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Theorem fta1blem 19658
Description: Lemma for fta1b 19659. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
fta1b.p  |-  P  =  (Poly1 `  R )
fta1b.b  |-  B  =  ( Base `  P
)
fta1b.d  |-  D  =  ( deg1  `  R )
fta1b.o  |-  O  =  (eval1 `  R )
fta1b.w  |-  W  =  ( 0g `  R
)
fta1b.z  |-  .0.  =  ( 0g `  P )
fta1blem.k  |-  K  =  ( Base `  R
)
fta1blem.t  |-  .X.  =  ( .r `  R )
fta1blem.x  |-  X  =  (var1 `  R )
fta1blem.s  |-  .x.  =  ( .s `  P )
fta1blem.1  |-  ( ph  ->  R  e.  CRing )
fta1blem.2  |-  ( ph  ->  M  e.  K )
fta1blem.3  |-  ( ph  ->  N  e.  K )
fta1blem.4  |-  ( ph  ->  ( M  .X.  N
)  =  W )
fta1blem.5  |-  ( ph  ->  M  =/=  W )
fta1blem.6  |-  ( ph  ->  ( ( M  .x.  X )  e.  ( B  \  {  .0.  } )  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
( D `  ( M  .x.  X ) ) ) )
Assertion
Ref Expression
fta1blem  |-  ( ph  ->  N  =  W )

Proof of Theorem fta1blem
StepHypRef Expression
1 fta1blem.3 . . . 4  |-  ( ph  ->  N  e.  K )
2 fta1b.o . . . . . . 7  |-  O  =  (eval1 `  R )
3 fta1b.p . . . . . . 7  |-  P  =  (Poly1 `  R )
4 fta1blem.k . . . . . . 7  |-  K  =  ( Base `  R
)
5 fta1b.b . . . . . . 7  |-  B  =  ( Base `  P
)
6 fta1blem.1 . . . . . . 7  |-  ( ph  ->  R  e.  CRing )
7 fta1blem.x . . . . . . . 8  |-  X  =  (var1 `  R )
82, 7, 4, 3, 5, 6, 1evl1vard 19520 . . . . . . 7  |-  ( ph  ->  ( X  e.  B  /\  ( ( O `  X ) `  N
)  =  N ) )
9 fta1blem.2 . . . . . . 7  |-  ( ph  ->  M  e.  K )
10 fta1blem.s . . . . . . 7  |-  .x.  =  ( .s `  P )
11 fta1blem.t . . . . . . 7  |-  .X.  =  ( .r `  R )
122, 3, 4, 5, 6, 1, 8, 9, 10, 11evl1vsd 19524 . . . . . 6  |-  ( ph  ->  ( ( M  .x.  X )  e.  B  /\  ( ( O `  ( M  .x.  X ) ) `  N )  =  ( M  .X.  N ) ) )
1312simprd 449 . . . . 5  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  N )  =  ( M  .X.  N ) )
14 fta1blem.4 . . . . 5  |-  ( ph  ->  ( M  .X.  N
)  =  W )
1513, 14eqtrd 2390 . . . 4  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  N )  =  W )
16 eqid 2358 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
17 eqid 2358 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
18 fvex 5622 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
194, 18eqeltri 2428 . . . . . . . 8  |-  K  e. 
