MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fta1blem Unicode version

Theorem fta1blem 20044
Description: Lemma for fta1b 20045. (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 19906 . . . . . . 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 19910 . . . . . 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 2436 . . . 4  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  N )  =  W )
16 eqid 2404 . . . . . . 7  |-  ( R  ^s  K )  =  ( R  ^s  K )
17 eqid 2404 . . . . . . 7  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
18 fvex 5701 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
194, 18eqeltri 2474 . . . . . . . 8  |-  K  e. 
_V
2019a1i 11 . . . . . . 7  |-  ( ph  ->  K  e.  _V )
212, 3, 16, 4evl1rhm 19902 . . . . . . . . . 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 15782 . . . . . . . . 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 5830 . . . . . . 7  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  e.  ( Base `  ( R  ^s  K ) ) )
2716, 4, 17, 6, 20, 26pwselbas 13666 . . . . . 6  |-  ( ph  ->  ( O `  ( M  .x.  X ) ) : K --> K )
28 ffn 5550 . . . . . 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 5810 . . . . 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 5701 . . . . . . . 8  |-  ( O `
 ( M  .x.  X ) )  e. 
_V
3433cnvex 5365 . . . . . . 7  |-  `' ( O `  ( M 
.x.  X ) )  e.  _V
35 imaexg 5176 . . . . . . 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 10193 . . . . . 6  |-  1  e.  NN0
3938a1i 11 . . . . 5  |-  ( ph  ->  1  e.  NN0 )
40 crngrng 15629 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  R  e.  Ring )
416, 40syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  R  e.  Ring )
427, 3, 5vr1cl 16566 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  X  e.  B )
4341, 42syl 16 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  B )
44 eqid 2404 . . . . . . . . . . . . 13  |-  (mulGrp `  P )  =  (mulGrp `  P )
4544, 5mgpbas 15609 . . . . . . . . . . . 12  |-  B  =  ( Base `  (mulGrp `  P ) )
46 eqid 2404 . . . . . . . . . . . 12  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
4745, 46mulg1 14852 . . . . . . . . . . 11  |-  ( X  e.  B  ->  (
1 (.g `  (mulGrp `  P
) ) X )  =  X )
4843, 47syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( 1 (.g `  (mulGrp `  P ) ) X )  =  X )
4948oveq2d 6056 . . . . . . . . 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 16621 . . . . . . . . . . . 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 16611 . . . . . . . . . . . . . 14  |-  ( R  e.  Ring  ->  (coe1 `  .0.  )  =  ( NN0  X. 
{ W } ) )
5641, 55syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  (coe1 `  .0.  )  =  ( NN0  X.  { W } ) )
5756fveq1d 5689 . . . . . . . . . . . 12  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  ( ( NN0  X.  { W } ) `  1
) )
58 fvex 5701 . . . . . . . . . . . . . . 15  |-  ( 0g
`  R )  e. 
_V
5951, 58eqeltri 2474 . . . . . . . . . . . . . 14  |-  W  e. 
