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Theorem cply1coe0bi 30978
Description: A polynomial is constant (i.e. a "lifted scalar") iff all but the first coefficient are zero. (Contributed by AV, 16-Nov-2019.)
Hypotheses
Ref Expression
cply1coe0.k  |-  K  =  ( Base `  R
)
cply1coe0.0  |-  .0.  =  ( 0g `  R )
cply1coe0.p  |-  P  =  (Poly1 `  R )
cply1coe0.b  |-  B  =  ( Base `  P
)
cply1coe0.a  |-  A  =  (algSc `  P )
Assertion
Ref Expression
cply1coe0bi  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( E. s  e.  K  M  =  ( A `  s )  <->  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  ) )
Distinct variable groups:    n, K    R, n    A, n, s    B, n, s    K, s    n, M, s    R, s    .0. , s
Allowed substitution hints:    P( n, s)    .0. ( n)

Proof of Theorem cply1coe0bi
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Ring )
21anim1i 568 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  s  e.  K
)  ->  ( R  e.  Ring  /\  s  e.  K ) )
32adantr 465 . . . . . 6  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  s  e.  K )  /\  M  =  ( A `  s ) )  -> 
( R  e.  Ring  /\  s  e.  K ) )
4 cply1coe0.k . . . . . . 7  |-  K  =  ( Base `  R
)
5 cply1coe0.0 . . . . . . 7  |-  .0.  =  ( 0g `  R )
6 cply1coe0.p . . . . . . 7  |-  P  =  (Poly1 `  R )
7 cply1coe0.b . . . . . . 7  |-  B  =  ( Base `  P
)
8 cply1coe0.a . . . . . . 7  |-  A  =  (algSc `  P )
94, 5, 6, 7, 8cply1coe0 30977 . . . . . 6  |-  ( ( R  e.  Ring  /\  s  e.  K )  ->  A. n  e.  NN  ( (coe1 `  ( A `  s )
) `  n )  =  .0.  )
103, 9syl 16 . . . . 5  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  s  e.  K )  /\  M  =  ( A `  s ) )  ->  A. n  e.  NN  ( (coe1 `  ( A `  s ) ) `  n )  =  .0.  )
11 fveq2 5775 . . . . . . . . 9  |-  ( M  =  ( A `  s )  ->  (coe1 `  M )  =  (coe1 `  ( A `  s
) ) )
1211fveq1d 5777 . . . . . . . 8  |-  ( M  =  ( A `  s )  ->  (
(coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  s ) ) `  n ) )
1312eqeq1d 2452 . . . . . . 7  |-  ( M  =  ( A `  s )  ->  (
( (coe1 `  M ) `  n )  =  .0.  <->  ( (coe1 `  ( A `  s ) ) `  n )  =  .0.  ) )
1413ralbidv 2815 . . . . . 6  |-  ( M  =  ( A `  s )  ->  ( A. n  e.  NN  ( (coe1 `  M ) `  n )  =  .0.  <->  A. n  e.  NN  (
(coe1 `  ( A `  s ) ) `  n )  =  .0.  ) )
1514adantl 466 . . . . 5  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  s  e.  K )  /\  M  =  ( A `  s ) )  -> 
( A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  <->  A. n  e.  NN  ( (coe1 `  ( A `  s ) ) `  n )  =  .0.  ) )
1610, 15mpbird 232 . . . 4  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  s  e.  K )  /\  M  =  ( A `  s ) )  ->  A. n  e.  NN  ( (coe1 `  M ) `  n )  =  .0.  )
1716ex 434 . . 3  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  s  e.  K
)  ->  ( M  =  ( A `  s )  ->  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  ) )
1817rexlimdva 2923 . 2  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( E. s  e.  K  M  =  ( A `  s )  ->  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  ) )
19 simpr 461 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  M  e.  B )
20 eqid 2450 . . . . . . . . . 10  |-  (Scalar `  P )  =  (Scalar `  P )
216ply1rng 17796 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
Ring )
226ply1lmod 17800 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
LMod )
23 eqid 2450 . . . . . . . . . 10  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
248, 20, 21, 22, 23, 7asclf 17500 . . . . . . . . 9  |-  ( R  e.  Ring  ->  A :
( Base `  (Scalar `  P
) ) --> B )
2524adantr 465 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  A : ( Base `  (Scalar `  P ) ) --> B )