_V
2019a1i 10 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
212, 3, 16, 4evl1rhm 19516 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
226, 21syl 15 . . . . . . . . 9  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
235, 17rhmf 15603 . . . . . . . . 9  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
2422, 23syl 15 . . . . . . . 8  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
2512simpld 445 . . . . . . . 8  |-  ( ph  ->  ( M  .x.  X
)  e.  B )
26 ffvelrn 5746 . . . . . . . 8  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  ( M 
.x.  X )  e.  B )  ->  ( O `  ( M  .x.  X ) )  e.  ( Base `  ( R  ^s  K ) ) )
2724, 25, 26syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  e.  ( Base `  ( R  ^s  K ) ) )
2816, 4, 17, 6, 20, 27pwselbas 13487 . . . . . 6  |-  ( ph  ->  ( O `  ( M  .x.  X ) ) : K --> K )
29 ffn 5472 . . . . . 6  |-  ( ( O `  ( M 
.x.  X ) ) : K --> K  -> 
( O `  ( M  .x.  X ) )  Fn  K )
3028, 29syl 15 . . . . 5  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  Fn  K )
31 fniniseg 5729 . . . . 5  |-  ( ( O `  ( M 
.x.  X ) )  Fn  K  ->  ( N  e.  ( `' ( O `  ( M 
.x.  X ) )
" { W }
)  <->  ( N  e.  K  /\  ( ( O `  ( M 
.x.  X ) ) `
 N )  =  W ) ) )
3230, 31syl 15 . . . 4  |-  ( ph  ->  ( N  e.  ( `' ( O `  ( M  .x.  X ) ) " { W } )  <->  ( N  e.  K  /\  (
( O `  ( M  .x.  X ) ) `
 N )  =  W ) ) )
331, 15, 32mpbir2and 888 . . 3  |-  ( ph  ->  N  e.  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )
34 fvex 5622 . . . . . . . 8  |-  ( O `
 ( M  .x.  X ) )  e. 
_V
3534cnvex 5291 . . . . . . 7  |-  `' ( O `  ( M 
.x.  X ) )  e.  _V
36 imaexg 5108 . . . . . . 7  |-  ( `' ( O `  ( M  .x.  X ) )  e.  _V  ->  ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  _V )
3735, 36ax-mp 8 . . . . . 6  |-  ( `' ( O `  ( M  .x.  X ) )
" { W }
)  e.  _V
3837a1i 10 . . . . 5  |-  ( ph  ->  ( `' ( O `
 ( M  .x.  X ) ) " { W } )  e. 
_V )
39 1nn0 10073 . . . . . 6  |-  1  e.  NN0
4039a1i 10 . . . . 5  |-  ( ph  ->  1  e.  NN0 )
41 crngrng 15450 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  R  e.  Ring )
426, 41syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  Ring )
437, 3, 5vr1cl 16393 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  X  e.  B )
4442, 43syl 15 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  B )
45 eqid 2358 . . . . . . . . . . . . 13  |-  (mulGrp `  P )  =  (mulGrp `  P )
4645, 5mgpbas 15430 . . . . . . . . . . . 12  |-  B  =  ( Base `  (mulGrp `  P ) )
47 eqid 2358 . . . . . . . . . . . 12  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4846, 47mulg1 14673 . . . . . . . . . . 11  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4944, 48syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
5049oveq2d 5961 . . . . . . . . 9  |-  ( ph  ->  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  ( M 
.x.  X ) )
51 fta1blem.5 . . . . . . . . . . 11  |-  ( ph  ->  M  =/=  W )
52 fta1b.w . . . . . . . . . . . . 13  |-  W  =  ( 0g `  R
)
5352, 4, 3, 7, 10, 45, 47coe1tmfv1 16449 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  M  e.  K  /\  1  e.  NN0 )  ->  (
(coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  M )
5442, 9, 40, 53syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  M )
55 fta1b.z . . . . . . . . . . . . . . 15  |-  .0.  =  ( 0g `  P )
563, 55, 52coe1z 16439 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  (coe1 `  .0.  )  =  ( NN0  X. 
{ W } ) )
5742, 56syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  (coe1 `  .0.  )  =  ( NN0  X.  { W } ) )
5857fveq1d 5610 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  ( ( NN0  X.  { W } ) `  1
) )
59 fvex 5622 . . . . . . . . . . . . . . 15  |-  ( 0g
`  R )  e. 
_V
6052, 59eqeltri 2428 . . . . . . . . . . . . . 14  |-  W  e. 
