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Theorem pwsmgp 17465
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 2454 . . . . . 6  |-  ( (Scalar `  R ) X_s ( I  X.  { R } ) )  =  ( (Scalar `  R
) X_s ( I  X.  { R } ) )
2 eqid 2454 . . . . . 6  |-  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )  =  (mulGrp `  (
(Scalar `  R ) X_s ( I  X.  { R } ) ) )
3 eqid 2454 . . . . . 6  |-  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) )  =  ( (Scalar `  R ) X_s (mulGrp 
o.  ( I  X.  { R } ) ) )
4 simpr 459 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  I  e.  W )
5 fvex 5858 . . . . . . 7  |-  (Scalar `  R )  e.  _V
65a1i 11 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  R )  e.  _V )
7 fnconstg 5755 . . . . . . 7  |-  ( R  e.  V  ->  (
I  X.  { R } )  Fn  I
)
87adantr 463 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { R } )  Fn  I
)
91, 2, 3, 4, 6, 8prdsmgp 17457 . . . . 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 457 . . . 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 2454 . . . . . . . 8  |-  (Scalar `  R )  =  (Scalar `  R )
1412, 13pwsval 14978 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
1514fveq2d 5852 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (mulGrp `  Y )  =  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
1611, 15syl5eq 2507 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  N  =  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) )
1716fveq2d 5852 . . . 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 5858 . . . . . . . . 9  |-  (mulGrp `  R )  e.  _V
2119, 20eqeltri 2538 . . . . . . . 8  |-  M  e. 
_V
22 eqid 2454 . . . . . . . . 9  |-  ( M  ^s  I )  =  ( M  ^s  I )
23 eqid 2454 . . . . . . . . 9  |-  (Scalar `  M )  =  (Scalar `  M )
2422, 23pwsval 14978 . . . . . . . 8  |-  ( ( M  e.  _V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2521, 4, 24sylancr 661 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2619, 13mgpsca 17346 . . . . . . . . . 10  |-  (Scalar `  R )  =  (Scalar `  M )
2726eqcomi 2467 . . . . . . . . 9  |-  (Scalar `  M )  =  (Scalar `  R )
2827a1i 11 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  M )  =  (Scalar `  R )
)
29 fnmgp 17341 . . . . . . . . . 10  |- mulGrp  Fn  _V
30 elex 3115 . . . . . . . . . . 11  |-  ( R  e.  V  ->  R  e.  _V )
3130adantr 463 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  e.  _V )
32 fcoconst 6044 . . . . . . . . . 10  |-  ( (mulGrp 
Fn  _V  /\  R  e. 
_V )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  {
(mulGrp `  R ) } ) )
3329, 31, 32sylancr 661 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  { (mulGrp `  R ) } ) )
3419sneqi 4027 . . . . . . . . . 10  |-  { M }  =  { (mulGrp `  R ) }
3534xpeq2i 5009 . . . . . . . . 9  |-  ( I  X.  { M }
)  =  ( I  X.  { (mulGrp `  R ) } )
3633, 35syl6reqr 2514 . . . . . . . 8  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { M } )  =  (mulGrp 
o.  ( I  X.  { R } ) ) )
3728, 36oveq12d 6288 . . . . . . 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 2495 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )
3918, 38syl5eq 2507 . . . . 5  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Z  =  ( (Scalar `  R ) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) )
4039fveq2d 5852 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( Base `  Z
)  =  ( Base `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
4110, 17, 403eqtr4d 2505 . . 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 2520 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  B  =  C )
459simprd 461 . . . 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 5852 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  N
)  =  ( +g  `  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
4739fveq2d 5852 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  Z
)  =  ( +g  `  ( (Scalar `  R
) X_s (mulGrp  o.  ( I  X.  { R } ) ) ) ) )
4845, 46, 473eqtr4d 2505 . . 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 2520 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  .+  =  .+b  )
5244, 51jca 530 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( B  =  C  /\  .+  =  .+b  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016    X. cxp 4986    o. ccom 4992    Fn wfn 5565   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787  Scalarcsca 14790   X_scprds 14938    ^s cpws 14939  mulGrpcmgp 17339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-plusg 14800  df-mulr 14801  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-hom 14811  df-cco 14812  df-prds 14940  df-pws 14942  df-mgp 17340
This theorem is referenced by:  pwsco1rhm  17585  pwsco2rhm  17586  pwsdiagrhm  17660  evl1expd  18579
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