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Theorem fta1blem 22994
Description: Lemma for fta1b 22995. (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 18860 . . . . . . 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 18867 . . . . . 6  |-  ( ph  ->  ( ( M  .x.  X )  e.  B  /\  ( ( O `  ( M  .x.  X ) ) `  N )  =  ( M  .X.  N ) ) )
1312simprd 464 . . . . 5  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  N )  =  ( M  .X.  N ) )
14 fta1blem.4 . . . . 5  |-  ( ph  ->  ( M  .X.  N
)  =  W )
1513, 14eqtrd 2470 . . . 4  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  N )  =  W )
16 eqid 2429 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
17 eqid 2429 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
18 fvex 5891 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
194, 18eqeltri 2513 . . . . . . . 8  |-  K  e. 
_V
2019a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
212, 3, 16, 4evl1rhm 18855 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
226, 21syl 17 . . . . . . . . 9  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
235, 17rhmf 17889 . . . . . . . . 9  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
2422, 23syl 17 . . . . . . . 8  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
2512simpld 460 . . . . . . . 8  |-  ( ph  ->  ( M  .x.  X
)  e.  B )
2624, 25ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  e.  ( Base `  ( R  ^s  K ) ) )
2716, 4, 17, 6, 20, 26pwselbas 15346 . . . . . 6  |-  ( ph  ->  ( O `  ( M  .x.  X ) ) : K --> K )
28 ffn 5746 . . . . . 6  |-  ( ( O `  ( M 
.x.  X ) ) : K --> K  -> 
( O `  ( M  .x.  X ) )  Fn  K )
2927, 28syl 17 . . . . 5  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  Fn  K )
30 fniniseg 6018 . . . . 5  |-  ( ( O `  ( M 
.x.  X ) )  Fn  K  ->  ( N  e.  ( `' ( O `  ( M 
.x.  X ) )
" { W }
)  <->  ( N  e.  K  /\  ( ( O `  ( M 
.x.  X ) ) `
 N )  =  W ) ) )
3129, 30syl 17 . . . 4  |-  ( ph  ->  ( N  e.  ( `' ( O `  ( M  .x.  X ) ) " { W } )  <->  ( N  e.  K  /\  (
( O `  ( M  .x.  X ) ) `
 N )  =  W ) ) )
321, 15, 31mpbir2and 930 . . 3  |-  ( ph  ->  N  e.  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )
33 fvex 5891 . . . . . . . 8  |-  ( O `
 ( M  .x.  X ) )  e. 
_V
3433cnvex 6754 . . . . . . 7  |-  `' ( O `  ( M 
.x.  X ) )  e.  _V
35 imaexg 6744 . . . . . . 7  |-  ( `' ( O `  ( M  .x.  X ) )  e.  _V  ->  ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  _V )
3634, 35ax-mp 5 . . . . . 6  |-  ( `' ( O `  ( M  .x.  X ) )
" { W }
)  e.  _V
3736a1i 11 . . . . 5  |-  ( ph  ->  ( `' ( O `
 ( M  .x.  X ) ) " { W } )  e. 
_V )
38 1nn0 10885 . . . . . 6  |-  1  e.  NN0
3938a1i 11 . . . . 5  |-  ( ph  ->  1  e.  NN0 )
40 crngring 17726 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  R  e.  Ring )
416, 40syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  Ring )
427, 3, 5vr1cl 18745 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  X  e.  B )
4341, 42syl 17 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  B )
44 eqid 2429 . . . . . . . . . . . . 13  |-  (mulGrp `  P )  =  (mulGrp `  P )
4544, 5mgpbas 17664 . . . . . . . . . . . 12  |-  B  =  ( Base `  (mulGrp `  P ) )
46 eqid 2429 . . . . . . . . . . . 12  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4745, 46mulg1 16716 . . . . . . . . . . 11  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4843, 47syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4948oveq2d 6321 . . . . . . . . 9  |-  ( ph  ->  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) )  =  ( M 
.x.  X ) )
50 fta1blem.5 . . . . . . . . . . 11  |-  ( ph  ->  M  =/=  W )
51 fta1b.w . . . . . . . . . . . . 13  |-  W  =  ( 0g `  R
)
5251, 4, 3, 7, 10, 44, 46coe1tmfv1 18802 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  M  e.  K  /\  1  e.  NN0 )  ->  (
(coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  M )
5341, 9, 39, 52syl3anc 1264 . . . . . . . . . . 11  |-  ( ph  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  M )
54 fta1b.z . . . . . . . . . . . . . . 15  |-  .0.  =  ( 0g `  P )
553, 54, 51coe1z 18791 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  (coe1 `  .0.  )  =  ( NN0  X. 
