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Theorem lply1binomsc 31001
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 2451 . . . . . 6  |-  (Scalar `  P )  =  (Scalar `  P )
3 crngrng 16770 . . . . . . . 8  |-  ( R  e.  CRing  ->  R  e.  Ring )
4 cply1binom.p . . . . . . . . 9  |-  P  =  (Poly1 `  R )
54ply1rng 17819 . . . . . . . 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 17823 . . . . . . . 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 2451 . . . . . 6  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
12 eqid 2451 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
131, 2, 7, 10, 11, 12asclf 17523 . . . . 5  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  S : ( Base `  (Scalar `  P ) ) --> (
Base `  P )
)
14 lply1binomsc.k . . . . . . 7  |-  K  =  ( Base `  R
)
154ply1sca 17824 . . . . . . . . 9  |-  ( R  e.  CRing  ->  R  =  (Scalar `  P ) )
16153ad2ant1 1009 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  R  =  (Scalar `  P )
)
1716fveq2d 5796 . . . . . . 7  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( Base `  R )  =  ( Base `  (Scalar `  P ) ) )
1814, 17syl5eq 2504 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  K  =  ( Base `  (Scalar `  P ) ) )
1918feq2d 5648 . . . . 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 5946 . . 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 31000 . . 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 1264 . 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 17771 . . . . . . . . . . 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 11597 . . . . . . . . . 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 5796 . . . . . . . . . . . . . 14  |-  ( R  e.  CRing  ->  ( Base `  R )  =  (
Base `  (Scalar `  P
) ) )
3714, 36syl5eq 2504 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  K  =  ( Base `  (Scalar `  P
) ) )
3837eleq2d 2521 . . . . . . . . . . . 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 2451 . . . . . . . . . . . . 13  |-  ( 1r
`  P )  =  ( 1r `  P
)
4312, 42rngidcl 16780 . . . . . . . . . . . 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 2451 . . . . . . . . . 10  |-  ( .s
`  P )  =  ( .s `  P
)
48 eqid 2451 . . . . . . . . . 10  |-  (mulGrp `  (Scalar `  P ) )  =  (mulGrp `  (Scalar `  P ) )
49 eqid 2451 . . . . . . . . . 10  |-  (.g `  (mulGrp `  (Scalar `  P )
) )  =  (.g `  (mulGrp `  (Scalar `  P
) ) )
5012, 2, 11, 47, 48, 49, 27, 28assamulgscm 30969 . . . . . . . . 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 1221 . . . . . . . 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 5796 . . . . . . . . . . . . . . . 16  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  =  (mulGrp `  (Scalar `  P ) ) )
5553, 54syl5eq 2504 . . . . . . . . . . . . . . 15  |-  ( R  e.  CRing  ->  H  =  (mulGrp `  (Scalar `  P
) ) )
5655fveq2d 5796 . . . . . . . . . . . . . 14  |-  ( R  e.  CRing  ->  (.g `  H
)  =  (.g `  (mulGrp `  (Scalar `  P )
) ) )
5752, 56syl5eq 2504 . . . . . . . . . . . . 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 2459 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  (.g `  (mulGrp `  (Scalar `  P ) ) )  =  E )
6160oveqd 6210 . . . . . . . . 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 16766 . . . . . . . . . . . 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 16711 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  G )
6627, 42rngidval 16719 . . . . . . . . . . 11  |-  ( 1r
`  P )  =  ( 0g `  G
)
6765, 28, 66mulgnn0z 15758 . . . . . . . . . 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 6211 . . . . . . . 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 2492 . . . . . . 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 17521 . . . . . . . . 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 6209 . . . . . . 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 16766 . . . . . . . . . . . . 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 16711 . . . . . . . . . . . . 13  |-  K  =  ( Base `  H
)
8078, 79syl6eleq 2549 . . . . . . . . . . . 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 2451 . . . . . . . . . . 11  |-  ( Base `  H )  =  (
Base `  H )
8483, 52mulgnn0cl 15754 . . . . . . . . . 10  |-  ( ( H  e.  Mnd  /\  ( N  -  k
)  e.  NN0  /\  A  e.  ( Base `  H ) )  -> 
( ( N  -  k ) E A )  e.  ( Base `  H ) )
8577, 35, 82, 84syl3anc 1219 . . . . . . . . 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 2459 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  (Scalar `  P )  =  R )
8887fveq2d 5796 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( Base `  (Scalar `  P ) )  =  ( Base `  R
) )
89 eqid 2451 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
9053, 89mgpbas 16711 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  H )
9188, 90syl6eq 2508 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( Base `  (Scalar `  P ) )  =  ( Base `  H
) )
9285, 91eleqtrrd 2542 . . . . . . . 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 17521 . . . . . . . 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 2502 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k )  .^  ( S `  A ) )  =  ( S `
 ( ( N  -  k ) E A ) ) )
9695oveq1d 6208 . . . . 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 6209 . . . 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 4479 . . 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 6209 . 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 2492 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 1370    e. wcel 1758    |-> cmpt 4451   -->wf 5515   ` cfv 5519  (class class class)co 6193   0cc0 9386    - cmin 9699   NN0cn0 10683   ...cfz 11547    _C cbc 12188   Basecbs 14285   +g cplusg 14349   .rcmulr 14350  Scalarcsca 14352   .scvsca 14353    gsumg cgsu 14490   Mndcmnd 15520  .gcmg 15525  mulGrpcmgp 16705   1rcur 16717   Ringcrg 16760   CRingccrg 16761   LModclmod 17063  AssAlgcasa 17496  algSccascl 17498  var1cv1 17748  Poly1cpl1 17749
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-ofr 6424  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-seq 11917  df-fac 12162  df-bc 12189  df-hash 12214  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-sca 14365  df-vsca 14366  df-tset 14368  df-ple 14369  df-0g 14491  df-gsum 14492  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mulg 15659  df-subg 15789  df-ghm 15856  df-cntz 15946  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-srg 16722  df-rng 16762  df-cring 16763  df-subrg 16978  df-lmod 17065  df-lss 17129  df-assa 17499  df-ascl 17501  df-psr 17538  df-mvr 17539  df-mpl 17540  df-opsr 17542  df-psr1 17752  df-vr1 17753  df-ply1 17754
This theorem is referenced by:  cpscmatgsumbin  31301
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