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Theorem efghgrp 25148
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.) (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 2467 . . . . . . . 8  |-  ( x  e.  X  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
43rnmpt 5248 . . . . . . 7  |-  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) )  =  { y  |  E. x  e.  X  y  =  ( exp `  ( A  x.  x ) ) }
52, 4eqtr4i 2499 . . . . . 6  |-  S  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )
6 df-ima 5012 . . . . . . . 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 5323 . . . . . . . . . 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 5230 . . . . . . . 8  |-  ( ph  ->  ran  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  |`  X )  =  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) ) )
116, 10syl5eq 2520 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ran  (
x  e.  X  |->  ( exp `  ( A  x.  x ) ) ) )
12 ax-addf 9572 . . . . . . . . . . 11  |-  +  :
( CC  X.  CC )
--> CC
1312fdmi 5736 . . . . . . . . . 10  |-  dom  +  =  ( CC  X.  CC )
1413a1i 11 . . . . . . . . 9  |-  ( ph  ->  dom  +  =  ( CC  X.  CC ) )
15 cnaddablo 25125 . . . . . . . . . . . 12  |-  +  e.  AbelOp
16 efghgrp.5 . . . . . . . . . . . 12  |-  (  +  |`  ( X  X.  X
) )  e.  (
SubGrpOp `  +  )
17 subgoablo 25086 . . . . . . . . . . . 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 25059 . . . . . . . . . 10  |-  ( (  +  |`  ( X  X.  X ) )  e. 
AbelOp  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
2119, 20syl 16 . . . . . . . . 9  |-  ( ph  ->  (  +  |`  ( X  X.  X ) )  e.  GrpOp )
22 eqid 2467 . . . . . . . . . 10  |-  (  +  |`  ( X  X.  X
) )  =  (  +  |`  ( X  X.  X ) )
2322resgrprn 25055 . . . . . . . . 9  |-  ( ( dom  +  =  ( CC  X.  CC )  /\  (  +  |`  ( X  X.  X ) )  e.  GrpOp  /\  X  C_  CC )  ->  X  =  ran  (  +  |`  ( X  X.  X ) ) )
2414, 21, 7, 23syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  X  =  ran  (  +  |`  ( X  X.  X ) ) )
2524imaeq2d 5337 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x )
) ) " X
)  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2611, 25eqtr3d 2510 . . . . . 6  |-  ( ph  ->  ran  ( x  e.  X  |->  ( exp `  ( A  x.  x )
) )  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
275, 26syl5eq 2520 . . . . 5  |-  ( ph  ->  S  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) ) )
2827, 27xpeq12d 5024 . . . 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 5273 . . 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 2520 . 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 25059 . . . . 5  |-  (  +  e.  AbelOp  ->  +  e.  GrpOp )
3315, 32ax-mp 5 . . . 4  |-  +  e.  GrpOp
3433, 13grporn 24987 . . 3  |-  CC  =  ran  +
35 efghgrp.3 . . . . 5  |-  ( ph  ->  A  e.  CC )
36 mulcl 9577 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
37 efcl 13683 . . . . . 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 2467 . . . 4  |-  ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) )  =  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )
4139, 40fmptd 6046 . . 3  |-  ( ph  ->  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) : CC --> CC )
42 ssid 3523 . . . 4  |-  CC  C_  CC
4342a1i 11 . . 3  |-  ( ph  ->  CC  C_  CC )
44 ax-mulf 9573 . . . . 5  |-  x.  :
( CC  X.  CC )
--> CC
45 ffn 5731 . . . . 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 ) )
48 cnrng 18251 . . . . . . . 8  |-fld  e.  Ring
49 rnggrp 17017 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Grp )
50 cnfldbas 18235 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
5150subgid 16017 . . . . . . . 8  |-  (fld  e.  Grp  ->  CC  e.  (SubGrp ` fld )
)
5248, 49, 51mp2b 10 . . . . . . 7  |-  CC  e.  (SubGrp ` fld )
5352jctr 542 . . . . . 6  |-  ( A  e.  CC  ->  ( A  e.  CC  /\  CC  e.  (SubGrp ` fld ) ) )
5440efgh 22753 . . . . . 6  |-  ( ( ( A  e.  CC  /\  CC  e.  (SubGrp ` fld )
)  /\  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 ) ) )
5553, 54syl3an1 1261 . . . . 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 ) ) )
56553expb 1197 . . . 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 ) ) )
5735, 56sylan 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 ) ) )
58 eqid 2467 . . 3  |-  ran  (  +  |`  ( X  X.  X ) )  =  ran  (  +  |`  ( X  X.  X ) )
59 eqid 2467 . . 3  |-  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) ) " ran  (  +  |`  ( X  X.  X ) ) )  =  ( ( x  e.  CC  |->  ( exp `  ( A  x.  x
) ) ) " ran  (  +  |`  ( X  X.  X ) ) )
60 eqid 2467 . . 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 ) ) ) ) )
6131, 34, 41, 43, 47, 57, 58, 59, 60, 19ghsubablo 25147 . 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 )
6230, 61eqeltrd 2555 1  |-  ( ph  ->  G  e.  AbelOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815    C_ wss 3476    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285   CCcc 9491    + caddc 9496    x. cmul 9498   expce 13662   Grpcgrp 15730  SubGrpcsubg 16009   Ringcrg 17012  ℂfldccnfld 18231   GrpOpcgr 24961   AbelOpcablo 25056   SubGrpOpcsubgo 25076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-rp 11222  df-ico 11536  df-fz 11674  df-fzo 11794  df-fl 11898  df-seq 12077  df-exp 12136  df-fac 12323  df-bc 12350  df-hash 12375  df-shft 12866  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-limsup 13260  df-clim 13277  df-rlim 13278  df-sum 13475  df-ef 13668  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-starv 14573  df-tset 14577  df-ple 14578  df-ds 14580  df-unif 14581  df-0g 14700  df-mnd 15735  df-grp 15871  df-subg 16012  df-cmn 16615  df-mgp 16956  df-rng 17014  df-cring 17015  df-cnfld 18232  df-grpo 24966  df-gid 24967  df-ginv 24968  df-gdiv 24969  df-ablo 25057  df-subgo 25077
This theorem is referenced by:  circgrpOLD  25149
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