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Theorem fta1blem 20091
Description: Lemma for fta1b 20092. (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 19953 . . . . . . 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 19957 . . . . . 6  |-  ( ph  ->  ( ( M  .x.  X )  e.  B  /\  ( ( O `  ( M  .x.  X ) ) `  N )  =  ( M  .X.  N ) ) )
1312simprd 450 . . . . 5  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  N )  =  ( M  .X.  N ) )
14 fta1blem.4 . . . . 5  |-  ( ph  ->  ( M  .X.  N
)  =  W )
1513, 14eqtrd 2468 . . . 4  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  N )  =  W )
16 eqid 2436 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
17 eqid 2436 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
18 fvex 5742 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
194, 18eqeltri 2506 . . . . . . . 8  |-  K  e. 
_V
2019a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
212, 3, 16, 4evl1rhm 19949 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
226, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
235, 17rhmf 15827 . . . . . . . . 9  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
2422, 23syl 16 . . . . . . . 8  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
2512simpld 446 . . . . . . . 8  |-  ( ph  ->  ( M  .x.  X
)  e.  B )
2624, 25ffvelrnd 5871 . . . . . . 7  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  e.  ( Base `  ( R  ^s  K ) ) )
2716, 4, 17, 6, 20, 26pwselbas 13711 . . . . . 6  |-  ( ph  ->  ( O `  ( M  .x.  X ) ) : K --> K )
28 ffn 5591 . . . . . 6  |-  ( ( O `  ( M 
.x.  X ) ) : K --> K  -> 
( O `  ( M  .x.  X ) )  Fn  K )
2927, 28syl 16 . . . . 5  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  Fn  K )
30 fniniseg 5851 . . . . 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 16 . . . 4  |-  ( ph  ->  ( N  e.  ( `' ( O `  ( M  .x.  X ) ) " { W } )  <->  ( N  e.  K  /\  (
( O `  ( M  .x.  X ) ) `
 N )  =  W ) ) )
321, 15, 31mpbir2and 889 . . 3  |-  ( ph  ->  N  e.  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )
33 fvex 5742 . . . . . . . 8  |-  ( O `
 ( M  .x.  X ) )  e. 
_V
3433cnvex 5406 . . . . . . 7  |-  `' ( O `  ( M 
.x.  X ) )  e.  _V
35 imaexg 5217 . . . . . . 7  |-  ( `' ( O `  ( M  .x.  X ) )  e.  _V  ->  ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  _V )
3634, 35ax-mp 8 . . . . . 6  |-  ( `' ( O `  ( M  .x.  X ) )
" { W }
)  e.  _V
3736a1i 11 . . . . 5  |-  ( ph  ->  ( `' ( O `
 ( M  .x.  X ) ) " { W } )  e. 
_V )
38 1nn0 10237 . . . . . 6  |-  1  e.  NN0
3938a1i 11 . . . . 5  |-  ( ph  ->  1  e.  NN0 )
40 crngrng 15674 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  R  e.  Ring )
416, 40syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  Ring )
427, 3, 5vr1cl 16611 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  X  e.  B )
4341, 42syl 16 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  B )
44 eqid 2436 . . . . . . . . . . . . 13  |-  (mulGrp `  P )  =  (mulGrp `  P )
4544, 5mgpbas 15654 . . . . . . . . . . . 12  |-  B  =  ( Base `  (mulGrp `  P ) )
46 eqid 2436 . . . . . . . . . . . 12  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4745, 46mulg1 14897 . . . . . . . . . . 11  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4843, 47syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4948oveq2d 6097 . . . . . . . . 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 16666 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  M  e.  K  /\  1  e.  NN0 )  ->  (
(coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  M )
5341, 9, 39, 52syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  M )
54 fta1b.z . . . . . . . . . . . . . . 15  |-  .0.  =  ( 0g `  P )
553, 54, 51coe1z 16656 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  (coe1 `  .0.  )  =  ( NN0  X. 
{ W } ) )
5641, 55syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (coe1 `  .0.  )  =  ( NN0  X.  { W } ) )
5756fveq1d 5730 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  ( ( NN0  X.  { W } ) `  1
) )
58 fvex 5742 . . . . . . . . . . . . . . 15  |-  ( 0g
`  R )  e. 