_V
6160fvconst2 5813 . . . . . . . . . . . . 13  |-  ( 1  e.  NN0  ->  ( ( NN0  X.  { W } ) `  1
)  =  W )
6239, 61ax-mp 8 . . . . . . . . . . . 12  |-  ( ( NN0  X.  { W } ) `  1
)  =  W
6358, 62syl6eq 2406 . . . . . . . . . . 11  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  W )
6451, 54, 633netr4d 2548 . . . . . . . . . 10  |-  ( ph  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =/=  (
(coe1 `  .0.  ) ` 
1 ) )
65 fveq2 5608 . . . . . . . . . . . 12  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  (coe1 `  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) ) )  =  (coe1 `  .0.  ) )
6665fveq1d 5610 . . . . . . . . . . 11  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  .0.  ) ` 
1 ) )
6766necon3i 2560 . . . . . . . . . 10  |-  ( ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =/=  (
(coe1 `  .0.  ) ` 
1 )  ->  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) )  =/=  .0.  )
6864, 67syl 15 . . . . . . . . 9  |-  ( ph  ->  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) )  =/=  .0.  )
6950, 68eqnetrrd 2541 . . . . . . . 8  |-  ( ph  ->  ( M  .x.  X
)  =/=  .0.  )
70 eldifsn 3825 . . . . . . . 8  |-  ( ( M  .x.  X )  e.  ( B  \  {  .0.  } )  <->  ( ( M  .x.  X )  e.  B  /\  ( M 
.x.  X )  =/= 
.0.  ) )
7125, 69, 70sylanbrc 645 . . . . . . 7  |-  ( ph  ->  ( M  .x.  X
)  e.  ( B 
\  {  .0.  }
) )
72 fta1blem.6 . . . . . . 7  |-  ( ph  ->  ( ( M  .x.  X )  e.  ( B  \  {  .0.  } )  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
( D `  ( M  .x.  X ) ) ) )
7371, 72mpd 14 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
( D `  ( M  .x.  X ) ) )
7450fveq2d 5612 . . . . . . 7  |-  ( ph  ->  ( D `  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) ) )  =  ( D `
 ( M  .x.  X ) ) )
75 fta1b.d . . . . . . . . 9  |-  D  =  ( deg1  `  R )
7675, 4, 3, 7, 10, 45, 47, 52deg1tm 19608 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( M  e.  K  /\  M  =/=  W )  /\  1  e.  NN0 )  -> 
( D `  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) ) )  =  1 )
7742, 9, 51, 40, 76syl121anc 1187 . . . . . . 7  |-  ( ph  ->  ( D `  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) ) )  =  1 )
7874, 77eqtr3d 2392 . . . . . 6  |-  ( ph  ->  ( D `  ( M  .x.  X ) )  =  1 )
7973, 78breqtrd 4128 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
1 )
80 hashbnd 11436 . . . . 5  |-  ( ( ( `' ( O `
 ( M  .x.  X ) ) " { W } )  e. 
_V  /\  1  e.  NN0 
/\  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
1 )  ->  ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  Fin )
8138, 40, 79, 80syl3anc 1182 . . . 4  |-  ( ph  ->  ( `' ( O `
 ( M  .x.  X ) ) " { W } )  e. 
