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Theorem lply1binomsc 18544
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 2454 . . . . . 6  |-  (Scalar `  P )  =  (Scalar `  P )
3 crngring 17404 . . . . . . . 8  |-  ( R  e.  CRing  ->  R  e.  Ring )
4 cply1binom.p . . . . . . . . 9  |-  P  =  (Poly1 `  R )
54ply1ring 18484 . . . . . . . 8  |-  ( R  e.  Ring  ->  P  e. 
Ring )
63, 5syl 16 . . . . . . 7  |-  ( R  e.  CRing  ->  P  e.  Ring )
763ad2ant1 1015 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  P  e.  Ring )
84ply1lmod 18488 . . . . . . . 8  |-  ( R  e.  Ring  ->  P  e. 
LMod )
93, 8syl 16 . . . . . . 7  |-  ( R  e.  CRing  ->  P  e.  LMod )
1093ad2ant1 1015 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  P  e.  LMod )
11 eqid 2454 . . . . . 6  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
12 eqid 2454 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
131, 2, 7, 10, 11, 12asclf 18181 . . . . 5  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  S : ( Base `  (Scalar `  P ) ) --> (
Base `  P )
)
14 lply1binomsc.k . . . . . . 7  |-  K  =  ( Base `  R
)
154ply1sca 18489 . . . . . . . . 9  |-  ( R  e.  CRing  ->  R  =  (Scalar `  P ) )
16153ad2ant1 1015 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  R  =  (Scalar `  P )
)
1716fveq2d 5852 . . . . . . 7  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( Base `  R )  =  ( Base `  (Scalar `  P ) ) )
1814, 17syl5eq 2507 . . . . . 6  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  K  =  ( Base `  (Scalar `  P ) ) )
1918feq2d 5700 . . . . 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 996 . . . 4  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  A  e.  K )
2220, 21ffvelrnd 6008 . . 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 18543 . . 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 1271 . 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 18433 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  P  e. AssAlg )
32313ad2ant1 1015 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  P  e. AssAlg )
3332adantr 463 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  P  e. AssAlg )
34 fznn0sub 11720 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  ( N  -  k )  e.  NN0 )
3534adantl 464 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( N  -  k )  e.  NN0 )
3615fveq2d 5852 . . . . . . . . . . . . . 14  |-  ( R  e.  CRing  ->  ( Base `  R )  =  (
Base `  (Scalar `  P
) ) )
3714, 36syl5eq 2507 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  K  =  ( Base `  (Scalar `  P
) ) )
3837eleq2d 2524 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  ( A  e.  K  <->  A  e.  ( Base `  (Scalar `  P
) ) ) )
3938biimpa 482 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  A  e.  K )  ->  A  e.  ( Base `  (Scalar `  P ) ) )
40393adant2 1013 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  A  e.  ( Base `  (Scalar `  P ) ) )
4140adantr 463 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  A  e.  (
Base `  (Scalar `  P
) ) )
42 eqid 2454 . . . . . . . . . . . . 13  |-  ( 1r
`  P )  =  ( 1r `  P
)
4312, 42ringidcl 17414 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  ( 1r
`  P )  e.  ( Base `  P
) )
446, 43syl 16 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  ( 1r `  P )  e.  (
Base `  P )
)
45443ad2ant1 1015 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  ( 1r `  P )  e.  ( Base `  P
) )
4645adantr 463 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( 1r `  P )  e.  (
Base `  P )
)
47 eqid 2454 . . . . . . . . . 10  |-  ( .s
`  P )  =  ( .s `  P
)
48 eqid 2454 . . . . . . . . . 10  |-  (mulGrp `  (Scalar `  P ) )  =  (mulGrp `  (Scalar `  P ) )
49 eqid 2454 . . . . . . . . . 10  |-  (.g `  (mulGrp `  (Scalar `  P )
) )  =  (.g `  (mulGrp `  (Scalar `  P
) ) )
5012, 2, 11, 47, 48, 49, 27, 28assamulgscm 18194 . . . . . . . . 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 1228 . . . . . . . 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 5852 . . . . . . . . . . . . . . . 16  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  =  (mulGrp `  (Scalar `  P ) ) )
5553, 54syl5eq 2507 . . . . . . . . . . . . . . 15  |-  ( R  e.  CRing  ->  H  =  (mulGrp `  (Scalar `  P
) ) )
5655fveq2d 5852 . . . . . . . . . . . . . 14  |-  ( R  e.  CRing  ->  (.g `  H
)  =  (.g `  (mulGrp `  (Scalar `  P )
) ) )
5752, 56syl5eq 2507 . . . . . . . . . . . . 13  |-  ( R  e.  CRing  ->  E  =  (.g
`  (mulGrp `  (Scalar `  P
) ) ) )
58573ad2ant1 1015 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  E  =  (.g `  (mulGrp `  (Scalar `  P ) ) ) )
5958adantr 463 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  E  =  (.g `  (mulGrp `  (Scalar `  P
) ) ) )
6059eqcomd 2462 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  (.g `  (mulGrp `  (Scalar `  P ) ) )  =  E )
6160oveqd 6287 . . . . . . . . 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 ) )
6227ringmgp 17399 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  G  e. 