_V
5951, 58eqeltri 2506 . . . . . . . . . . . . . 14  |-  W  e. 
_V
6059fvconst2 5947 . . . . . . . . . . . . 13  |-  ( 1  e.  NN0  ->  ( ( NN0  X.  { W } ) `  1
)  =  W )
6138, 60ax-mp 8 . . . . . . . . . . . 12  |-  ( ( NN0  X.  { W } ) `  1
)  =  W
6257, 61syl6eq 2484 . . . . . . . . . . 11  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  W )
6350, 53, 623netr4d 2628 . . . . . . . . . 10  |-  ( ph  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =/=  (
(coe1 `  .0.  ) ` 
1 ) )
64 fveq2 5728 . . . . . . . . . . . 12  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  (coe1 `  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) ) )  =  (coe1 `  .0.  ) )
6564fveq1d 5730 . . . . . . . . . . 11  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  .0.  ) ` 
1 ) )
6665necon3i 2643 . . . . . . . . . 10  |-  ( ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =/=  (
(coe1 `  .0.  ) ` 
1 )  ->  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) )  =/=  .0.  )
6763, 66syl 16 . . . . . . . . 9  |-  ( ph  ->  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) )  =/=  .0.  )
6849, 67eqnetrrd 2621 . . . . . . . 8  |-  ( ph  ->  ( M  .x.  X
)  =/=  .0.  )
69 eldifsn 3927 . . . . . . . 8  |-  ( ( M  .x.  X )  e.  ( B  \  {  .0.  } )  <->  ( ( M  .x.  X )  e.  B  /\  ( M 
.x.  X )  =/= 
.0.  ) )
7025, 68, 69sylanbrc 646 . . . . . . 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 5732 . . . . . . 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 20041 . . . . . . . 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 1189 . . . . . . 7  |-  ( ph  ->  ( D `  ( M  .x.  ( 1 (.g `  (mulGrp `  P )
) X ) ) )  =  1 )
7773, 76eqtr3d 2470 . . . . . 6  |-  ( ph  ->  ( D `  ( M  .x.  X ) )  =  1 )
7872, 77breqtrd 4236 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
1 )
79 hashbnd 11624 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( `' ( O `
 ( M  .x.  X ) ) " { W } )  e. 
Fin )
814, 51rng0cl 15685 . . . . . . 7  |-  ( R  e.  Ring  ->  W  e.  K )
8241, 81syl 16 . . . . . 6  |-  ( ph  ->  W  e.  K )
83 eqid 2436 . . . . . . . . . . . . 13  |-  (algSc `  P )  =  (algSc `  P )
843, 83, 4, 5ply1sclf 16677 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  (algSc `  P ) : K --> B )
8541, 84syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (algSc `  P ) : K --> B )
8685, 9ffvelrnd 5871 . . . . . . . . . 10  |-  ( ph  ->  ( (algSc `  P
) `  M )  e.  B )
87 eqid 2436 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( .r `  P
)
88 eqid 2436 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
895, 87, 88rhmmul 15828 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( (algSc `  P
) `  M )
( .r `  P
) X ) )  =  ( ( O `
 ( (algSc `  P ) `  M
) ) ( .r
`  ( R  ^s  K
) ) ( O `
 X ) ) )
913ply1assa 16597 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  P  e. AssAlg )
926, 91syl 16 . . . . . . . . . . 11  |-  ( ph  ->  P  e. AssAlg )
933ply1sca 16647 . . . . . . . . . . . . . . 15  |-  ( R  e.  CRing  ->  R  =  (Scalar `  P ) )
946, 93syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  R  =  (Scalar `  P ) )
9594fveq2d 5732 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
964, 95syl5eq 2480 . . . . . . . . . . . 12  |-  ( ph  ->  K  =  ( Base `  (Scalar `  P )
) )
979, 96eleqtrd 2512 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ( Base `  (Scalar `  P )
) )
98 eqid 2436 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
99 eqid 2436 . . . . . . . . . . . 12  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
10083, 98, 99, 5, 87, 10asclmul1 16398 . . . . . . . . . . 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 1184 . . . . . . . . . 10  |-  ( ph  ->  ( ( (algSc `  P ) `  M
) ( .r `  P ) X )  =  ( M  .x.  X ) )
102101fveq2d 5732 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( (algSc `  P
