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Theorem lply1binomsc 30775
Description: The binomial theorem for linear polynomials (monic polynomials of degree 1) over commutative rings, expressed by an element of this ring:  ( X  +  A ) ^ N is the sum from  k  =  0 to  N of  ( N  _C  k )  x.  (
( A ^ ( N  -  k )
)  x.  ( X ^ k ) ). (Contributed by AV, 25-Aug-2019.)
Hypotheses
Ref Expression
cply1binom.p  |-  P  =  (Poly1 `  R )
cply1binom.x  |-  X  =  (var1 `  R )
cply1binom.a  |-  .+  =  ( +g  `  P )
cply1binom.m  |-  .X.  =  ( .r `  P )
cply1binom.t  |-  .x.  =  (.g
`  P )
cply1binom.g  |-  G  =  (mulGrp `  P )
cply1binom.e  |-  .^  =  (.g
`  G )
lply1binomsc.k  |-  K  =  ( Base `  R
)
lply1binomsc.s  |-  S  =  (algSc `  P )
lply1binomsc.h  |-  H  =  (mulGrp `  R )
lply1binomsc.e  |-  E  =  (.g `  H )
Assertion
Ref Expression
lply1binomsc  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( N  .^  ( X  .+  ( S `  A ) ) )  =  ( P  gsumg  ( k  e.  ( 0 ... N ) 
|->  ( ( N  _C  k )  .x.  (
( S `  (
( N  -  k
) E A ) )  .X.  ( k  .^  X ) ) ) ) ) )
Distinct variable groups:    A, k    k, K    k, N    P, k    R, k    k, X    .X. , k    .x. , k    .^ , k    .+ , k    S, k
Allowed substitution hints:    E( k)    G( k)    H( k)

Proof of Theorem lply1binomsc
StepHypRef Expression
1 lply1binomsc.s . . . . . 6  |-  S  =  (algSc `  P )
2 eqid 2438 . . . . . 6  |-  (Scalar `  P )  =  (Scalar `  P )
3 crngrng 16643 . . . . . . . 8  |-  ( R  e.  CRing  ->  R  e.  Ring )
4 cply1binom.p . . . . . . . . 9  |-  P  =  (Poly1 `  R )
54ply1rng 17678 . . . . . . . 8  |-  ( R  e.  Ring  ->  P  e. 
Ring )
63, 5syl 16 . . . . . . 7  |-  ( R  e.  CRing  ->  P  e.  Ring )
763ad2ant1 1009 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  P  e.  Ring )
84ply1lmod 17682 . . . . . . . 8  |-  ( R  e.  Ring  ->  P  e. 
LMod )
93, 8syl 16 . . . . . . 7  |-  ( R  e.  CRing  ->  P  e.  LMod )
1093ad2ant1 1009 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  P  e.  LMod )
11 eqid 2438 . . . . . 6  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
12 eqid 2438 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
131, 2, 7, 10, 11, 12asclf 17385 . . . . 5  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  S : ( Base `  (Scalar `  P ) ) --> (
Base `  P )
)
14 lply1binomsc.k . . . . . . 7  |-  K  =  ( Base `  R
)
154ply1sca 17683 . . . . . . . . 9  |-  ( R  e.  CRing  ->  R  =  (Scalar `  P ) )
16153ad2ant1 1009 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  R  =  (Scalar `  P )
)
1716fveq2d 5690 . . . . . . 7  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( Base `  R )  =  ( Base `  (Scalar `  P ) ) )
1814, 17syl5eq 2482 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  K  =  ( Base `  (Scalar `  P ) ) )
1918feq2d 5542 . . . . 5  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( S : K --> ( Base `  P )  <->  S :
( Base `  (Scalar `  P
) ) --> ( Base `  P ) ) )
2013, 19mpbird 232 . . . 4  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  S : K --> ( Base `  P
) )
21 simp3 990 . . . 4  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  A  e.  K )
2220, 21ffvelrnd 5839 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( S `  A )  e.  ( Base `  P
) )
23 cply1binom.x . . . 4  |-  X  =  (var1 `  R )
24 cply1binom.a . . . 4  |-  .+  =  ( +g  `  P )
25 cply1binom.m . . . 4  |-  .X.  =  ( .r `  P )
26 cply1binom.t . . . 4  |-  .x.  =  (.g
`  P )
27 cply1binom.g . . . 4  |-  G  =  (mulGrp `  P )
28 cply1binom.e . . . 4  |-  .^  =  (.g
`  G )
294, 23, 24, 25, 26, 27, 28, 12lply1binom 30774 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  ( S `
 A )  e.  ( Base `  P
) )  ->  ( N  .^  ( X  .+  ( S `  A ) ) )  =  ( P  gsumg  ( k  e.  ( 0 ... N ) 
|->  ( ( N  _C  k )  .x.  (
( ( N  -  k )  .^  ( S `  A )
)  .X.  ( k  .^  X ) ) ) ) ) )
