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Theorem efghgrp 21056
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 2296 . . . . . . . 8  |-  ( x  e.  X  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
43rnmpt 4941 . . . . . . 7  |-  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) )  =  { y  |  E. x  e.  X  y  =  ( exp `  ( A  x.  x ) ) }
52, 4eqtr4i 2319 . . . . . 6  |-  S  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
6 df-ima 4718 . . . . . . . 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 5016 . . . . . . . . . 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 4922 . . . . . . . 8  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  |`  X )  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) ) )
116, 10syl5eq 2340 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) ) )
12 ax-addf 8832 . . . . . . . . . . 11  |-  +  :
( CC  X.  CC )
--> CC
1312fdmi 5410 . . . . . . . . . 10  |-  dom  +  =  ( CC  X.  CC )
1413a1i 10 . . . . . . . . 9  |-  ( ph  ->  dom  +  =  ( CC  X.  CC ) )
15 cnaddablo 21033 . . . . . . . . . . . 12  |-  +  e.  AbelOp
16 efghgrp.5 . . . . . . . . . . . 12  |-  (  +  |`  ( X  X.  X
) )  e.  (
SubGrpOp `  +  )
17 subgoablo 20994 . . . . . . . . . . . 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 20967 . . . . . . . . . 10  |-  ( (  +  |`  ( X  X.  X ) )  e. 
AbelOp  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
2119, 20syl 15 . . . . . . . . 9  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
22 eqid 2296 . . . . . . . . . 10  |-  (  +  |`  ( X  X.  X
) )  =  (  +  |`  ( X  X.  X ) )
2322resgrprn 20963 . . . . . . . . 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 5028 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2611, 25eqtr3d 2330 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
275, 26syl5eq 2340 . . . . 5  |-  ( ph  ->  S  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2827, 27xpeq12d 4730 . . . 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 4971 . . 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 2340 . 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 20967 . . . . 5  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
3315, 32ax-mp 8 . . . 4  |-  +  e.  GrpOp
3433, 13grporn 20895 . . 3  |-  CC  =  ran  +
35 efghgrp.3 . . . . 5  |-  ( ph  ->  A  e.  CC )
36 mulcl 8837 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
37 efcl 12380 . . . . . 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 2296 . . . 4  |-  ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )
4139, 40fmptd 5700 . . 3  |-  ( ph  ->  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) : CC --> CC )
42 ssid 3210 . . . 4  |-  CC  C_  CC
4342a1i 10 . . 3  |-  ( ph  ->  CC  C_  CC )
44 ax-mulf 8833 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
45 ffn 5405 . . . . 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 19919 . . . . 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 2296 . . 3  |-  ran  (  +  |`  ( X  X.  X ) )  =  ran  (  +  |`  ( X  X.  X ) )
52 eqid 2296 . . 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 2296 . . 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 21055 . 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 2370 1  |-  ( ph  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    C_ wss 3165    e. cmpt 4093    X. cxp 4703   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    + caddc 8756    x. cmul 8758   expce 12359   GrpOpcgr 20869   AbelOpcablo 20964   SubGrpOpcsubgo 20984
This theorem is referenced by:  circgrp  21057
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-subgo 20985
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