_V
6059fvconst2 5906 . . . . . . . . . . . . 13  |-  ( 1  e.  NN0  ->  ( ( NN0  X.  { W } ) `  1
)  =  W )
6138, 60ax-mp 8 . . . . . . . . . . . 12  |-  ( ( NN0  X.  { W } ) `  1
)  =  W
6257, 61syl6eq 2452 . . . . . . . . . . 11  |-  ( ph  ->  ( (coe1 `  .0.  ) ` 
1 )  =  W )
6350, 53, 623netr4d 2594 . . . . . . . . . 10  |-  ( ph  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =/=  (
(coe1 `  .0.  ) ` 
1 ) )
64 fveq2 5687 . . . . . . . . . . . 12  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  (coe1 `  ( M  .x.  (
1 (.g `  (mulGrp `  P
) ) X ) ) )  =  (coe1 `  .0.  ) )
6564fveq1d 5689 . . . . . . . . . . 11  |-  ( ( M  .x.  ( 1 (.g `  (mulGrp `  P
) ) X ) )  =  .0.  ->  ( (coe1 `  ( M  .x.  ( 1 (.g `  (mulGrp `  P ) ) X ) ) ) ` 
1 )  =  ( (coe1 `  .0.  ) ` 
1 ) )
6665necon3i 2606 . . . . . . . . . 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 2587 . . . . . . . 8  |-  ( ph  ->  ( M  .x.  X
)  =/=  .0.  )
69 eldifsn 3887 . . . . . . . 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 5691 . . . . . . 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 19994 . . . . . . . 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 2438 . . . . . 6  |-  ( ph  ->  ( D `  ( M  .x.  X ) )  =  1 )
7872, 77breqtrd 4196 . . . . 5  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  <_ 
1 )
79 hashbnd 11579 . . . . 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 15640 . . . . . . 7  |-  ( R  e.  Ring  ->  W  e.  K )
8241, 81syl 16 . . . . . 6  |-  ( ph  ->  W  e.  K )
83 eqid 2404 . . . . . . . . . . . . 13  |-  (algSc `  P )  =  (algSc `  P )
843, 83, 4, 5ply1sclf 16632 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  (algSc `  P ) : K --> B )
8541, 84syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (algSc `  P ) : K --> B )
8685, 9ffvelrnd 5830 . . . . . . . . . 10  |-  ( ph  ->  ( (algSc `  P
) `  M )  e.  B )
87 eqid 2404 . . . . . . . . . . 11  |-  ( .r
`  P )  =  ( .r `  P
)
88 eqid 2404 . . . . . . . . . . 11  |-  ( .r
`  ( R  ^s  K
) )  =  ( .r `  ( R  ^s  K ) )
895, 87, 88rhmmul 15783 . . . . . . . . . 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 16552 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  P  e. AssAlg )
926, 91syl 16 . . . . . . . . . . 11  |-  ( ph  ->  P  e. AssAlg )
933ply1sca 16602 . . . . . . . . . . . . . . 15  |-  ( R  e.  CRing  ->  R  =  (Scalar `  P ) )
946, 93syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  R  =  (Scalar `  P ) )
9594fveq2d 5691 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
964, 95syl5eq 2448 . . . . . . . . . . . 12  |-  ( ph  ->  K  =  ( Base `  (Scalar `  P )
) )
979, 96eleqtrd 2480 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  ( Base `  (Scalar `  P )
) )
98 eqid 2404 . . . . . . . . . . . 12  |-  (Scalar `  P )  =  (Scalar `  P )
99 eqid 2404 . . . . . . . . . . . 12  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
10083, 98, 99, 5, 87, 10asclmul1 16353 . . . . . . . . . . 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 5691 . . . . . . . . 9  |-  ( ph  ->  ( O `  (
( (algSc `  P
) `  M )
( .r `  P
) X ) )  =  ( O `  ( M  .x.  X ) ) )
10324, 86ffvelrnd 5830 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  (
(algSc `  P ) `  M ) )  e.  ( Base `  ( R  ^s  K ) ) )
10424, 43ffvelrnd 5830 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  X
)  e.  ( Base `  ( R  ^s  K ) ) )
10516, 17, 6, 20, 103, 104, 11, 88pwsmulrval 13668 . . . . . . . . . 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 19903 . . . . . . . . . . . 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 19905 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( O `  X )  =  (  _I  |`  K )
)
1096, 108syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  X
)  =  (  _I  |`  K ) )
110107, 109oveq12d 6058 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) )  o F  .X.  ( O `  X ) )  =  ( ( K  X.  { M } )  o F  .X.  (  _I  |`  K ) ) )
111105, 110eqtrd 2436 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  ( (algSc `  P ) `  M ) ) ( .r `  ( R  ^s  K ) ) ( O `  X ) )  =  ( ( K  X.  { M } )  o F 