3022, 29syld3an3 1263 . 2  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( N  .^  ( X  .+  ( S `  A ) ) )  =  ( P  gsumg  ( k  e.  ( 0 ... N ) 
|->  ( ( N  _C  k )  .x.  (
( ( N  -  k )  .^  ( S `  A )
)  .X.  ( k  .^  X ) ) ) ) ) )
314ply1assa 17630 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  P  e. AssAlg )
32313ad2ant1 1009 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  P  e. AssAlg )
3332adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  P  e. AssAlg )
34 fznn0sub 11479 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  ( N  -  k )  e.  NN0 )
3534adantl 466 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( N  -  k )  e.  NN0 )
3615fveq2d 5690 . . . . . . . . . . . . . 14  |-  ( R  e.  CRing  ->  ( Base `  R )  =  (
Base `  (Scalar `  P
) ) )
3714, 36syl5eq 2482 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  K  =  ( Base `  (Scalar `  P
) ) )
3837eleq2d 2505 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( A  e.  K  <->  A  e.  ( Base `  (Scalar `  P
) ) ) )
3938biimpa 484 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  A  e.  K )  ->  A  e.  ( Base `  (Scalar `  P ) ) )
40393adant2 1007 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  A  e.  ( Base `  (Scalar `  P ) ) )
4140adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  A  e.  (
Base `  (Scalar `  P
) ) )
42 eqid 2438 . . . . . . . . . . . . 13  |-  ( 1r
`  P )  =  ( 1r `  P
)
4312, 42rngidcl 16653 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  ( 1r
`  P )  e.  ( Base `  P
) )
446, 43syl 16 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  ( 1r `  P )  e.  (
Base `  P )
)
45443ad2ant1 1009 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( 1r `  P )  e.  ( Base `  P
) )
4645adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( 1r `  P )  e.  (
Base `  P )
)
47 eqid 2438 . . . . . . . . . 10  |-  ( .s
`  P )  =  ( .s `  P
)
48 eqid 2438 . . . . . . . . . 10  |-  (mulGrp `  (Scalar `  P ) )  =  (mulGrp `  (Scalar `  P ) )
49 eqid 2438 . . . . . . . . . 10  |-  (.g `  (mulGrp `  (Scalar `  P )
) )  =  (.g `  (mulGrp `  (Scalar `  P
) ) )
5012, 2, 11, 47, 48, 49, 27, 28assamulgscm 30761 . . . . . . . . 9  |-  ( ( P  e. AssAlg  /\  (
( N  -  k
)  e.  NN0  /\  A  e.  ( Base `  (Scalar `  P )
)  /\  ( 1r `  P )  e.  (
Base `  P )
) )  ->  (
( N  -  k
)  .^  ( A
( .s `  P
) ( 1r `  P ) ) )  =  ( ( ( N  -  k ) (.g `  (mulGrp `  (Scalar `  P ) ) ) A ) ( .s
`  P ) ( ( N  -  k
)  .^  ( 1r `  P ) ) ) )
5133, 35, 41, 46, 50syl13anc 1220 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k )  .^  ( A ( .s `  P ) ( 1r
`  P ) ) )  =  ( ( ( N  -  k
) (.g `  (mulGrp `  (Scalar `  P ) ) ) A ) ( .s
`  P ) ( ( N  -  k
)  .^  ( 1r `  P ) ) ) )
52 lply1binomsc.e . . . . . . . . . . . . . 14  |-  E  =  (.g `  H )
53 lply1binomsc.h . . . . . . . . . . . . . . . 16  |-  H  =  (mulGrp `  R )
5415fveq2d 5690 . . . . . . . . . . . . . . . 16  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  =  (mulGrp `  (Scalar `  P ) ) )
5553, 54syl5eq 2482 . . . . . . . . . . . . . . 15  |-  ( R  e.  CRing  ->  H  =  (mulGrp `  (Scalar `  P
) ) )
5655fveq2d 5690 . . . . . . . . . . . . . 14  |-  ( R  e.  CRing  ->  (.g `  H
)  =  (.g `  (mulGrp `  (Scalar `  P )
) ) )
5752, 56syl5eq 2482 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  E  =  (.g
`  (mulGrp `  (Scalar `  P
) ) ) )
58573ad2ant1 1009 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  E  =  (.g `  (mulGrp `  (Scalar `  P ) ) ) )
5958adantr 465 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  E  =  (.g `  (mulGrp `  (Scalar `  P
) ) ) )
6059eqcomd 2443 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  (.g `  (mulGrp `  (Scalar `  P ) ) )  =  E )
6160oveqd 6103 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k ) (.g `  (mulGrp `  (Scalar `  P
) ) ) A )  =  ( ( N  -  k ) E A ) )
6227rngmgp 16639 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  G  e. 