Fin )
824, 52rng0cl 15461 . . . . . . 7  |-  ( R  e.  Ring  ->  W  e.  K )
8342, 82syl 15 . . . . . 6  |-  ( ph  ->  W  e.  K )
84 eqid 2358 . . . . . . . . . . . . 13  |-  (algSc `  P )  =  (algSc `  P )
853, 84, 4, 5ply1sclf 16460 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  (algSc `  P ) : K --> B )
8642, 85syl 15 . . . . . . . . . . 11  |-  ( ph  ->  (algSc `  P ) : K --> B )
87 ffvelrn 5746 . . . . . . . . . . 11  |-  ( ( (algSc `  P ) : K --> B  /\  M  e.  K )  ->  (
(algSc `  P ) `  M )  e.  B
)
8886, 9, 87syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( (algSc `  P
) `  M )  e.  B )
89 eqid 2358 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( .r `  P
)
90 eqid 2358 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
915, 89, 90rhmmul 15604 . . . . . . . . . 10  |-  ( ( O  e.  ( P RingHom 
( R  ^s  K ) )  /\  ( (algSc `  P ) `  M
)  e.  B  /\  X  e.  B )  ->  ( O `  (
( (algSc `  P
) `  M )
( .r `  P
) X ) )  =  ( ( O `
 ( (algSc `  P ) `  M
) ) ( .r
`  ( R  ^s  K
) ) ( O `
 X ) ) )
9222, 88, 44, 91syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( (algSc `  P
) `  M )
( .r `  P
) X ) )  =  ( ( O `
 ( (algSc `  P ) `  M
) ) ( .r
`  ( R  ^s  K
) ) ( O `
 X ) ) )
933ply1assa 16377 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  P  e. AssAlg )
946, 93syl 15 . . . . . . . . . . 11  |-  ( ph  ->  P  e. AssAlg )
953ply1sca 16430 . . . . . . . . . . . . . . 15  |-  ( R  e.  CRing  ->  R  =  (Scalar `  P ) )
966, 95syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  R  =  (Scalar `  P ) )
9796fveq2d 5612 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
984, 97syl5eq 2402 . . . . . . . . . . . 12  |-  ( ph  ->  K  =  ( Base `  (Scalar `  P )
) )
999, 98eleqtrd 2434 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ( Base `  (Scalar `  P )
) )
100 eqid 2358 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
101 eqid 2358 . . . . . . . . . . . 12  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
10284, 100, 101, 5, 89, 10asclmul1 16178 . . . . . . . . . . 11  |-  ( ( P  e. AssAlg  /\  M  e.  ( Base `  (Scalar `  P ) )  /\  X  e.  B )  ->  ( ( (algSc `  P ) `  M
) ( .r `  P ) X )  =  ( M  .x.  X ) )
10394, 99, 44, 102syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( ( (algSc `  P ) `  M
) ( .r `  P ) X )  =  ( M  .x.  X ) )
104103fveq2d 5612 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( (algSc `  P
) `  M )
( .r `  P
) X ) )  =  ( O `  ( M  .x.  X ) ) )
105 ffvelrn 5746 . . . . . . . . . . . 12  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  ( (algSc `  P ) `  M
)  e.  B )  ->  ( O `  ( (algSc `  P ) `  M ) )  e.  ( Base `  ( R  ^s  K ) ) )
10624, 88, 105syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  (
(algSc `  P ) `  M ) )  e.  ( Base `  ( R  ^s  K ) ) )
107 ffvelrn 5746 . . . . . . . . . . . 12  |-  ( ( O : B --> ( Base `  ( R  ^s  K ) )  /\  X  e.  B )  ->  ( O `  X )  e.  ( Base `  ( R  ^s  K ) ) )
10824, 44, 107syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  X
)  e.  ( Base `  ( R  ^s  K ) ) )
10916, 17, 6, 20, 106, 108, 11, 90pwsmulrval 13489 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) ) ( .r `  ( R  ^s  K ) ) ( O `  X ) )  =  ( ( O `  ( (algSc `  P ) `  M
) )  o F 
.X.  ( O `  X ) ) )
1102, 3, 4, 84evl1sca 19517 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  M  e.  K )  ->  ( O `  ( (algSc `  P ) `  M
) )  =  ( K  X.  { M } ) )
1116, 9, 110syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  (
(algSc `  P ) `  M ) )  =  ( K  X.  { M } ) )
1122, 7, 4evl1var 19519 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( O `  X )  =  (  _I  |`  K )
)
1136, 112syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  X
)  =  (  _I  |`  K ) )
114111, 113oveq12d 5963 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) )  o F  .X.  ( O `  X ) )  =  ( ( K  X.  { M } )  o F  .X.  (  _I  |`  K ) ) )
115109, 114eqtrd 2390 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) ) ( .r `  ( R  ^s  K ) ) ( O `  X ) )  =  ( ( K  X.  { M } )  o F 
.X.  (  _I  |`  K ) ) )
