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Theorem cply1coe0bi 18537
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 455 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Ring )
21anim1i 566 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  s  e.  K
)  ->  ( R  e.  Ring  /\  s  e.  K ) )
32adantr 463 . . . . . 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 18536 . . . . . 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 5848 . . . . . . . . 9  |-  ( M  =  ( A `  s )  ->  (coe1 `  M )  =  (coe1 `  ( A `  s
) ) )
1211fveq1d 5850 . . . . . . . 8  |-  ( M  =  ( A `  s )  ->  (
(coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  s ) ) `  n ) )
1312eqeq1d 2456 . . . . . . 7  |-  ( M  =  ( A `  s )  ->  (
( (coe1 `  M ) `  n )  =  .0.  <->  ( (coe1 `  ( A `  s ) ) `  n )  =  .0.  ) )
1413ralbidv 2893 . . . . . 6  |-  ( M  =  ( A `  s )  ->  ( A. n  e.  NN  ( (coe1 `  M ) `  n )  =  .0.  <->  A. n  e.  NN  (
(coe1 `  ( A `  s ) ) `  n )  =  .0.  ) )
1514adantl 464 . . . . 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 432 . . 3  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  s  e.  K
)  ->  ( M  =  ( A `  s )  ->  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  ) )
1817rexlimdva 2946 . 2  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( E. s  e.  K  M  =  ( A `  s )  ->  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  ) )
19 simpr 459 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  M  e.  B )
20 0nn0 10806 . . . . . 6  |-  0  e.  NN0
21 eqid 2454 . . . . . . 7  |-  (coe1 `  M
)  =  (coe1 `  M
)
2221, 7, 6, 4coe1fvalcl 18446 . . . . . 6  |-  ( ( M  e.  B  /\  0  e.  NN0 )  -> 
( (coe1 `  M ) ` 
0 )  e.  K
)
2319, 20, 22sylancl 660 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(coe1 `  M ) ` 
0 )  e.  K
)
2423adantr 463 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  (
(coe1 `  M ) ` 
0 )  e.  K
)
25 fveq2 5848 . . . . . 6  |-  ( s  =  ( (coe1 `  M
) `  0 )  ->  ( A `  s
)  =  ( A `
 ( (coe1 `  M
) `  0 )
) )
2625eqeq2d 2468 . . . . 5  |-  ( s  =  ( (coe1 `  M
) `  0 )  ->  ( M  =  ( A `  s )  <-> 
M  =  ( A `
 ( (coe1 `  M
) `  0 )
) ) )
2726adantl 464 . . . 4  |-  ( ( ( ( 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
) ) ) )
28 eqid 2454 . . . . . . . . . 10  |-  (Scalar `  P )  =  (Scalar `  P )
296ply1ring 18484 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
Ring )
306ply1lmod 18488 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
LMod )
31 eqid 2454 . . . . . . . . . 10  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
328, 28, 29, 30, 31, 7asclf 18181 . . . . . . . . 9  |-  ( R  e.  Ring  ->  A :
( Base `  (Scalar `  P
) ) --> B )
3332adantr 463 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  A : ( Base `  (Scalar `  P ) ) --> B )
34 eqid 2454 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
3521, 7, 6, 34coe1fvalcl 18446 . . . . . . . . . 10  |-  ( ( M  e.  B  /\  0  e.  NN0 )  -> 
( (coe1 `  M ) ` 
0 )  e.  (
Base `  R )
)
3619, 20, 35sylancl 660 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(coe1 `  M ) ` 
0 )  e.  (
Base `  R )
)
376ply1sca 18489 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  =  (Scalar `  P )
)
3837eqcomd 2462 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  (Scalar `  P )  =  R )
3938fveq2d 5852 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  ( Base `  (Scalar `  P )
)  =  ( Base `  R ) )
4039adantr 463 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( Base `  (Scalar `  P
) )  =  (
Base `  R )
)
4136, 40eleqtrrd 2545 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(coe1 `  M ) ` 
0 )  e.  (
Base `  (Scalar `  P
) ) )
4233, 41ffvelrnd 6008 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( A `  ( (coe1 `  M ) `  0
) )  e.  B
)
431, 19, 423jca 1174 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( R  e.  Ring  /\  M  e.  B  /\  ( A `  ( (coe1 `  M ) `  0
) )  e.  B
) )
4443adantr 463 . . . . 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
) )
45 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  (
(coe1 `  M ) `  n )  =  .0.  )  ->  ( (coe1 `  M ) `  n
)  =  .0.  )
466, 8, 4, 5coe1scl 18523 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Ring  /\  (
(coe1 `  M ) ` 
0 )  e.  K
)  ->  (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) )  =  ( k  e.  NN0  |->  if ( k  =  0 ,  ( (coe1 `  M
) `  0 ) ,  .0.  ) ) )
4723, 46syldan 468 . . . . . . . . . . . . . 14  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) )  =  ( k  e.  NN0  |->  if ( k  =  0 ,  ( (coe1 `  M
) `  0 ) ,  .0.  ) ) )
4847adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) )  =  ( k  e. 
