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

Theorem pwsmgp 15679
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 2404 . . . . . 6  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
2 eqid 2404 . . . . . 6  |-  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  (mulGrp `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
3 eqid 2404 . . . . . 6  |-  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) )  =  ( (Scalar `  R ) X_s (mulGrp 
o.  ( I  X.  { R } ) ) )
4 simpr 448 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  I  e.  W )
5 fvex 5701 . . . . . . 7  |-  (Scalar `  R )  e.  _V
65a1i 11 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  R )  e.  _V )
7 fnconstg 5590 . . . . . . 7  |-  ( R  e.  V  ->  (
I  X.  { R } )  Fn  I
)
87adantr 452 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { R } )  Fn  I
)
91, 2, 3, 4, 6, 8prdsmgp 15671 . . . . 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 446 . . . 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 2404 . . . . . . . 8  |-  (Scalar `  R )  =  (Scalar `  R )
1412, 13pwsval 13663 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
1514fveq2d 5691 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (mulGrp `  Y )  =  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
1611, 15syl5eq 2448 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  N  =  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
1716fveq2d 5691 . . . 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 5701 . . . . . . . . 9  |-  (mulGrp `  R )  e.  _V
2119, 20eqeltri 2474 . . . . . . . 8  |-  M  e. 
_V
22 eqid 2404 . . . . . . . . 9  |-  ( M  ^s  I )  =  ( M  ^s  I )
23 eqid 2404 . . . . . . . . 9  |-  (Scalar `  M )  =  (Scalar `  M )
2422, 23pwsval 13663 . . . . . . . 8  |-  ( ( M  e.  _V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2521, 4, 24sylancr 645 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2619, 13mgpsca 15610 . . . . . . . . . 10  |-  (Scalar `  R )  =  (Scalar `  M )
2726eqcomi 2408 . . . . . . . . 9  |-  (Scalar `  M )  =  (Scalar `  R )
2827a1i 11 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  M )  =  (Scalar `  R )
)
29 fnmgp 15605 . . . . . . . . . 10  |- mulGrp  Fn  _V
30 elex 2924 . . . . . . . . . . 11  |-  ( R  e.  V  ->  R  e.  _V )
3130adantr 452 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  e.  _V )
32 fcoconst 5864 . . . . . . . . . 10  |-  ( (mulGrp 
Fn  _V  /\  R  e. 
_V )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  {
(mulGrp `  R ) } ) )
3329, 31, 32sylancr 645 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  { (mulGrp `  R ) } ) )
3419sneqi 3786 . . . . . . . . . 10  |-  { M }  =  { (mulGrp `  R ) }
3534xpeq2i 4858 . . . . . . . . 9  |-  ( I  X.  { M }
)  =  ( I  X.  { (mulGrp `  R ) } )
3633, 35syl6reqr 2455 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { M } )  =  (mulGrp 
o.  ( I  X.  { R } ) ) )
3728, 36oveq12d 6058 . . . . . . 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 2436 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )
3918, 38syl5eq 2448 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Z  =  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )
4039fveq2d 5691 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  Z
)  =  ( Base `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
4110, 17, 403eqtr4d 2446 . . 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 2461 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  C )
459simprd 450 . . . 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 5691 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  N
)  =  ( +g  `  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
4739fveq2d 5691 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  Z
)  =  ( +g  `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
4845, 46, 473eqtr4d 2446 . . 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 2461 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  .+  =  .+b  )
5244, 51jca 519 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( B  =  C  /\  .+  =  .+b  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774    X. cxp 4835    o. ccom 4841    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   X_scprds 13624    ^s cpws 13625  mulGrpcmgp 15603
This theorem is referenced by:  pwsco1rhm  15788  pwsco2rhm  15789  pwsdiagrhm  15856  evl1expd  19911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-mgp 15604
  Copyright terms: Public domain W3C validator