11692, 104, 1153eqtr3d 2398 . . . . . . . 8  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  =  ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) )
117116fveq1d 5610 . . . . . . 7  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  ( ( ( K  X.  { M } )  o F 
.X.  (  _I  |`  K ) ) `  W ) )
118 fnconstg 5512 . . . . . . . . . 10  |-  ( M  e.  K  ->  ( K  X.  { M }
)  Fn  K )
1199, 118syl 15 . . . . . . . . 9  |-  ( ph  ->  ( K  X.  { M } )  Fn  K
)
120 fnresi 5443 . . . . . . . . . 10  |-  (  _I  |`  K )  Fn  K
121120a1i 10 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  K )  Fn  K )
122 fnfvof 6177 . . . . . . . . 9  |-  ( ( ( ( K  X.  { M } )  Fn  K  /\  (  _I  |`  K )  Fn  K
)  /\  ( K  e.  _V  /\  W  e.  K ) )  -> 
( ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) `  W )  =  ( ( ( K  X.  { M } ) `  W
)  .X.  ( (  _I  |`  K ) `  W ) ) )
123119, 121, 20, 83, 122syl22anc 1183 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) `  W )  =  ( ( ( K  X.  { M } ) `  W
)  .X.  ( (  _I  |`  K ) `  W ) ) )
124 fvconst2g 5811 . . . . . . . . . . 11  |-  ( ( M  e.  K  /\  W  e.  K )  ->  ( ( K  X.  { M } ) `  W )  =  M )
1259, 83, 124syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( K  X.  { M } ) `  W )  =  M )
126 fvresi 5795 . . . . . . . . . . 11  |-  ( W  e.  K  ->  (
(  _I  |`  K ) `
 W )  =  W )
12783, 126syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( (  _I  |`  K ) `
 W )  =  W )
128125, 127oveq12d 5963 . . . . . . . . 9  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  ( M  .X.  W ) )
1294, 11, 52rngrz 15477 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  M  e.  K )  ->  ( M  .X.  W )  =  W )
13042, 9, 129syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( M  .X.  W
)  =  W )
131128, 130eqtrd 2390 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  W )
132123, 131eqtrd 2390 . . . . . . 7  |-  ( ph  ->  ( ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) `  W )  =  W )
133117, 132eqtrd 2390 . . . . . 6  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  W )
134 fniniseg 5729 . . . . . . 7  |-  ( ( O `  ( M 
.x.  X ) )  Fn  K  ->  ( W  e.  ( `' ( O `  ( M 
.x.  X ) )
" { W }
)  <->  ( W  e.  K  /\  ( ( O `  ( M 
.x.  X ) ) `
 W )  =  W ) ) )
13530, 134syl 15 . . . . . 6  |-  ( ph  ->  ( W  e.  ( `' ( O `  ( M  .x.  X ) ) " { W } )  <->  ( W  e.  K  /\  (
( O `  ( M  .x.  X ) ) `
 W )  =  W ) ) )
13683, 133, 135mpbir2and 888 . . . . 5  |-  ( ph  ->  W  e.  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )
137136snssd 3839 . . . 4  |-  ( ph  ->  { W }  C_  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
138 hashsng 11449 . . . . . . 7  |-  ( W  e.  K  ->  ( # `
 { W }
)  =  1 )
13983, 138syl 15 . . . . . 6  |-  ( ph  ->  ( # `  { W } )  =  1 )
140 ssdomg 6995 . . . . . . . . . 10  |-  ( ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  _V  ->  ( { W }  C_  ( `' ( O `
 ( M  .x.  X ) ) " { W } )  ->  { W }  ~<_  ( `' ( O `  ( M  .x.  X ) )
" { W }
) ) )
14137, 137, 140mpsyl 59 . . . . . . . . 9  |-  ( ph  ->  { W }  ~<_  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )
142 snfi 7029 . . . . . . . . . 10  |-  { W }  e.  Fin
143 hashdom 11454 . . . . . . . . . 10  |-  ( ( { W }  e.  Fin  /\  ( `' ( O `  ( M 
.x.  X ) )
" { W }
)  e.  _V )  ->  ( ( # `  { W } )  <_  ( # `
 ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~<_  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
144142, 37, 143mp2an 653 . . . . . . . . 9  |-  ( (
# `  { W } )  <_  ( # `
 ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~<_  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )
145141, 144sylibr 203 . . . . . . . 8  |-  ( ph  ->  ( # `  { W } )  <_  ( # `
 ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
146139, 145eqbrtrrd 4126 . . . . . . 7  |-  ( ph  ->  1  <_  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) ) )
147 hashcl 11443 . . . . . . . . . 10  |-  ( ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  Fin  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e. 
