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Theorem efghgrp 23872
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 2443 . . . . . . . 8  |-  ( x  e.  X  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
43rnmpt 5097 . . . . . . 7  |-  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) )  =  { y  |  E. x  e.  X  y  =  ( exp `  ( A  x.  x ) ) }
52, 4eqtr4i 2466 . . . . . 6  |-  S  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
6 df-ima 4865 . . . . . . . 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 5168 . . . . . . . . . 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 5079 . . . . . . . 8  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  |`  X )  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) ) )
116, 10syl5eq 2487 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) ) )
12 ax-addf 9373 . . . . . . . . . . 11  |-  +  :
( CC  X.  CC )
--> CC
1312fdmi 5576 . . . . . . . . . 10  |-  dom  +  =  ( CC  X.  CC )
1413a1i 11 . . . . . . . . 9  |-  ( ph  ->  dom  +  =  ( CC  X.  CC ) )
15 cnaddablo 23849 . . . . . . . . . . . 12  |-  +  e.  AbelOp
16 efghgrp.5 . . . . . . . . . . . 12  |-  (  +  |`  ( X  X.  X
) )  e.  (
SubGrpOp `  +  )
17 subgoablo 23810 . . . . . . . . . . . 12  |-  ( (  +  e.  AbelOp  /\  (  +  |`  ( X  X.  X ) )  e.  ( SubGrpOp `  +  )
)  ->  (  +  |`  ( X  X.  X
) )  e.  AbelOp )
1815, 16, 17mp2an 672 . . . . . . . . . . 11  |-  (  +  |`  ( X  X.  X
) )  e.  AbelOp
1918a1i 11 . . . . . . . . . 10  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  AbelOp )
20 ablogrpo 23783 . . . . . . . . . 10  |-  ( (  +  |`  ( X  X.  X ) )  e. 
AbelOp  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
2119, 20syl 16 . . . . . . . . 9  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
22 eqid 2443 . . . . . . . . . 10  |-  (  +  |`  ( X  X.  X
) )  =  (  +  |`  ( X  X.  X ) )
2322resgrprn 23779 . . . . . . . . 9  |-  ( ( dom  +  =  ( CC  X.  CC )  /\  (  +  |`  ( X  X.  X ) )  e.  GrpOp  /\  X  C_  CC )  ->  X  =  ran  (  +  |`  ( X  X.  X ) ) )
2414, 21, 7, 23syl3anc 1218 . . . . . . . 8  |-  ( ph  ->  X  =  ran  (  +  |`  ( X  X.  X ) ) )
2524imaeq2d 5181 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2611, 25eqtr3d 2477 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
275, 26syl5eq 2487 . . . . 5  |-  ( ph  ->  S  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2827, 27xpeq12d 4877 . . . 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 5122 . . 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 2487 . 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 23783 . . . . 5  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
3315, 32ax-mp 5 . . . 4  |-  +  e.  GrpOp
3433, 13grporn 23711 . . 3  |-  CC  =  ran  +
35 efghgrp.3 . . . . 5  |-  ( ph  ->  A  e.  CC )
36 mulcl 9378 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
37 efcl 13380 . . . . . 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 471 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( exp `  ( A  x.  x
) )  e.  CC )
40 eqid 2443 . . . 4  |-  ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )
4139, 40fmptd 5879 . . 3  |-  ( ph  ->  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) : CC --> CC )
42 ssid 3387 . . . 4  |-  CC  C_  CC
4342a1i 11 . . 3  |-  ( ph  ->  CC  C_  CC )
44 ax-mulf 9374 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
45 ffn 5571 . . . . 5  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
4644, 45ax-mp 5 . . . 4  |-  x.  Fn  ( CC  X.  CC )
4746a1i 11 . . 3  |-  ( ph  ->  x.  Fn  ( CC 
X.  CC ) )
4840efgh 22009 . . . . 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 1188 . . . 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 471 . . 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 2443 . . 3  |-  ran  (  +  |`  ( X  X.  X ) )  =  ran  (  +  |`  ( X  X.  X ) )
52 eqid 2443 . . 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 2443 . . 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 23871 . 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 2517 1  |-  ( ph  ->  G  e.  AbelOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2728    C_ wss 3340    e. cmpt 4362    X. cxp 4850   dom cdm 4852   ran crn 4853    |` cres 4854   "cima 4855    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   CCcc 9292    + caddc 9297    x. cmul 9299   expce 13359   GrpOpcgr 23685   AbelOpcablo 23780   SubGrpOpcsubgo 23800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  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-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-ico 11318  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-fac 12064  df-bc 12091  df-hash 12116  df-shft 12568  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-limsup 12961  df-clim 12978  df-rlim 12979  df-sum 13176  df-ef 13365  df-grpo 23690  df-gid 23691  df-ginv 23692  df-gdiv 23693  df-ablo 23781  df-subgo 23801
This theorem is referenced by:  circgrp  23873
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