  =  ran  (  +  |`  ( X  X.  X ) ) )
2414, 21, 7, 23syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  X  =  ran  (  +  |`  ( X  X.  X ) ) )
2524imaeq2d 5143 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2611, 25eqtr3d 2421 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
275, 26syl5eq 2431 . . . . 5  |-  ( ph  ->  S  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2827, 27xpeq12d 4843 . . . 4  |-  ( ph  ->  ( S  X.  S
)  =  ( ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) )  X.  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) ) )
2928reseq2d 5086 . . 3  |-  ( ph  ->  (  x.  |`  ( S  X.  S ) )  =  (  x.  |`  (
( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " ran  (  +  |`  ( X  X.  X ) ) )  X.  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) ) ) )
301, 29syl5eq 2431 . 2  |-  ( ph  ->  G  =  (  x.  |`  ( ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) ) " ran  (  +  |`  ( X  X.  X ) ) )  X.  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) ) ) )
3116a1i 11 . . 3  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  ( SubGrpOp `  +  ) )
32 ablogrpo 21720 . . . . 5  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
3315, 32ax-mp 8 . . . 4  |-  +  e.  GrpOp
3433, 13grporn 21648 . . 3  |-  CC  =  ran  +
35 efghgrp.3 . . . . 5  |-  ( ph  ->  A  e.  CC )
36 mulcl 9007 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
37 efcl 12612 . . . . . 6  |-  ( ( A  x.  x )  e.  CC  ->  ( exp `  ( A  x.  x ) )  e.  CC )
3836, 37syl 16 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( exp `  ( A  x.  x )
)  e.  CC )
3935, 38sylan 458 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( exp `  ( A  x.  x
) )  e.  CC )
40 eqid 2387 . . . 4  |-  ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )
4139, 40fmptd 5832 . . 3  |-  ( ph  ->  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) : CC --> CC )
42 ssid 3310 . . . 4  |-  CC  C_  CC
4342a1i 11 . . 3  |-  ( ph  ->  CC  C_  CC )
44 ax-mulf 9003 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
45 ffn 5531 . . . . 5  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
4644, 45ax-mp 8 . . . 4  |-  x.  Fn  ( CC  X.  CC )
4746a1i 11 . . 3  |-  ( ph  ->  x.  Fn  ( CC 
X.  CC ) )
4840efgh 20310 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) `  ( y  +  z ) )  =  ( ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) `  y )  x.  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) ) `  z ) ) )
49483expb 1154 . . . 4  |-  ( ( A  e.  CC  /\  ( y  e.  CC  /\  z  e.  CC ) )  ->  ( (
x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) `  ( y  +  z ) )  =  ( ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) `  y )  x.  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) ) `  z ) ) )
5035, 49sylan 458 . . 3  |-  ( (
ph  /\  ( y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) `  (
y  +  z ) )  =  ( ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) `  y )  x.  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) ) `  z ) ) )
51 eqid 2387 . . 3  |-  ran  (  +  |`  ( X  X.  X ) )  =  ran  (  +  |`  ( X  X.  X ) )
52 eqid 2387 . . 3  |-  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) )  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) ) " ran  (  +  |`  ( X  X.  X ) ) )
53 eqid 2387 . . 3  |-  (  x.  |`  ( ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) ) " ran  (  +  |`  ( X  X.  X ) ) )  X.  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) ) )  =  (  x.  |`  ( (
( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) )  X.  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) ) )
5431, 34, 41, 43, 47, 50, 51, 52, 53, 19ghsubablo 21808 . 2  |-  ( ph  ->  (  x.  |`  (
( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " ran  (  +  |`  ( X  X.  X ) ) )  X.  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) ) )  e.  AbelOp )
5530, 54eqeltrd 2461 1  |-  ( ph  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   E.wrex 2650    C_ wss 3263    e. cmpt 4207    X. cxp 4816   dom cdm 4818   ran crn 4819    |` cres 4820   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921    + caddc 8926    x. cmul 8928   expce 12591   GrpOpcgr 21622   AbelOpcablo 21717   SubGrpOpcsubgo 21737
This theorem is referenced by:  circgrp  21810
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-ico 10854  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-sum 12407  df-ef 12597  df-grpo 21627  df-gid 21628  df-ginv 21629  df-gdiv 21630  df-ablo 21718  df-subgo 21738
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