MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwsmgp Unicode version

Theorem pwsmgp 15401
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 2283 . . . . . 6  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
2 eqid 2283 . . . . . 6  |-  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  (mulGrp `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
3 eqid 2283 . . . . . 6  |-  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) )  =  ( (Scalar `  R ) X_s (mulGrp 
o.  ( I  X.  { R } ) ) )
4 simpr 447 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  I  e.  W )
5 fvex 5539 . . . . . . 7  |-  (Scalar `  R )  e.  _V
65a1i 10 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  R )  e.  _V )
7 fnconstg 5429 . . . . . . 7  |-  ( R  e.  V  ->  (
I  X.  { R } )  Fn  I
)
87adantr 451 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { R } )  Fn  I
)
91, 2, 3, 4, 6, 8prdsmgp 15393 . . . . 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 445 . . . 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 2283 . . . . . . . 8  |-  (Scalar `  R )  =  (Scalar `  R )
1412, 13pwsval 13385 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
1514fveq2d 5529 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (mulGrp `  Y )  =  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
1611, 15syl5eq 2327 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  N  =  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
1716fveq2d 5529 . . . 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 5539 . . . . . . . . 9  |-  (mulGrp `  R )  e.  _V
2119, 20eqeltri 2353 . . . . . . . 8  |-  M  e. 
_V
22 eqid 2283 . . . . . . . . 9  |-  ( M  ^s  I )  =  ( M  ^s  I )
23 eqid 2283 . . . . . . . . 9  |-  (Scalar `  M )  =  (Scalar `  M )
2422, 23pwsval 13385 . . . . . . . 8  |-  ( ( M  e.  _V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2521, 4, 24sylancr 644 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2619, 13mgpsca 15332 . . . . . . . . . 10  |-  (Scalar `  R )  =  (Scalar `  M )
2726eqcomi 2287 . . . . . . . . 9  |-  (Scalar `  M )  =  (Scalar `  R )
2827a1i 10 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  M )  =  (Scalar `  R )
)
29 fnmgp 15327 . . . . . . . . . 10  |- mulGrp  Fn  _V
30 elex 2796 . . . . . . . . . . 11  |-  ( R  e.  V  ->  R  e.  _V )
3130adantr 451 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  e.  _V )
32 fcoconst 5695 . . . . . . . . . 10  |-  ( (mulGrp 
Fn  _V  /\  R  e. 
_V )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  {
(mulGrp `  R ) } ) )
3329, 31, 32sylancr 644 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  { (mulGrp `  R ) } ) )
3419sneqi 3652 . . . . . . . . . 10  |-  { M }  =  { (mulGrp `  R ) }
3534xpeq2i 4710 . . . . . . . . 9  |-  ( I  X.  { M }
)  =  ( I  X.  { (mulGrp `  R ) } )
3633, 35syl6reqr 2334 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { M } )  =  (mulGrp 
o.  ( I  X.  { R } ) ) )
3728, 36oveq12d 5876 . . . . . . 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 2315 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )
3918, 38syl5eq 2327 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Z  =  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )
4039fveq2d 5529 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  Z
)  =  ( Base `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
4110, 17, 403eqtr4d 2325 . . 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 2340 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  C )
459simprd 449 . . . 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 5529 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  N
)  =  ( +g  `  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
4739fveq2d 5529 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  Z
)  =  ( +g  `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
4845, 46, 473eqtr4d 2325 . . 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 2340 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  .+  =  .+b  )
5244, 51jca 518 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( B  =  C  /\  .+  =  .+b  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687    o. ccom 4693    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   X_scprds 13346    ^s cpws 13347  mulGrpcmgp 15325
This theorem is referenced by:  pwsco1rhm  15510  pwsco2rhm  15511  pwsdiagrhm  15578  evl1expd  19421
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-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
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-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-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-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-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-mgp 15326
  Copyright terms: Public domain W3C validator