) `  M )
( .r `  P
) X ) )  =  ( O `  ( M  .x.  X ) ) )
10324, 86ffvelrnd 5871 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  (
(algSc `  P ) `  M ) )  e.  ( Base `  ( R  ^s  K ) ) )
10424, 43ffvelrnd 5871 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  X
)  e.  ( Base `  ( R  ^s  K ) ) )
10516, 17, 6, 20, 103, 104, 11, 88pwsmulrval 13713 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) ) ( .r `  ( R  ^s  K ) ) ( O `  X ) )  =  ( ( O `  ( (algSc `  P ) `  M
) )  o F 
.X.  ( O `  X ) ) )
1062, 3, 4, 83evl1sca 19950 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  M  e.  K )  ->  ( O `  ( (algSc `  P ) `  M
) )  =  ( K  X.  { M } ) )
1076, 9, 106syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  (
(algSc `  P ) `  M ) )  =  ( K  X.  { M } ) )
1082, 7, 4evl1var 19952 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( O `  X )  =  (  _I  |`  K )
)
1096, 108syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  X
)  =  (  _I  |`  K ) )
110107, 109oveq12d 6099 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) )  o F  .X.  ( O `  X ) )  =  ( ( K  X.  { M } )  o F  .X.  (  _I  |`  K ) ) )
111105, 110eqtrd 2468 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) ) ( .r `  ( R  ^s  K ) ) ( O `  X ) )  =  ( ( K  X.  { M } )  o F 
.X.  (  _I  |`  K ) ) )
11290, 102, 1113eqtr3d 2476 . . . . . . . 8  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  =  ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) )
113112fveq1d 5730 . . . . . . 7  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  ( ( ( K  X.  { M } )  o F 
.X.  (  _I  |`  K ) ) `  W ) )
114 fnconstg 5631 . . . . . . . . . 10  |-  ( M  e.  K  ->  ( K  X.  { M }
)  Fn  K )
1159, 114syl 16 . . . . . . . . 9  |-  ( ph  ->  ( K  X.  { M } )  Fn  K
)
116 fnresi 5562 . . . . . . . . . 10  |-  (  _I  |`  K )  Fn  K
117116a1i 11 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  K )  Fn  K )
118 fnfvof 6317 . . . . . . . . 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 ) ) )
119115, 117, 20, 82, 118syl22anc 1185 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) `  W )  =  ( ( ( K  X.  { M } ) `  W
)  .X.  ( (  _I  |`  K ) `  W ) ) )
120 fvconst2g 5945 . . . . . . . . . . 11  |-  ( ( M  e.  K  /\  W  e.  K )  ->  ( ( K  X.  { M } ) `  W )  =  M )
1219, 82, 120syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( K  X.  { M } ) `  W )  =  M )
122 fvresi 5924 . . . . . . . . . . 11  |-  ( W  e.  K  ->  (
(  _I  |`  K ) `
 W )  =  W )
12382, 122syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( (  _I  |`  K ) `
 W )  =  W )
124121, 123oveq12d 6099 . . . . . . . . 9  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  ( M  .X.  W ) )
1254, 11, 51rngrz 15701 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  M  e.  K )  ->  ( M  .X.  W )  =  W )
12641, 9, 125syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( M  .X.  W
)  =  W )
127124, 126eqtrd 2468 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  W )
128119, 127eqtrd 2468 . . . . . . 7  |-  ( ph  ->  ( ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) `  W )  =  W )
129113, 128eqtrd 2468 . . . . . 6  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  W )
130 fniniseg 5851 . . . . . . 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 16 . . . . . 6  |-  ( ph  ->  ( W  e.  ( `' ( O `  ( M  .x.  X ) ) " { W } )  <->  ( W  e.  K  /\  (
( O `  ( M  .x.  X ) ) `
 W )  =  W ) ) )
13282, 129, 131mpbir2and 889 . . . . 5  |-  ( ph  ->  W  e.  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )
133132snssd 3943 . . . 4  |-  ( ph  ->  { W }  C_  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
134 hashsng 11647 . . . . . . 7  |-  ( W  e.  K  ->  ( # `
 { W }
)  =  1 )