26 0nn0 10681 . . . . . . . . . 10  |-  0  e.  NN0
27 eqid 2450 . . . . . . . . . . 11  |-  (coe1 `  M
)  =  (coe1 `  M
)
28 eqid 2450 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
2927, 7, 6, 28coe1fvalcl 30958 . . . . . . . . . 10  |-  ( ( M  e.  B  /\  0  e.  NN0 )  -> 
( (coe1 `  M ) ` 
0 )  e.  (
Base `  R )
)
3019, 26, 29sylancl 662 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(coe1 `  M ) ` 
0 )  e.  (
Base `  R )
)
316ply1sca 17801 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  =  (Scalar `  P )
)
3231eqcomd 2457 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  (Scalar `  P )  =  R )
3332fveq2d 5779 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( Base `  (Scalar `  P )
)  =  ( Base `  R ) )
3433adantr 465 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( Base `  (Scalar `  P
) )  =  (
Base `  R )
)
3530, 34eleqtrrd 2539 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(coe1 `  M ) ` 
0 )  e.  (
Base `  (Scalar `  P
) ) )
3625, 35ffvelrnd 5929 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( A `  ( (coe1 `  M ) `  0
) )  e.  B
)
371, 19, 363jca 1168 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( R  e.  Ring  /\  M  e.  B  /\  ( A `  ( (coe1 `  M ) `  0
) )  e.  B
) )
3837adantr 465 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  ( R  e.  Ring  /\  M  e.  B  /\  ( A `  ( (coe1 `  M ) `  0
) )  e.  B
) )
39 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  (
(coe1 `  M ) `  n )  =  .0.  )  ->  ( (coe1 `  M ) `  n
)  =  .0.  )
4027, 7, 6, 4coe1fvalcl 30958 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  B  /\  0  e.  NN0 )  -> 
( (coe1 `  M ) ` 
0 )  e.  K
)
4119, 26, 40sylancl 662 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(coe1 `  M ) ` 
0 )  e.  K
)
426, 8, 4, 5coe1scl 17834 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  Ring  /\  (
(coe1 `  M ) ` 
0 )  e.  K
)  ->  (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) )  =  ( k  e.  NN0  |->  if ( k  =  0 ,  ( (coe1 `  M
) `  0 ) ,  .0.  ) ) )
431, 41, 42syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) )  =  ( k  e.  NN0  |->  if ( k  =  0 ,  ( (coe1 `  M
) `  0 ) ,  .0.  ) ) )
4443adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) )  =  ( k  e. 
NN0  |->  if ( k  =  0 ,  ( (coe1 `  M ) ` 
0 ) ,  .0.  ) ) )
45 nnne0 10441 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  n  =/=  0 )
4645neneqd 2648 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  -.  n  =  0 )
4746adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  -.  n  = 
0 )
4847adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  k  =  n )  ->  -.  n  =  0 )
49 eqeq1 2453 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  n  ->  (
k  =  0  <->  n  =  0 ) )
5049notbid 294 . . . . . . . . . . . . . . . . 17  |-  ( k  =  n  ->  ( -.  k  =  0  <->  -.  n  =  0 ) )
5150adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  k  =  n )  ->  ( -.  k  =  0  <->  -.  n  =  0 ) )
5248, 51mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  k  =  n )  ->  -.  k  =  0 )
53 iffalse 3883 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  0  ->  if ( k  =  0 ,  ( (coe1 `  M
) `  0 ) ,  .0.  )  =  .0.  )
5452, 53syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  k  =  n )  ->  if ( k  =  0 ,  ( (coe1 `  M
) `  0 ) ,  .0.  )  =  .0.  )
55 nnnn0 10673 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  e.  NN0 )
5655adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  n  e.  NN0 )
57 fvex 5785 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  e. 
