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Theorem fta1blem 19554
Description: Lemma for fta1b 19555. (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 19416 . . . . . . 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 19420 . . . . . 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 2315 . . . 4  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  N )  =  W )
16 eqid 2283 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
17 eqid 2283 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
18 fvex 5539 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
194, 18eqeltri 2353 . . . . . . . 8  |-  K  e. 
_V
2019a1i 10 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
212, 3, 16, 4evl1rhm 19412 . . . . . . . . . 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 15504 . . . . . . . . 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 5663 . . . . . . . 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 13388 . . . . . 6  |-  ( ph  ->  ( O `  ( M  .x.  X ) ) : K --> K )
29 ffn 5389 . . . . . 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 5646 . . . . 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 5539 . . . . . . . 8  |-  ( O `
 ( M  .x.  X ) )  e. 
_V
3534cnvex 5209 . . . . . . 7  |-  `' ( O `  ( M 
.x.  X ) )  e.  _V
36 imaexg 5026 . . . . . . 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 9981 . . . . . 6  |-  1  e.  NN0
4039a1i 10 . . . . 5  |-  ( ph  ->  1  e.  NN0 )
41 crngrng 15351 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  R  e.  Ring )
426, 41syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  Ring )
437, 3, 5vr1cl 16294 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  X  e.  B )
4442, 43syl 15 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  B )
45 eqid 2283 . . . . . . . . . . . . 13  |-  (mulGrp `  P )  =  (mulGrp `  P )
4645, 5mgpbas 15331 . . . . . . . . . . . 12  |-  B  =  ( Base `  (mulGrp `  P ) )
47 eqid 2283 . . . . . . . . . . . 12  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4846, 47mulg1 14574 . . . . . . . . . . 11  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4944, 48syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
5049oveq2d 5874 . . . . . . . . 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 16350 . . . . . . . . . . . 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 16340 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  (coe1 `  .0.  )  =  ( NN0  X. 
{ W } ) )
5742, 56syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  (coe1 `  .0.  )  =  ( NN0  X.  { W } ) )
5857fveq1d 5527 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  ( ( NN0  X.  { W } ) `  1
) )
59 fvex 5539 . . . . . . . . . . . . . . 15  |-  ( 0g
`  R )  e. 
_V
6052, 59eqeltri 2353 . . . . . . . . . . . . . 14  |-  W  e. 
_V
6160fvconst2 5729 . . . . . . . . . . . . 13  |-  ( 1  e.  NN0  ->  ( ( NN0  X.  { W } ) `  1
)  =  W )
6239, 61ax-mp 8 . . . . . . . . . . . 12  |-  ( ( NN0  X.  { W } ) `  1
)  =  W
6358, 62syl6eq 2331 . . . . . . . . . . 11  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  W )
6451, 54, 633netr4d 2473 . . . . . . . . . 10  |-  ( ph  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =/=  (
(coe1 `  .0.  ) ` 
1 ) )
65 fveq2 5525 . . . . . . . . . . . 12  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  (coe1 `  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) ) )  =  (coe1 `  .0.  ) )
6665fveq1d 5527 . . . . . . . . . . 11  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  .0.  ) ` 
1 ) )
6766necon3i 2485 . . . . . . . . . 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 2466 . . . . . . . 8  |-  ( ph  ->  ( M  .x.  X
)  =/=  .0.  )
70 eldifsn 3749 . . . . . . . 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 5529 . . . . . . 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 19504 . . . . . . . 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 2317 . . . . . 6  |-  ( ph  ->  ( D `  ( M  .x.  X ) )  =  1 )
7973, 78breqtrd 4047 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
1 )
80 hashbnd 11343 . . . . 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 15362 . . . . . . 7  |-  ( R  e.  Ring  ->  W  e.  K )
8342, 82syl 15 . . . . . 6  |-  ( ph  ->  W  e.  K )
84 eqid 2283 . . . . . . . . . . . . 13  |-  (algSc `  P )  =  (algSc `  P )
853, 84, 4, 5ply1sclf 16361 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  (algSc `  P ) : K --> B )
8642, 85syl 15 . . . . . . . . . . 11  |-  ( ph  ->  (algSc `  P ) : K --> B )
87 ffvelrn 5663 . . . . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( .r `  P
)
90 eqid 2283 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
915, 89, 90rhmmul 15505 . . . . . . . . . 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 16278 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  P  e. AssAlg )
946, 93syl 15 . . . . . . . . . . 11  |-  ( ph  ->  P  e. AssAlg )
953ply1sca 16331 . . . . . . . . . . . . . . 15  |-  ( R  e.  CRing  ->  R  =  (Scalar `  P ) )
966, 95syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  R  =  (Scalar `  P ) )
9796fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
984, 97syl5eq 2327 . . . . . . . . . . . 12  |-  ( ph  ->  K  =  ( Base `  (Scalar `  P )
) )
999, 98eleqtrd 2359 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ( Base `  (Scalar `  P )
) )