{ W } ) )
5641, 55syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  (coe1 `  .0.  )  =  ( NN0  X.  { W } ) )
5756fveq1d 5883 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  ( ( NN0  X.  { W } ) `  1
) )
58 fvex 5891 . . . . . . . . . . . . . . 15  |-  ( 0g
`  R )  e. 
_V
5951, 58eqeltri 2513 . . . . . . . . . . . . . 14  |-  W  e. 
_V
6059fvconst2 6135 . . . . . . . . . . . . 13  |-  ( 1  e.  NN0  ->  ( ( NN0  X.  { W } ) `  1
)  =  W )
6138, 60ax-mp 5 . . . . . . . . . . . 12  |-  ( ( NN0  X.  { W } ) `  1
)  =  W
6257, 61syl6eq 2486 . . . . . . . . . . 11  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  W )
6350, 53, 623netr4d 2736 . . . . . . . . . 10  |-  ( ph  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =/=  (
(coe1 `  .0.  ) ` 
1 ) )
64 fveq2 5881 . . . . . . . . . . . 12  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  (coe1 `  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) ) )  =  (coe1 `  .0.  ) )
6564fveq1d 5883 . . . . . . . . . . 11  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  .0.  ) ` 
1 ) )
6665necon3i 2671 . . . . . . . . . 10  |-  ( ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =/=  (
(coe1 `  .0.  ) ` 
1 )  ->  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) )  =/=  .0.  )
6763, 66syl 17 . . . . . . . . 9  |-  ( ph  ->  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) )  =/=  .0.  )
6849, 67eqnetrrd 2725 . . . . . . . 8  |-  ( ph  ->  ( M  .x.  X
)  =/=  .0.  )
69 eldifsn 4128 . . . . . . . 8  |-  ( ( M  .x.  X )  e.  ( B  \  {  .0.  } )  <->  ( ( M  .x.  X )  e.  B  /\  ( M 
.x.  X )  =/= 
.0.  ) )
7025, 68, 69sylanbrc 668 . . . . . . 7  |-  ( ph  ->  ( M  .x.  X
)  e.  ( B 
\  {  .0.  }
) )
71 fta1blem.6 . . . . . . 7  |-  ( ph  ->  ( ( M  .x.  X )  e.  ( B  \  {  .0.  } )  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
( D `  ( M  .x.  X ) ) ) )
7270, 71mpd 15 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
( D `  ( M  .x.  X ) ) )
7349fveq2d 5885 . . . . . . 7  |-  ( ph  ->  ( D `  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) ) )  =  ( D `
 ( M  .x.  X ) ) )
74 fta1b.d . . . . . . . . 9  |-  D  =  ( deg1  `  R )
7574, 4, 3, 7, 10, 44, 46, 51deg1tm 22944 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( M  e.  K  /\  M  =/=  W )  /\  1  e.  NN0 )  -> 
( D `  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) ) )  =  1 )
7641, 9, 50, 39, 75syl121anc 1269 . . . . . . 7  |-  ( ph  ->  ( D `  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) ) )  =  1 )
7773, 76eqtr3d 2472 . . . . . 6  |-  ( ph  ->  ( D `  ( M  .x.  X ) )  =  1 )
7872, 77breqtrd 4450 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
1 )
79 hashbnd 12518 . . . . 5  |-  ( ( ( `' ( O `
 ( M  .x.  X ) ) " { W } )  e. 