13582, 134syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { W } )  =  1 )
136 ssdomg 7153 . . . . . . . . . 10  |-  ( ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  _V  ->  ( { W }  C_  ( `' ( O `
 ( M  .x.  X ) ) " { W } )  ->  { W }  ~<_  ( `' ( O `  ( M  .x.  X ) )
" { W }
) ) )
13736, 133, 136mpsyl 61 . . . . . . . . 9  |-  ( ph  ->  { W }  ~<_  ( `' ( O `  ( M  .x.  X ) )
" { W }
) )
138 snfi 7187 . . . . . . . . . 10  |-  { W }  e.  Fin
139 hashdom 11653 . . . . . . . . . 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 654 . . . . . . . . 9  |-  ( (
# `  { W } )  <_  ( # `
 ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~<_  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )
141137, 140sylibr 204 . . . . . . . 8  |-  ( ph  ->  ( # `  { W } )  <_  ( # `
 ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
142135, 141eqbrtrrd 4234 . . . . . . 7  |-  ( ph  ->  1  <_  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) ) )
143 hashcl 11639 . . . . . . . . . 10  |-  ( ( `' ( O `  ( M  .x.  X ) ) " { W } )  e.  Fin  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e. 
NN0 )
14480, 143syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e. 
NN0 )
145144nn0red 10275 . . . . . . . 8  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e.  RR )
146 1re 9090 . . . . . . . 8  |-  1  e.  RR
147 letri3 9160 . . . . . . . 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 644 . . . . . . 7  |-  ( ph  ->  ( ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  =  1  <->  ( ( # `  ( `' ( O `
 ( M  .x.  X ) ) " { W } ) )  <_  1  /\  1  <_  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) ) ) ) )
14978, 142, 148mpbir2and 889 . . . . . 6  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  =  1 )
150135, 149eqtr4d 2471 . . . . 5  |-  ( ph  ->  ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
151 hashen 11631 . . . . . 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 645 . . . . 5  |-  ( ph  ->  ( ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) )  <->  { W }  ~~  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
153150, 152mpbid 202 . . . 4  |-  ( ph  ->  { W }  ~~  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
154 fisseneq 7320 . . . 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 1184 . . 3  |-  ( ph  ->  { W }  =  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
15632, 155eleqtrrd 2513 . 2  |-  ( ph  ->  N  e.  { W } )
157 elsni 3838 . 2  |-  ( N  e.  { W }  ->  N  =  W )
158156, 157syl 16 1  |-  ( ph  ->  N  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    \ cdif 3317    C_ wss 3320   {csn 3814   class class class wbr 4212    _I cid 4493    X. cxp 4876   `'ccnv 4877    |` cres 4880   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303    ~~ cen 7106    ~<_ cdom 7107   Fincfn 7109   RRcr 8989   1c1 8991    <_ cle 9121   NN0cn0 10221   #chash 11618   Basecbs 13469   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533    ^s cpws 13670   0gc0g 13723  .gcmg 14689  mulGrpcmgp 15648   Ringcrg 15660   CRingccrg 15661   RingHom crh 15817  AssAlgcasa 16369  algSccascl 16371  var1cv1 16570  Poly1cpl1 16571  eval1ce1 16573  coe1cco1 16574   deg1 cdg1 19977
This theorem is referenced by:  fta1b  20092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-ofr 6306  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-prds 13671  df-pws 13673  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-rnghom 15819  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-assa 16372  df-asp 16373  df-ascl 16374  df-psr 16417  df-mvr 16418  df-mpl 16419  df-evls 16420  df-evl 16421  df-opsr 16425  df-psr1 16576  df-vr1 16577  df-ply1 16578  df-evl1 16580  df-coe1 16581  df-cnfld 16704  df-mdeg 19978  df-deg1 19979
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