100 eqid 2283 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
101 eqid 2283 . . . . . . . . . . . 12  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
10284, 100, 101, 5, 89, 10asclmul1 16079 . . . . . . . . . . 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 5529 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( (algSc `  P
) `  M )
( .r `  P
) X ) )  =  ( O `  ( M  .x.  X ) ) )
105 ffvelrn 5663 . . . . . . . . . . . 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 5663 . . . . . . . . . . . 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 13390 . . . . . . . . . 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 19413 . . . . . . . . . . . 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 19415 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( O `  X )  =  (  _I  |`  K )
)
1136, 112syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  X
)  =  (  _I  |`  K ) )
114111, 113oveq12d 5876 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) )  o F  .X.  ( O `  X ) )  =  ( ( K  X.  { M } )  o F  .X.  (  _I  |`  K ) ) )
115109, 114eqtrd 2315 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) ) ( .r `  ( R  ^s  K ) ) ( O `  X ) )  =  ( ( K  X.  { M } )  o F 
.X.  (  _I  |`  K ) ) )
11692, 104, 1153eqtr3d 2323 . . . . . . . 8  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  =  ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) )
117116fveq1d 5527 . . . . . . 7  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  ( ( ( K  X.  { M } )  o F 
.X.  (  _I  |`  K ) ) `  W ) )
118 fnconstg 5429 . . . . . . . . . 10  |-  ( M  e.  K  ->  ( K  X.  { M }
)  Fn  K )
1199, 118syl 15 . . . . . . . . 9  |-  ( ph  ->  ( K  X.  { M } )  Fn  K
)
120 fnresi 5361 . . . . . . . . . 10  |-  (  _I  |`  K )  Fn  K
121120a1i 10 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  K )  Fn  K )
122 fnfvof 6090 . . . . . . . . 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 5727 . . . . . . . . . . 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 5711 . . . . . . . . . . 11  |-  ( W  e.  K  ->  (
(  _I  |`  K ) `
 W )  =  W )
12783, 126syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( (  _I  |`  K ) `
 W )  =  W )
128125, 127oveq12d 5876 . . . . . . . . 9  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  ( M  .X.  W ) )
1294, 11, 52rngrz 15378 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  M  e.  K )  ->  ( M  .X.  W )  =  W )
13042, 9, 129syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( M  .X.  W
)  =  W )
131128, 130eqtrd 2315 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  W )
132123, 131eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) `  W )  =  W )
133117, 132eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  W )
134 fniniseg 5646 . . . . . . 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 3760 . . . 4  |-  ( ph  ->  { W }  C_  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
138 hashsng 11356 . . . . . . 7  |-  ( W  e.  K  ->  ( # `
 { W }
)  =  1 )
13983, 138syl 15 . . . . . 6  |-  ( ph  ->  ( # `  { W } )  =  1 )
140 ssdomg 6907 . . . . . . . . . 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 6941 . . . . . . . . . 10  |-  { W }  e.  Fin
143 hashdom 11361 . . . . . . . . . 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 4045 . . . . . . 7  |-  ( ph  ->  1  <_  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) ) )
147 hashcl 11350 . . . . . . . . . 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 10019 . . . . . . . 8  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e.  RR )
150 1re 8837 . . . . . . . 8  |-  1  e.  RR
151 letri3 8907 . . . . . . . 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 2318 . . . . 5  |-  ( ph  ->  ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
155 hashen 11346 . . . . . 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 7074 . . . 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 2360 . 2  |-  ( ph  ->  N  e.  { W } )
161 elsni 3664 . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   class class class wbr 4023    _I cid 4304    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    ~~ cen 6860    ~<_ cdom 6861   Fincfn 6863   RRcr 8736   1c1 8738    <_ cle 8868   NN0cn0 9965   #chash 11337   Basecbs 13148   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212    ^s cpws 13347   0gc0g 13400  .gcmg 14366  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338   RingHom crh 15494  AssAlgcasa 16050  algSccascl 16052  var1cv1 16251  Poly1cpl1 16252  eval1ce1 16254  coe1cco1 16255   deg1 cdg1 19440
This theorem is referenced by:  fta1b  19555
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-inf2 7342  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  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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-rmo 2551  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-iin 3908  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  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-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  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-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-evls 16101  df-evl 16102  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-evl1 16261  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442
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