_V  /\  1  e.  NN0 
/\  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
1 )  ->  ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  Fin )
8037, 39, 78, 79syl3anc 1264 . . . 4  |-  ( ph  ->  ( `' ( O `
 ( M  .x.  X ) ) " { W } )  e. 
Fin )
814, 51ring0cl 17737 . . . . . . 7  |-  ( R  e.  Ring  ->  W  e.  K )
8241, 81syl 17 . . . . . 6  |-  ( ph  ->  W  e.  K )
83 eqid 2429 . . . . . . . . . . . . 13  |-  (algSc `  P )  =  (algSc `  P )
843, 83, 4, 5ply1sclf 18813 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  (algSc `  P ) : K --> B )
8541, 84syl 17 . . . . . . . . . . 11  |-  ( ph  ->  (algSc `  P ) : K --> B )
8685, 9ffvelrnd 6038 . . . . . . . . . 10  |-  ( ph  ->  ( (algSc `  P
) `  M )  e.  B )
87 eqid 2429 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( .r `  P
)
88 eqid 2429 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
895, 87, 88rhmmul 17890 . . . . . . . . . 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 ) ) )
9022, 86, 43, 89syl3anc 1264 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( (algSc `  P
) `  M )
( .r `  P
) X ) )  =  ( ( O `
 ( (algSc `  P ) `  M
) ) ( .r
`  ( R  ^s  K
) ) ( O `
 X ) ) )
913ply1assa 18727 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  P  e. AssAlg )
926, 91syl 17 . . . . . . . . . . 11  |-  ( ph  ->  P  e. AssAlg )
933ply1sca 18781 . . . . . . . . . . . . . . 15  |-  ( R  e.  CRing  ->  R  =  (Scalar `  P ) )
946, 93syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  R  =  (Scalar `  P ) )
9594fveq2d 5885 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
964, 95syl5eq 2482 . . . . . . . . . . . 12  |-  ( ph  ->  K  =  ( Base `  (Scalar `  P )
) )
979, 96eleqtrd 2519 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ( Base `  (Scalar `  P )
) )
98 eqid 2429 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
99 eqid 2429 . . . . . . . . . . . 12  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
10083, 98, 99, 5, 87, 10asclmul1 18498 . . . . . . . . . . 11  |-  ( ( P  e. AssAlg  /\  M  e.  ( Base `  (Scalar `  P ) )  /\  X  e.  B )  ->  ( ( (algSc `  P ) `  M
) ( .r `  P ) X )  =  ( M  .x.  X ) )
10192, 97, 43, 100syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  ( ( (algSc `  P ) `  M
) ( .r `  P ) X )  =  ( M  .x.  X ) )
102101fveq2d 5885 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( (algSc `  P
) `  M )
( .r `  P
) X ) )  =  ( O `  ( M  .x.  X ) ) )
10324, 86ffvelrnd 6038 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  (
(algSc `  P ) `  M ) )  e.  ( Base `  ( R  ^s  K ) ) )
10424, 43ffvelrnd 6038 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  X
)  e.  ( Base `  ( R  ^s  K ) ) )
10516, 17, 6, 20, 103, 104, 11, 88pwsmulrval 15348 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) ) ( .r `  ( R  ^s  K ) ) ( O `  X ) )  =  ( ( O `  ( (algSc `  P ) `  M
) )  oF 
.X.  ( O `  X ) ) )
1062, 3, 4, 83evl1sca 18857 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  M  e.  K )  ->  ( O `  ( (algSc `  P ) `  M
) )  =  ( K  X.  { M } ) )
1076, 9, 106syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  (
(algSc `  P ) `  M ) )  =  ( K  X.  { M } ) )
1082, 7, 4evl1var 18859 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( O `  X )  =  (  _I  |`  K )
)
1096, 108syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  X
)  =  (  _I  |`  K ) )
110107, 109oveq12d 6323 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) )  oF  .X.  ( O `  X ) )  =  ( ( K  X.  { M } )  oF  .X.  (  _I  |`  K ) ) )
111105, 110eqtrd 2470 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) ) ( .r `  ( R  ^s  K ) ) ( O `  X ) )  =  ( ( K  X.  { M } )  oF 
.X.  (  _I  |`  K ) ) )
11290, 102, 1113eqtr3d 2478 . . . . . . . 8  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  =  ( ( K  X.  { M }
)  oF  .X.  (  _I  |`  K ) ) )