Mnd )
636, 62syl 16 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  G  e.  Mnd )
64633ad2ant1 1015 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  G  e.  Mnd )
6527, 12mgpbas 17342 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  G )
6627, 42ringidval 17350 . . . . . . . . . . 11  |-  ( 1r
`  P )  =  ( 0g `  G
)
6765, 28, 66mulgnn0z 16361 . . . . . . . . . 10  |-  ( ( G  e.  Mnd  /\  ( N  -  k
)  e.  NN0 )  ->  ( ( N  -  k )  .^  ( 1r `  P ) )  =  ( 1r `  P ) )
6864, 34, 67syl2an 475 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k )  .^  ( 1r `  P ) )  =  ( 1r
`  P ) )
6961, 68oveq12d 6288 . . . . . . . 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 2495 . . . . . . 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 18179 . . . . . . . . 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 6286 . . . . . . 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
) ) ) )
7453ringmgp 17399 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  H  e. 
Mnd )
753, 74syl 16 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  H  e.  Mnd )
76753ad2ant1 1015 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  H  e.  Mnd )
7776adantr 463 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  H  e.  Mnd )
78 simpr 459 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  A  e.  K )  ->  A  e.  K )
7953, 14mgpbas 17342 . . . . . . . . . . . . 13  |-  K  =  ( Base `  H
)
8078, 79syl6eleq 2552 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  A  e.  K )  ->  A  e.  ( Base `  H
) )
81803adant2 1013 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  ->  A  e.  ( Base `  H
) )
8281adantr 463 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  A  e.  (
Base `  H )
)
83 eqid 2454 . . . . . . . . . . 11  |-  ( Base `  H )  =  (
Base `  H )
8483, 52mulgnn0cl 16357 . . . . . . . . . 10  |-  ( ( H  e.  Mnd  /\  ( N  -  k
)  e.  NN0  /\  A  e.  ( Base `  H ) )  -> 
( ( N  -  k ) E A )  e.  ( Base `  H ) )
8577, 35, 82, 84syl3anc 1226 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k ) E A )  e.  (
Base `  H )
)
8616adantr 463 . . . . . . . . . . . 12  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  R  =  (Scalar `  P ) )
8786eqcomd 2462 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  (Scalar `  P )  =  R )
8887fveq2d 5852 . . . . . . . . . 10  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( Base `  (Scalar `  P ) )  =  ( Base `  R
) )
89 eqid 2454 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
9053, 89mgpbas 17342 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  H )
9188, 90syl6eq 2511 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( Base `  (Scalar `  P ) )  =  ( Base `  H
) )
9285, 91eleqtrrd 2545 . . . . . . . 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 18179 . . . . . . . 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 2505 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  N  e.  NN0  /\  A  e.  K )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  -  k )  .^  ( S `  A ) )  =  ( S `
 ( ( N  -  k ) E A ) ) )
9695oveq1d 6285 . . . . 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 6286 . . . 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 4525 . . 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 6286 . 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 2495 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270   0cc0 9481    - cmin 9796   NN0cn0 10791   ...cfz 11675    _C cbc 12362   Basecbs 14716   +g cplusg 14784   .rcmulr 14785  Scalarcsca 14787   .scvsca 14788    gsumg cgsu 14930   Mndcmnd 16118  .gcmg 16255  mulGrpcmgp 17336   1rcur 17348   Ringcrg 17393   CRingccrg 17394   LModclmod 17707  AssAlgcasa 18153  algSccascl 18155  var1cv1 18410  Poly1cpl1 18411
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-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-div 10203  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-rp 11222  df-fz 11676  df-fzo 11800  df-seq 12090  df-fac 12336  df-bc 12363  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-cring 17396  df-subrg 17622  df-lmod 17709  df-lss 17774  df-assa 18156  df-ascl 18158  df-psr 18200  df-mvr 18201  df-mpl 18202  df-opsr 18204  df-psr1 18414  df-vr1 18415  df-ply1 18416
This theorem is referenced by:  chpscmatgsumbin  19512
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