.X.  (  _I  |`  K ) ) )
11290, 102, 1113eqtr3d 2444 . . . . . . . 8  |-  ( ph  ->  ( O `  ( M  .x.  X ) )  =  ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) )
113112fveq1d 5689 . . . . . . 7  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  ( ( ( K  X.  { M } )  o F 
.X.  (  _I  |`  K ) ) `  W ) )
114 fnconstg 5590 . . . . . . . . . 10  |-  ( M  e.  K  ->  ( K  X.  { M }
)  Fn  K )
1159, 114syl 16 . . . . . . . . 9  |-  ( ph  ->  ( K  X.  { M } )  Fn  K
)
116 fnresi 5521 . . . . . . . . . 10  |-  (  _I  |`  K )  Fn  K
117116a1i 11 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  K )  Fn  K )
118 fnfvof 6276 . . . . . . . . 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 5904 . . . . . . . . . . 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 5883 . . . . . . . . . . 11  |-  ( W  e.  K  ->  (
(  _I  |`  K ) `
 W )  =  W )
12382, 122syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( (  _I  |`  K ) `
 W )  =  W )
124121, 123oveq12d 6058 . . . . . . . . 9  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  ( M  .X.  W ) )
1254, 11, 51rngrz 15656 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  M  e.  K )  ->  ( M  .X.  W )  =  W )
12641, 9, 125syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( M  .X.  W
)  =  W )
127124, 126eqtrd 2436 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  X.  { M }
) `  W )  .X.  ( (  _I  |`  K ) `
 W ) )  =  W )
128119, 127eqtrd 2436 . . . . . . 7  |-  ( ph  ->  ( ( ( K  X.  { M }
)  o F  .X.  (  _I  |`  K ) ) `  W )  =  W )
129113, 128eqtrd 2436 . . . . . 6  |-  ( ph  ->  ( ( O `  ( M  .x.  X ) ) `  W )  =  W )
130 fniniseg 5810 . . . . . . 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 3903 . . . 4  |-  ( ph  ->  { W }  C_  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )
134 hashsng 11602 . . . . . . 7  |-  ( W  e.  K  ->  ( # `
 { W }
)  =  1 )
13582, 134syl 16 . . . . . 6  |-  ( ph  ->  ( # `  { W } )  =  1 )
136 ssdomg 7112 . . . . . . . . . 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 7146 . . . . . . . . . 10  |-  { W }  e.  Fin
139 hashdom 11608 . . . . . . . . . 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 4194 . . . . . . 7  |-  ( ph  ->  1  <_  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) ) )
143 hashcl 11594 . . . . . . . . . 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 10231 . . . . . . . 8  |-  ( ph  ->  ( # `  ( `' ( O `  ( M  .x.  X ) ) " { W } ) )  e.  RR )
146 1re 9046 . . . . . . . 8  |-  1  e.  RR
147 letri3 9116 . . . . . . . 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 2439 . . . . 5  |-  ( ph  ->  ( # `  { W } )  =  (
# `  ( `' ( O `  ( M 
.x.  X ) )
" { W }
) ) )
151 hashen 11586 . . . . . 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 7279 . . . 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 2481 . 2  |-  ( ph  ->  N  e.  { W } )
157 elsni 3798 . 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 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774   class class class wbr 4172    _I cid 4453    X. cxp 4835   `'ccnv 4836    |` cres 4839   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262    ~~ cen 7065    ~<_ cdom 7066   Fincfn 7068   RRcr 8945   1c1 8947    <_ cle 9077   NN0cn0 10177   #chash 11573   Basecbs 13424   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488    ^s cpws 13625   0gc0g 13678  .gcmg 14644  mulGrpcmgp 15603   Ringcrg 15615   CRingccrg 15616   RingHom crh 15772  AssAlgcasa 16324  algSccascl 16326  var1cv1 16525  Poly1cpl1 16526  eval1ce1 16528  coe1cco1 16529   deg1 cdg1 19930
This theorem is referenced by:  fta1b  20045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-rnghom 15774  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-assa 16327  df-asp 16328  df-ascl 16329  df-psr 16372  df-mvr 16373  df-mpl 16374  df-evls 16375  df-evl 16376  df-opsr 16380  df-psr1 16531  df-vr1 16532  df-ply1 16533  df-evl1 16535  df-coe1 16536  df-cnfld 16659  df-mdeg 19931  df-deg1 19932
  Copyright terms: Public domain W3C validator