113112fveq1d 5883 . . . . . . 7  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  ( ( ( K  X.  { M } )  oF 
.X.  (  _I  |`  K ) ) `  W ) )
114 fnconstg 5788 . . . . . . . . . 10  |-  ( M  e.  K  ->  ( K  X.  { M }
)  Fn  K )
1159, 114syl 17 . . . . . . . . 9  |-  ( ph  ->  ( K  X.  { M } )  Fn  K
)
116 fnresi 5711 . . . . . . . . . 10  |-  (  _I  |`  K )  Fn  K
117116a1i 11 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  K )  Fn  K )
118 fnfvof 6559 . . . . . . . . 9  |-  ( ( ( ( K  X.  { M } )  Fn  K  /\  (  _I  |`  K )  Fn  K
)  /\  ( K  e.  _V  /\  W  e.  K ) )  -> 
( ( ( K  X.  { M }
)  oF  .X.  (  _I  |`  K ) ) `  W )  =  ( ( ( K  X.  { M } ) `  W
)  .X.  ( (  _I  |`  K ) `  W ) ) )
119115, 117, 20, 82, 118syl22anc 1265 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  X.  { M }
)  oF  .X.  (  _I  |`  K ) ) `  W )  =  ( ( ( K  X.  { M } ) `  W
)  .X.  ( (  _I  |`  K ) `  W ) ) )
120 fvconst2g 6133 . . . . . . . . . . 11  |-  ( ( M  e.  K  /\  W  e.  K )  ->  ( ( K  X.  { M } ) `  W )  =  M )
1219, 82, 120syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( ( K  X.  { M } ) `  W )  =  M )
122 fvresi 6105 . . . . . . . . . . 11  |-  ( W  e.  K  ->  (
(  _I  |`  K ) `
 W )  =  W )
12382, 122syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( (  _I  |`  K ) `
 W )  =  W )
124121, 123oveq12d 6323 . . . . . . . . 9  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  ( M  .X.  W ) )
1254, 11, 51ringrz 17753 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  M  e.  K )  ->  ( M  .X.  W )  =  W )
12641, 9, 125syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( M  .X.  W
)  =  W )
127124, 126eqtrd 2470 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  W )
128119, 127eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( ( ( K  X.  { M }
)  oF  .X.  (  _I  |`  K ) ) `  W )  =  W )
129113, 128eqtrd 2470 . . . . . 6  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  W )
130 fniniseg 6018 . . . . . . 7  |-  ( ( O `  ( M 
.x.  X ) )  Fn  K  ->  ( W  e.  ( `' ( O `  ( M 
.x.  X ) )
" { W }
)  <->  ( W  e.  K  /\  ( ( O `  ( M 
.x.  X ) ) `
 W )  =  W ) ) )
13129, 130syl 17 . . . . . 6  |-  ( ph  ->  ( W  e.  ( `' ( O `  ( M  .x.  X ) ) " { W } )  <->  ( W  e.  K  /\  (
( O `  ( M  .x.  X ) ) `
 W )  =  W ) ) )
13282, 129, 131mpbir2and 930 . . . . 5  |-  ( ph  ->  W  e.  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )
133132snssd 4148 . . . 4  |-  ( ph  ->  { W }  C_  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
134 hashsng 12546 . . . . . . 7  |-  ( W  e.  K  ->  ( # `
 { W }
)  =  1 )
13582, 134syl 17 . . . . . 6  |-  ( ph  ->  ( # `  { W } )  =  1 )
136 ssdomg 7622 . . . . . . . . . 10  |-  ( ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  _V  ->  ( { W }  C_  ( `' ( O `
 ( M  .x.  X ) ) " { W } )  ->  { W }  ~<_  ( `' ( O `  ( M  .x.  X ) )
" { W }
) ) )
13736, 133, 136mpsyl 65 . . . . . . . . 9  |-  ( ph  ->  { W }  ~<_  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )
138 snfi 7657 . . . . . . . . . 10  |-  { W }  e.  Fin
139 hashdom 12555 . . . . . . . . . 10  |-  ( ( { W }  e.  Fin  /\  ( `' ( O `  ( M 
.x.  X ) )
" { W }
)  e.  _V )  ->  ( ( # `  { W } )  <_  ( # `
 ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~<_  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
140138, 36, 139mp2an 676 . . . . . . . . 9  |-  ( (
# `  { W } )  <_  ( # `
 ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~<_  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )
141137, 140sylibr 215 . . . . . . . 8  |-  ( ph  ->  ( # `  { W } )  <_  ( # `
 ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
142135, 141eqbrtrrd 4448 . . . . . . 7  |-  ( ph  ->  1  <_  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) ) )
143 hashcl 12535 . . . . . . . . . 10  |-  ( ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  Fin  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e. 
