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Theorem efghgrp 21040
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 2283 . . . . . . . 8  |-  ( x  e.  X  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
43rnmpt 4925 . . . . . . 7  |-  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) )  =  { y  |  E. x  e.  X  y  =  ( exp `  ( A  x.  x ) ) }
52, 4eqtr4i 2306 . . . . . 6  |-  S  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
6 df-ima 4702 . . . . . . . 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 5000 . . . . . . . . . 10  |-  ( X 
C_  CC  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )  |`  X )  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) ) )
97, 8syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) )  |`  X )  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) ) )
109rneqd 4906 . . . . . . . 8  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  |`  X )  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) ) )
116, 10syl5eq 2327 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) ) )
12 ax-addf 8816 . . . . . . . . . . 11  |-  +  :
( CC  X.  CC )
--> CC
1312fdmi 5394 . . . . . . . . . 10  |-  dom  +  =  ( CC  X.  CC )
1413a1i 10 . . . . . . . . 9  |-  ( ph  ->  dom  +  =  ( CC  X.  CC ) )
15 cnaddablo 21017 . . . . . . . . . . . 12  |-  +  e.  AbelOp
16 efghgrp.5 . . . . . . . . . . . 12  |-  (  +  |`  ( X  X.  X
) )  e.  (
SubGrpOp `  +  )
17 subgoablo 20978 . . . . . . . . . . . 12  |-  ( (  +  e.  AbelOp  /\  (  +  |`  ( X  X.  X ) )  e.  ( SubGrpOp `  +  )
)  ->  (  +  |`  ( X  X.  X
) )  e.  AbelOp )
1815, 16, 17mp2an 653 . . . . . . . . . . 11  |-  (  +  |`  ( X  X.  X
) )  e.  AbelOp
1918a1i 10 . . . . . . . . . 10  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  AbelOp )
20 ablogrpo 20951 . . . . . . . . . 10  |-  ( (  +  |`  ( X  X.  X ) )  e. 
AbelOp  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
2119, 20syl 15 . . . . . . . . 9  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
22 eqid 2283 . . . . . . . . . 10  |-  (  +  |`  ( X  X.  X
) )  =  (  +  |`  ( X  X.  X ) )
2322resgrprn 20947 . . . . . . . . 9  |-  ( ( dom  +  =  ( CC  X.  CC )  /\  (  +  |`  ( X  X.  X ) )  e.  GrpOp  /\  X  C_  CC )  ->  X  =  ran  (  +  |`  ( X  X.  X ) ) )
2414, 21, 7, 23syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  X  =  ran  (  +  |`  ( X  X.  X ) ) )
2524imaeq2d 5012 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2611, 25eqtr3d 2317 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
275, 26syl5eq 2327 . . . . 5  |-  ( ph  ->  S  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2827, 27xpeq12d 4714 . . . 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 4955 . . 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 2327 . 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 10 . . 3  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  ( SubGrpOp `  +  ) )
32 ablogrpo 20951 . . . . 5  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
3315, 32ax-mp 8 . . . 4  |-  +  e.  GrpOp
3433, 13grporn 20879 . . 3  |-  CC  =  ran  +
35 efghgrp.3 . . . . 5  |-  ( ph  ->  A  e.  CC )
36 mulcl 8821 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
37 efcl 12364 . . . . . 6  |-  ( ( A  x.  x )  e.  CC  ->  ( exp `  ( A  x.  x ) )  e.  CC )
3836, 37syl 15 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( exp `  ( A  x.  x )
)  e.  CC )
3935, 38sylan 457 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( exp `  ( A  x.  x
) )  e.  CC )
40 eqid 2283 . . . 4  |-  ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )
4139, 40fmptd 5684 . . 3  |-  ( ph  ->  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) : CC --> CC )
42 ssid 3197 . . . 4  |-  CC  C_  CC
4342a1i 10 . . 3  |-  ( ph  ->  CC  C_  CC )
44 ax-mulf 8817 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
45 ffn 5389 . . . . 5  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
4644, 45ax-mp 8 . . . 4  |-  x.  Fn  ( CC  X.  CC )
4746a1i 10 . . 3  |-  ( ph  ->  x.  Fn  ( CC 
X.  CC ) )
4840efgh 19903 . . . . 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 1152 . . . 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 457 . . 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 2283 . . 3  |-  ran  (  +  |`  ( X  X.  X ) )  =  ran  (  +  |`  ( X  X.  X ) )
52 eqid 2283 . . 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 2283 . . 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 21039 . 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 2357 1  |-  ( ph  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742   expce 12343   GrpOpcgr 20853   AbelOpcablo 20948   SubGrpOpcsubgo 20968
This theorem is referenced by:  circgrp  21041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969
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