Mnd )
636, 62syl 16 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  G  e.  Mnd )
64633ad2ant1 1009 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  G  e.  Mnd )
6527, 12mgpbas 16585 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  G )
6627, 42rngidval 16593 . . . . . . . . . . 11  |-  ( 1r
`  P )  =  ( 0g `  G
)
6765, 28, 66mulgnn0z 15638 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  ( N  -  k
)  e.  NN0 )  ->  ( ( N  -  k )  .^  ( 1r `  P ) )  =  ( 1r `  P ) )
6864, 34, 67syl2an 477 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k )  .^  ( 1r `  P ) )  =  ( 1r
`  P ) )
6961, 68oveq12d 6104 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( ( N  -  k ) (.g `  (mulGrp `  (Scalar `  P ) ) ) A ) ( .s
`  P ) ( ( N  -  k
)  .^  ( 1r `  P ) ) )  =  ( ( ( N  -  k ) E A ) ( .s `  P ) ( 1r `  P
) ) )
7051, 69eqtrd 2470 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k )  .^  ( A ( .s `  P ) ( 1r
`  P ) ) )  =  ( ( ( N  -  k
) E A ) ( .s `  P
) ( 1r `  P ) ) )
711, 2, 11, 47, 42asclval 17383 . . . . . . . . 9  |-  ( A  e.  ( Base `  (Scalar `  P ) )  -> 
( S `  A
)  =  ( A ( .s `  P
) ( 1r `  P ) ) )
7241, 71syl 16 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( S `  A )  =  ( A ( .s `  P ) ( 1r
`  P ) ) )
7372oveq2d 6102 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k )  .^  ( S `  A ) )  =  ( ( N  -  k ) 
.^  ( A ( .s `  P ) ( 1r `  P
) ) ) )
7453rngmgp 16639 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  H  e. 