NN0 )
14480, 143syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e. 
NN0 )
145144nn0red 10926 . . . . . . . 8  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e.  RR )
146 1re 9641 . . . . . . . 8  |-  1  e.  RR
147 letri3 9718 . . . . . . . 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 }
) ) ) ) )
148145, 146, 147sylancl 666 . . . . . . 7  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  =  1  <->  ( ( # `  ( `' ( O `
 ( M  .x.  X ) ) " { W } ) )  <_  1  /\  1  <_  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) ) ) ) )
14978, 142, 148mpbir2and 930 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  =  1 )
150135, 149eqtr4d 2473 . . . . 5  |-  ( ph  ->  ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
151 hashen 12527 . . . . . 6  |-  ( ( { W }  e.  Fin  /\  ( `' ( O `  ( M 
.x.  X ) )
" { W }
)  e.  Fin )  ->  ( ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~~  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
152138, 80, 151sylancr 667 . . . . 5  |-  ( ph  ->  ( ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~~  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
153150, 152mpbid 213 . . . 4  |-  ( ph  ->  { W }  ~~  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
154 fisseneq 7789 . . . 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 }
) )
15580, 133, 153, 154syl3anc 1264 . . 3  |-  ( ph  ->  { W }  =  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
15632, 155eleqtrrd 2520 . 2  |-  ( ph  ->  N  e.  { W } )
157 elsni 4027 . 2  |-  ( N  e.  { W }  ->  N  =  W )
158156, 157syl 17 1  |-  ( ph  ->  N  =  W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   _Vcvv 3087    \ cdif 3439    C_ wss 3442   {csn 4002   class class class wbr 4426    _I cid 4764    X. cxp 4852   `'ccnv 4853    |` cres 4856   "cima 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543    ~~ cen 7574    ~<_ cdom 7575   Fincfn 7577   RRcr 9537   1c1 9539    <_ cle 9675   NN0cn0 10869   #chash 12512   Basecbs 15084   .rcmulr 15153  Scalarcsca 15155   .scvsca 15156   0gc0g 15297    ^s cpws 15304  .gcmg 16623  mulGrpcmgp 17658   Ringcrg 17715   CRingccrg 17716   RingHom crh 17875  AssAlgcasa 18468  algSccascl 18470  var1cv1 18704  Poly1cpl1 18705  coe1cco1 18706  eval1ce1 18838   deg1 cdg1 22880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-ofr 6546  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-0g 15299  df-gsum 15300  df-prds 15305  df-pws 15307  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-subg 16765  df-ghm 16832  df-cntz 16922  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-srg 17675  df-ring 17717  df-cring 17718  df-rnghom 17878  df-subrg 17941  df-lmod 18028  df-lss 18091  df-lsp 18130  df-assa 18471  df-asp 18472  df-ascl 18473  df-psr 18515  df-mvr 18516  df-mpl 18517  df-opsr 18519  df-evls 18664  df-evl 18665  df-psr1 18708  df-vr1 18709  df-ply1 18710  df-coe1 18711  df-evl1 18840  df-cnfld 18906  df-mdeg 22881  df-deg1 22882
This theorem is referenced by:  fta1b  22995
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