NN0  |->  if ( k  =  0 ,  ( (coe1 `  M ) ` 
0 ) ,  .0.  ) ) )
49 nnne0 10564 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  n  =/=  0 )
5049neneqd 2656 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  -.  n  =  0 )
5150adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  -.  n  = 
0 )
5251adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  k  =  n )  ->  -.  n  =  0 )
53 eqeq1 2458 . . . . . . . . . . . . . . . . 17  |-  ( k  =  n  ->  (
k  =  0  <->  n  =  0 ) )
5453notbid 292 . . . . . . . . . . . . . . . 16  |-  ( k  =  n  ->  ( -.  k  =  0  <->  -.  n  =  0 ) )
5554adantl 464 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  k  =  n )  ->  ( -.  k  =  0  <->  -.  n  =  0 ) )
5652, 55mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  k  =  n )  ->  -.  k  =  0 )
5756iffalsed 3940 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  k  =  n )  ->  if ( k  =  0 ,  ( (coe1 `  M
) `  0 ) ,  .0.  )  =  .0.  )
58 nnnn0 10798 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  n  e.  NN0 )
5958adantl 464 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  n  e.  NN0 )
60 fvex 5858 . . . . . . . . . . . . . . 15  |-  ( 0g
`  R )  e. 
_V
615, 60eqeltri 2538 . . . . . . . . . . . . . 14  |-  .0.  e.  _V
6261a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  .0.  e.  _V )
6348, 57, 59, 62fvmptd 5936 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  ( (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) ) `  n )  =  .0.  )
6463eqcomd 2462 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  .0.  =  (
(coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n ) )
6564adantr 463 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  (
(coe1 `  M ) `  n )  =  .0.  )  ->  .0.  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n ) )
6645, 65eqtrd 2495 . . . . . . . . 9  |-  ( ( ( ( R  e. 
Ring  /\  M  e.  B
)  /\  n  e.  NN )  /\  (
(coe1 `  M ) `  n )  =  .0.  )  ->  ( (coe1 `  M ) `  n
)  =  ( (coe1 `  ( A `  (
(coe1 `  M ) ` 
0 ) ) ) `
 n ) )
6766ex 432 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  n  e.  NN )  ->  ( ( (coe1 `  M ) `  n
)  =  .0.  ->  ( (coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n ) ) )
6867ralimdva 2862 . . . . . . 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 ) ) )
6968imp 427 . . . . . 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 ) )
706, 8, 4ply1sclid 18524 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
(coe1 `  M ) ` 
0 )  e.  K
)  ->  ( (coe1 `  M ) `  0
)  =  ( (coe1 `  ( A `  (
(coe1 `  M ) ` 
0 ) ) ) `
 0 ) )
7123, 70syldan 468 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(coe1 `  M ) ` 
0 )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) )
7271adantr 463 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  (
(coe1 `  M ) ` 
0 )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) )
73 df-n0 10792 . . . . . . . 8  |-  NN0  =  ( NN  u.  { 0 } )
7473raleqi 3055 . . . . . . 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 ) )
75 c0ex 9579 . . . . . . . 8  |-  0  e.  _V
76 fveq2 5848 . . . . . . . . . 10  |-  ( n  =  0  ->  (
(coe1 `  M ) `  n )  =  ( (coe1 `  M ) ` 
0 ) )
77 fveq2 5848 . . . . . . . . . 10  |-  ( n  =  0  ->  (
(coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 0 ) )
7876, 77eqeq12d 2476 . . . . . . . . 9  |-  ( n  =  0  ->  (
( (coe1 `  M ) `  n )  =  ( (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) ) `
 n )  <->  ( (coe1 `  M ) `  0
)  =  ( (coe1 `  ( A `  (
(coe1 `  M ) ` 
0 ) ) ) `
 0 ) ) )
7978ralunsn 4223 . . . . . . . 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 ) ) ) )
8075, 79mp1i 12 . . . . . . 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 ) ) ) )
8174, 80syl5bb 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 ) ) ) )
8269, 72, 81mpbir2and 920 . . . . 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 ) )
83 eqid 2454 . . . . . 6  |-  (coe1 `  ( A `  ( (coe1 `  M ) `  0
) ) )  =  (coe1 `  ( A `  ( (coe1 `  M ) ` 
0 ) ) )
846, 7, 21, 83eqcoe1ply1eq 18534 . . . . 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 ) ) ) )
8544, 82, 84sylc 60 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  M  =  ( A `  ( (coe1 `  M ) ` 
0 ) ) )
8624, 27, 85rspcedvd 3212 . . 3  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  A. n  e.  NN  ( (coe1 `  M
) `  n )  =  .0.  )  ->  E. s  e.  K  M  =  ( A `  s ) )
8786ex 432 . 2  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( A. n  e.  NN  ( (coe1 `  M ) `  n )  =  .0. 
->  E. s  e.  K  M  =  ( A `  s ) ) )
8818, 87impbid 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    u. cun 3459   ifcif 3929   {csn 4016    |-> cmpt 4497   -->wf 5566   ` cfv 5570   0cc0 9481   NNcn 10531   NN0cn0 10791   Basecbs 14716  Scalarcsca 14787   0gc0g 14929   Ringcrg 17393  algSccascl 18155  Poly1cpl1 18411  coe1cco1 18412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-ofr 6514  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-tset 14803  df-ple 14804  df-0g 14931  df-gsum 14932  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-subg 16397  df-ghm 16464  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-srg 17353  df-ring 17395  df-subrg 17622  df-lmod 17709  df-lss 17774  df-ascl 18158  df-psr 18200  df-mvr 18201  df-mpl 18202  df-opsr 18204  df-psr1 18414  df-vr1 18415  df-ply1 18416  df-coe1 18417
This theorem is referenced by:  cpmatel2  19381
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