Mnd )
753, 74syl 16 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  H  e.  Mnd )
76753ad2ant1 1009 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  H  e.  Mnd )
7776adantr 465 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  H  e.  Mnd )
78 simpr 461 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  A  e.  K )  ->  A  e.  K )
7953, 14mgpbas 16585 . . . . . . . . . . . . 13  |-  K  =  ( Base `  H
)
8078, 79syl6eleq 2528 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  A  e.  K )  ->  A  e.  ( Base `  H
) )
81803adant2 1007 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  A  e.  ( Base `  H
) )
8281adantr 465 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  A  e.  (
Base `  H )
)
83 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  H )  =  (
Base `  H )
8483, 52mulgnn0cl 15634 . . . . . . . . . 10  |-  ( ( H  e.  Mnd  /\  ( N  -  k
)  e.  NN0  /\  A  e.  ( Base `  H ) )  -> 
( ( N  -  k ) E A )  e.  ( Base `  H ) )
8577, 35, 82, 84syl3anc 1218 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k ) E A )  e.  (
Base `  H )
)
8616adantr 465 . . . . . . . . . . . 12  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  R  =  (Scalar `  P ) )
8786eqcomd 2443 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  (Scalar `  P )  =  R )
8887fveq2d 5690 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( Base `  (Scalar `  P ) )  =  ( Base `  R
) )
89 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
9053, 89mgpbas 16585 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  H )
9188, 90syl6eq 2486 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( Base `  (Scalar `  P ) )  =  ( Base `  H
) )
9285, 91eleqtrrd 2515 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k ) E A )  e.  (
Base `  (Scalar `  P
) ) )
931, 2, 11, 47, 42asclval 17383 . . . . . . . 8  |-  ( ( ( N  -  k
) E A )  e.  ( Base `  (Scalar `  P ) )  -> 
( S `  (
( N  -  k
) E A ) )  =  ( ( ( N  -  k
) E A ) ( .s `  P
) ( 1r `  P ) ) )
9492, 93syl 16 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( S `  ( ( N  -  k ) E A ) )  =  ( ( ( N  -  k ) E A ) ( .s `  P ) ( 1r
`  P ) ) )
9570, 73, 943eqtr4d 2480 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k )  .^  ( S `  A ) )  =  ( S `
 ( ( N  -  k ) E A ) ) )
9695oveq1d 6101 . . . . 5  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( ( N  -  k ) 
.^  ( S `  A ) )  .X.  ( k  .^  X
) )  =  ( ( S `  (
( N  -  k
) E A ) )  .X.  ( k  .^  X ) ) )
9796oveq2d 6102 . . . 4  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  _C  k )  .x.  ( ( ( N  -  k )  .^  ( S `  A ) )  .X.  ( k  .^  X ) ) )  =  ( ( N  _C  k )  .x.  ( ( S `  ( ( N  -  k ) E A ) )  .X.  (
k  .^  X )
) ) )
9897mpteq2dva 4373 . . 3  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  (
k  e.  ( 0 ... N )  |->  ( ( N  _C  k
)  .x.  ( (
( N  -  k
)  .^  ( S `  A ) )  .X.  ( k  .^  X
) ) ) )  =  ( k  e.  ( 0 ... N
)  |->  ( ( N  _C  k )  .x.  ( ( S `  ( ( N  -  k ) E A ) )  .X.  (
k  .^  X )
) ) ) )
9998oveq2d 6102 . 2  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( P  gsumg  ( k  e.  ( 0 ... N ) 
|->  ( ( N  _C  k )  .x.  (
( ( N  -  k )  .^  ( S `  A )
)  .X.  ( k  .^  X ) ) ) ) )  =  ( P  gsumg  ( k  e.  ( 0 ... N ) 
|->  ( ( N  _C  k )  .x.  (
( S `  (
( N  -  k
) E A ) )  .X.  ( k  .^  X ) ) ) ) ) )
10030, 99eqtrd 2470 1  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( N  .^  ( X  .+  ( S `  A ) ) )  =  ( P  gsumg  ( k  e.  ( 0 ... N ) 
|->  ( ( N  _C  k )  .x.  (
( S `  (
( N  -  k
) E A ) )  .X.  ( k  .^  X ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    e. cmpt 4345   -->wf 5409   ` cfv 5413  (class class class)co 6086   0cc0 9274    - cmin 9587   NN0cn0 10571   ...cfz 11429    _C cbc 12070   Basecbs 14166   +g cplusg 14230   .rcmulr 14231  Scalarcsca 14233   .scvsca 14234    gsumg cgsu 14371   Mndcmnd 15401  .gcmg 15406  mulGrpcmgp 16579   1rcur 16591   Ringcrg 16633   CRingccrg 16634   LModclmod 16926  AssAlgcasa 17358  algSccascl 17360  var1cv1 17607  Poly1cpl1 17608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-ofr 6316  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-fac 12044  df-bc 12071  df-hash 12096  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-tset 14249  df-ple 14250  df-0g 14372  df-gsum 14373  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-ghm 15736  df-cntz 15826  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-srg 16596  df-rng 16635  df-cring 16636  df-subrg 16841  df-lmod 16928  df-lss 16991  df-assa 17361  df-ascl 17363  df-psr 17400  df-mvr 17401  df-mpl 17402  df-opsr 17404  df-psr1 17611  df-vr1 17612  df-ply1 17613
This theorem is referenced by: (None)
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