NN0 )
14881, 147syl 15 . . . . . . . . 9  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e. 
NN0 )
149148nn0red 10111 . . . . . . . 8  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e.  RR )
150 1re 8927 . . . . . . . 8  |-  1  e.  RR
151 letri3 8997 . . . . . . . 8  |-  ( ( ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e.  RR  /\  1  e.  RR )  ->  (
( # `  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )  =  1  <-> 
( ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
1  /\  1  <_  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) ) ) )
152149, 150, 151sylancl 643 . . . . . . 7  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  =  1  <->  ( ( # `  ( `' ( O `
 ( M  .x.  X ) ) " { W } ) )  <_  1  /\  1  <_  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) ) ) ) )
15379, 146, 152mpbir2and 888 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  =  1 )
154139, 153eqtr4d 2393 . . . . 5  |-  ( ph  ->  ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
155 hashen 11439 . . . . . 6  |-  ( ( { W }  e.  Fin  /\  ( `' ( O `  ( M 
.x.  X ) )
" { W }
)  e.  Fin )  ->  ( ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~~  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
156142, 81, 155sylancr 644 . . . . 5  |-  ( ph  ->  ( ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~~  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
157154, 156mpbid 201 . . . 4  |-  ( ph  ->  { W }  ~~  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
158 fisseneq 7162 . . . 4  |-  ( ( ( `' ( O `
 ( M  .x.  X ) ) " { W } )  e. 
Fin  /\  { W }  C_  ( `' ( O `  ( M 
.x.  X ) )
" { W }
)  /\  { W }  ~~  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  ->  { W }  =  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )
15981, 137, 157, 158syl3anc 1182 . . 3  |-  ( ph  ->  { W }  =  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
16033, 159eleqtrrd 2435 . 2  |-  ( ph  ->  N  e.  { W } )
161 elsni 3740 . 2  |-  ( N  e.  { W }  ->  N  =  W )
162160, 161syl 15 1  |-  ( ph  ->  N  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864    \ cdif 3225    C_ wss 3228   {csn 3716   class class class wbr 4104    _I cid 4386    X. cxp 4769   `'ccnv 4770    |` cres 4773   "cima 4774    Fn wfn 5332   -->wf 5333   ` cfv 5337  (class class class)co 5945    o Fcof 6163    ~~ cen 6948    ~<_ cdom 6949   Fincfn 6951   RRcr 8826   1c1 8828    <_ cle 8958   NN0cn0 10057   #chash 11430   Basecbs 13245   .rcmulr 13306  Scalarcsca 13308   .scvsca 13309    ^s cpws 13446   0gc0g 13499  .gcmg 14465  mulGrpcmgp 15424   Ringcrg 15436   CRingccrg 15437   RingHom crh 15593  AssAlgcasa 16149  algSccascl 16151  var1cv1 16350  Poly1cpl1 16351  eval1ce1 16353  coe1cco1 16354   deg1 cdg1 19544
This theorem is referenced by:  fta1b  19659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-ofr 6166  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-fz 10875  df-fzo 10963  df-seq 11139  df-hash 11431  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-prds 13447  df-pws 13449  df-0g 13503  df-gsum 13504  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-mhm 14514  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-subg 14717  df-ghm 14780  df-cntz 14892  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-cring 15440  df-ur 15441  df-rnghom 15595  df-subrg 15642  df-lmod 15728  df-lss 15789  df-lsp 15828  df-assa 16152  df-asp 16153  df-ascl 16154  df-psr 16197  df-mvr 16198  df-mpl 16199  df-evls 16200  df-evl 16201  df-opsr 16205  df-psr1 16356  df-vr1 16357  df-ply1 16358  df-evl1 16360  df-coe1 16361  df-cnfld 16483  df-mdeg 19545  df-deg1 19546
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