_V
585, 57eqeltri 2532 . . . . . . . . . . . . . . 15  |-  .0.  e.  _V
5958a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  .0.  e.  _V )
6044, 54, 56, 59fvmptd 5864 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  ( (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) ) `  n )  =  .0.  )
6160eqcomd 2457 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  .0.  =  (
(coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n ) )
6261adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  (
(coe1 `  M ) `  n )  =  .0.  )  ->  .0.  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n ) )
6339, 62eqtrd 2490 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  (
(coe1 `  M ) `  n )  =  .0.  )  ->  ( (coe1 `  M ) `  n
)  =  ( (coe1 `  ( A `  (
(coe1 `  M ) ` 
0 ) ) ) `
 n ) )
6463ex 434 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  ( ( (coe1 `  M ) `  n
)  =  .0.  ->  ( (coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n ) ) )
6564ralimdva 2875 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( A. n  e.  NN  ( (coe1 `  M ) `  n )  =  .0. 
->  A. n  e.  NN  ( (coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n ) ) )
6665imp 429 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) ) `  n ) )
676, 8, 4ply1sclid 17835 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
(coe1 `  M ) ` 
0 )  e.  K
)  ->  ( (coe1 `  M ) `  0
)  =  ( (coe1 `  ( A `  (
(coe1 `  M ) ` 
0 ) ) ) `
 0 ) )
681, 41, 67syl2anc 661 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(coe1 `  M ) ` 
0 )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) )
6968adantr 465 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  (
(coe1 `  M ) ` 
0 )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) )
7066, 69jca 532 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  ( A. n  e.  NN  ( (coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n )  /\  ( (coe1 `  M ) ` 
0 )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) ) )
71 df-n0 10667 . . . . . . . 8  |-  NN0  =  ( NN  u.  { 0 } )
7271raleqi 3003 . . . . . . 7  |-  ( A. n  e.  NN0  ( (coe1 `  M ) `  n
)  =  ( (coe1 `  ( A `  (
(coe1 `  M ) ` 
0 ) ) ) `
 n )  <->  A. n  e.  ( NN  u.  {
0 } ) ( (coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n ) )
73 c0ex 9467 . . . . . . . . 9  |-  0  e.  _V
7473a1i 11 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  0  e.  _V )
75 fveq2 5775 . . . . . . . . . 10  |-  ( n  =  0  ->  (
(coe1 `  M ) `  n )  =  ( (coe1 `  M ) ` 
0 ) )
76 fveq2 5775 . . . . . . . . . 10  |-  ( n  =  0  ->  (
(coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) )
7775, 76eqeq12d 2471 . . . . . . . . 9  |-  ( n  =  0  ->  (
( (coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n )  <->  ( (coe1 `  M ) `  0
)  =  ( (coe1 `  ( A `  (
(coe1 `  M ) ` 
0 ) ) ) `
 0 ) ) )
7877ralunsn 4163 . . . . . . . 8  |-  ( 0  e.  _V  ->  ( A. n  e.  ( NN  u.  { 0 } ) ( (coe1 `  M
) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) ) `  n )  <->  ( A. n  e.  NN  (
(coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n )  /\  ( (coe1 `  M ) ` 
0 )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) ) ) )
7974, 78syl 16 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  ( A. n  e.  ( NN  u.  { 0 } ) ( (coe1 `  M
) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) ) `  n )  <->  ( A. n  e.  NN  (
(coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n )  /\  ( (coe1 `  M ) ` 
0 )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) ) ) )
8072, 79syl5bb 257 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  ( A. n  e.  NN0  ( (coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n )  <->  ( A. n  e.  NN  (
(coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n )  /\  ( (coe1 `  M ) ` 
0 )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) ) ) )
8170, 80mpbird 232 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  A. n  e.  NN0  ( (coe1 `  M
) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) ) `  n ) )
82 eqid 2450 . . . . . 6  |-  (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) )  =  (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) )
836, 7, 27, 82eqcoe1ply1eq 30963 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B  /\  ( A `  ( (coe1 `  M ) `  0
) )  e.  B
)  ->  ( A. n  e.  NN0  ( (coe1 `  M ) `  n
)  =  ( (coe1 `  ( A `  (
(coe1 `  M ) ` 
0 ) ) ) `
 n )  ->  M  =  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) )
8438, 81, 83sylc 60 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  M  =  ( A `  ( (coe1 `  M ) ` 
0 ) ) )
8541adantr 465 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  (
(coe1 `  M ) ` 
0 )  e.  K
)
86 fveq2 5775 . . . . . . 7  |-  ( s  =  ( (coe1 `  M
) `  0 )  ->  ( A `  s
)  =  ( A `
 ( (coe1 `  M
) `  0 )
) )
8786eqeq2d 2463 . . . . . 6  |-  ( s  =  ( (coe1 `  M
) `  0 )  ->  ( M  =  ( A `  s )  <-> 
M  =  ( A `
 ( (coe1 `  M
) `  0 )
) ) )
8887adantl 466 . . . . 5  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  /\  s  =  ( (coe1 `  M
) `  0 )
)  ->  ( M  =  ( A `  s )  <->  M  =  ( A `  ( (coe1 `  M ) `  0
) ) ) )
8985, 88rspcedv 3159 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  ( M  =  ( A `  ( (coe1 `  M ) ` 
0 ) )  ->  E. s  e.  K  M  =  ( A `  s ) ) )
9084, 89mpd 15 . . 3  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  E. s  e.  K  M  =  ( A `  s ) )
9190ex 434 . 2  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( A. n  e.  NN  ( (coe1 `  M ) `  n )  =  .0. 
->  E. s  e.  K  M  =  ( A `  s ) ) )
9218, 91impbid 191 1  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( E. s  e.  K  M  =  ( A `  s )  <->  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757   A.wral 2792   E.wrex 2793   _Vcvv 3054    u. cun 3410   ifcif 3875   {csn 3961    |-> cmpt 4434   -->wf 5498   ` cfv 5502   0cc0 9369   NNcn 10409   NN0cn0 10666   Basecbs 14262  Scalarcsca 14329   0gc0g 14466   Ringcrg 16737  algSccascl 17475  Poly1cpl1 17726  coe1cco1 17727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-inf2 7934  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-se 4764  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-isom 5511  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-of 6406  df-ofr 6407  df-om 6563  df-1st 6663  df-2nd 6664  df-supp 6777  df-recs 6918  df-rdg 6952  df-1o 7006  df-2o 7007  df-oadd 7010  df-er 7187  df-map 7302  df-pm 7303  df-ixp 7350  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-fsupp 7708  df-oi 7811  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-5 10470  df-6 10471  df-7 10472  df-8 10473  df-9 10474  df-10 10475  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-seq 11894  df-hash 12191  df-struct 14264  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-mulr 14340  df-sca 14342  df-vsca 14343  df-tset 14345  df-ple 14346  df-0g 14468  df-gsum 14469  df-mre 14612  df-mrc 14613  df-acs 14615  df-mnd 15503  df-mhm 15552  df-submnd 15553  df-grp 15633  df-minusg 15634  df-sbg 15635  df-mulg 15636  df-subg 15766  df-ghm 15833  df-cntz 15923  df-cmn 16369  df-abl 16370  df-mgp 16683  df-ur 16695  df-srg 16699  df-rng 16739  df-subrg 16955  df-lmod 17042  df-lss 17106  df-ascl 17478  df-psr 17515  df-mvr 17516  df-mpl 17517  df-opsr 17519  df-psr1 17729  df-vr1 17730  df-ply1 17731  df-coe1 17732
This theorem is referenced by:  cpmatel2  31162
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