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Theorem pwsmgp 16700
Description: The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.)
Hypotheses
Ref Expression
pwsmgp.y  |-  Y  =  ( R  ^s  I )
pwsmgp.m  |-  M  =  (mulGrp `  R )
pwsmgp.z  |-  Z  =  ( M  ^s  I )
pwsmgp.n  |-  N  =  (mulGrp `  Y )
pwsmgp.b  |-  B  =  ( Base `  N
)
pwsmgp.c  |-  C  =  ( Base `  Z
)
pwsmgp.p  |-  .+  =  ( +g  `  N )
pwsmgp.q  |-  .+b  =  ( +g  `  Z )
Assertion
Ref Expression
pwsmgp  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( B  =  C  /\  .+  =  .+b  ) )

Proof of Theorem pwsmgp
StepHypRef Expression
1 eqid 2438 . . . . . 6  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
2 eqid 2438 . . . . . 6  |-  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  (mulGrp `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
3 eqid 2438 . . . . . 6  |-  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) )  =  ( (Scalar `  R ) X_s (mulGrp 
o.  ( I  X.  { R } ) ) )
4 simpr 461 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  I  e.  W )
5 fvex 5696 . . . . . . 7  |-  (Scalar `  R )  e.  _V
65a1i 11 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  R )  e.  _V )
7 fnconstg 5593 . . . . . . 7  |-  ( R  e.  V  ->  (
I  X.  { R } )  Fn  I
)
87adantr 465 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { R } )  Fn  I
)
91, 2, 3, 4, 6, 8prdsmgp 16692 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( ( Base `  (mulGrp `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )  =  ( Base `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )  /\  ( +g  `  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )  =  ( +g  `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) ) )
109simpld 459 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  (mulGrp `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )  =  ( Base `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
11 pwsmgp.n . . . . . 6  |-  N  =  (mulGrp `  Y )
12 pwsmgp.y . . . . . . . 8  |-  Y  =  ( R  ^s  I )
13 eqid 2438 . . . . . . . 8  |-  (Scalar `  R )  =  (Scalar `  R )
1412, 13pwsval 14416 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
1514fveq2d 5690 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (mulGrp `  Y )  =  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
1611, 15syl5eq 2482 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  N  =  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
1716fveq2d 5690 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  N
)  =  ( Base `  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
18 pwsmgp.z . . . . . 6  |-  Z  =  ( M  ^s  I )
19 pwsmgp.m . . . . . . . . 9  |-  M  =  (mulGrp `  R )
20 fvex 5696 . . . . . . . . 9  |-  (mulGrp `  R )  e.  _V
2119, 20eqeltri 2508 . . . . . . . 8  |-  M  e. 
_V
22 eqid 2438 . . . . . . . . 9  |-  ( M  ^s  I )  =  ( M  ^s  I )
23 eqid 2438 . . . . . . . . 9  |-  (Scalar `  M )  =  (Scalar `  M )
2422, 23pwsval 14416 . . . . . . . 8  |-  ( ( M  e.  _V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2521, 4, 24sylancr 663 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2619, 13mgpsca 16588 . . . . . . . . . 10  |-  (Scalar `  R )  =  (Scalar `  M )
2726eqcomi 2442 . . . . . . . . 9  |-  (Scalar `  M )  =  (Scalar `  R )
2827a1i 11 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  M )  =  (Scalar `  R )
)
29 fnmgp 16583 . . . . . . . . . 10  |- mulGrp  Fn  _V
30 elex 2976 . . . . . . . . . . 11  |-  ( R  e.  V  ->  R  e.  _V )
3130adantr 465 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  e.  _V )
32 fcoconst 5875 . . . . . . . . . 10  |-  ( (mulGrp 
Fn  _V  /\  R  e. 
_V )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  {
(mulGrp `  R ) } ) )
3329, 31, 32sylancr 663 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  { (mulGrp `  R ) } ) )
3419sneqi 3883 . . . . . . . . . 10  |-  { M }  =  { (mulGrp `  R ) }
3534xpeq2i 4856 . . . . . . . . 9  |-  ( I  X.  { M }
)  =  ( I  X.  { (mulGrp `  R ) } )
3633, 35syl6reqr 2489 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { M } )  =  (mulGrp 
o.  ( I  X.  { R } ) ) )
3728, 36oveq12d 6104 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( (Scalar `  M
) X_s ( I  X.  { M } ) )  =  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )
3825, 37eqtrd 2470 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )
3918, 38syl5eq 2482 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Z  =  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )
4039fveq2d 5690 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  Z
)  =  ( Base `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
4110, 17, 403eqtr4d 2480 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  N
)  =  ( Base `  Z ) )
42 pwsmgp.b . . 3  |-  B  =  ( Base `  N
)
43 pwsmgp.c . . 3  |-  C  =  ( Base `  Z
)
4441, 42, 433eqtr4g 2495 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  C )
459simprd 463 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  (mulGrp `  ( (Scalar `  R
) X_s ( I  X.  { R } ) ) ) )  =  ( +g  `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
4616fveq2d 5690 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  N
)  =  ( +g  `  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
4739fveq2d 5690 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  Z
)  =  ( +g  `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
4845, 46, 473eqtr4d 2480 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  N
)  =  ( +g  `  Z ) )
49 pwsmgp.p . . 3  |-  .+  =  ( +g  `  N )
50 pwsmgp.q . . 3  |-  .+b  =  ( +g  `  Z )
5148, 49, 503eqtr4g 2495 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  .+  =  .+b  )
5244, 51jca 532 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( B  =  C  /\  .+  =  .+b  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   {csn 3872    X. cxp 4833    o. ccom 4839    Fn wfn 5408   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230  Scalarcsca 14233   X_scprds 14376    ^s cpws 14377  mulGrpcmgp 16581
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-hom 14254  df-cco 14255  df-prds 14378  df-pws 14380  df-mgp 16582
This theorem is referenced by:  pwsco1rhm  16810  pwsco2rhm  16811  pwsdiagrhm  16878  evl1expd  17759
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