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Theorem pwsmgp 15726
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 449 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  I  e.  W )
5 fvex 5744 . . . . . . 7  |-  (Scalar `  R )  e.  _V
65a1i 11 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (Scalar `  R )  e.  _V )
7 fnconstg 5633 . . . . . . 7  |-  ( R  e.  V  ->  (
I  X.  { R } )  Fn  I
)
87adantr 453 . . . . . 6  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( I  X.  { R } )  Fn  I
)
91, 2, 3, 4, 6, 8prdsmgp 15718 . . . . 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 447 . . . 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 13710 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) )
1514fveq2d 5734 . . . . . 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 5734 . . . 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 5744 . . . . . . . . 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 13710 . . . . . . . 8  |-  ( ( M  e.  _V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2521, 4, 24sylancr 646 . . . . . . 7  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( M  ^s  I )  =  ( (Scalar `  M ) X_s ( I  X.  { M } ) ) )
2619, 13mgpsca 15657 . . . . . . . . . 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 15652 . . . . . . . . . 10  |- mulGrp  Fn  _V
30 elex 2966 . . . . . . . . . . 11  |-  ( R  e.  V  ->  R  e.  _V )
3130adantr 453 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  I  e.  W )  ->  R  e.  _V )
32 fcoconst 5907 . . . . . . . . . 10  |-  ( (mulGrp 
Fn  _V  /\  R  e. 
_V )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  {
(mulGrp `  R ) } ) )
3329, 31, 32sylancr 646 . . . . . . . . 9  |-  ( ( R  e.  V  /\  I  e.  W )  ->  (mulGrp  o.  ( I  X.  { R } ) )  =  ( I  X.  { (mulGrp `  R ) } ) )
3419sneqi 3828 . . . . . . . . . 10  |-  { M }  =  { (mulGrp `  R ) }
3534xpeq2i 4901 . . . . . . . . 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 6101 . . . . . . 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 5734 . . . 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 451 . . . 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 5734 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( +g  `  N
)  =  ( +g  `  (mulGrp `  ( (Scalar `  R ) X_s ( I  X.  { R } ) ) ) ) )
4739fveq2d 5734 . . . 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 520 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( B  =  C  /\  .+  =  .+b  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    X. cxp 4878    o. ccom 4884    Fn wfn 5451   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531  Scalarcsca 13534   X_scprds 13671    ^s cpws 13672  mulGrpcmgp 15650
This theorem is referenced by:  pwsco1rhm  15835  pwsco2rhm  15836  pwsdiagrhm  15903  evl1expd  19960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-hom 13555  df-cco 13556  df-prds 13673  df-pws 13675  df-mgp 15651
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