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Theorem deg1mhm 27188
Description: Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Hypotheses
Ref Expression
deg1mhm.d  |-  D  =  ( deg1  `  R )
deg1mhm.b  |-  B  =  ( Base `  P
)
deg1mhm.p  |-  P  =  (Poly1 `  R )
deg1mhm.z  |-  .0.  =  ( 0g `  P )
deg1mhm.y  |-  Y  =  ( (mulGrp `  P
)s  ( B  \  {  .0.  } ) )
deg1mhm.n  |-  N  =  (flds  NN0 )
Assertion
Ref Expression
deg1mhm  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) )  e.  ( Y MndHom  N ) )

Proof of Theorem deg1mhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 deg1mhm.p . . . . . 6  |-  P  =  (Poly1 `  R )
21ply1domn 19906 . . . . 5  |-  ( R  e. Domn  ->  P  e. Domn )
3 deg1mhm.b . . . . . . 7  |-  B  =  ( Base `  P
)
4 deg1mhm.z . . . . . . 7  |-  .0.  =  ( 0g `  P )
5 eqid 2380 . . . . . . 7  |-  (mulGrp `  P )  =  (mulGrp `  P )
63, 4, 5isdomn3 27185 . . . . . 6  |-  ( P  e. Domn 
<->  ( P  e.  Ring  /\  ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P )
) ) )
76simprbi 451 . . . . 5  |-  ( P  e. Domn  ->  ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P
) ) )
82, 7syl 16 . . . 4  |-  ( R  e. Domn  ->  ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P
) ) )
9 deg1mhm.y . . . . 5  |-  Y  =  ( (mulGrp `  P
)s  ( B  \  {  .0.  } ) )
109submmnd 14674 . . . 4  |-  ( ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P ) )  ->  Y  e.  Mnd )
118, 10syl 16 . . 3  |-  ( R  e. Domn  ->  Y  e.  Mnd )
12 nn0subm 16670 . . . 4  |-  NN0  e.  (SubMnd ` fld )
13 deg1mhm.n . . . . 5  |-  N  =  (flds  NN0 )
1413submmnd 14674 . . . 4  |-  ( NN0 
e.  (SubMnd ` fld )  ->  N  e. 
Mnd )
1512, 14mp1i 12 . . 3  |-  ( R  e. Domn  ->  N  e.  Mnd )
1611, 15jca 519 . 2  |-  ( R  e. Domn  ->  ( Y  e. 
Mnd  /\  N  e.  Mnd ) )
17 deg1mhm.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
1817, 1, 3deg1xrf 19864 . . . . . . 7  |-  D : B
--> RR*
19 ffn 5524 . . . . . . 7  |-  ( D : B --> RR*  ->  D  Fn  B )
2018, 19ax-mp 8 . . . . . 6  |-  D  Fn  B
21 difss 3410 . . . . . 6  |-  ( B 
\  {  .0.  }
)  C_  B
22 fnssres 5491 . . . . . 6  |-  ( ( D  Fn  B  /\  ( B  \  {  .0.  } )  C_  B )  ->  ( D  |`  ( B  \  {  .0.  }
) )  Fn  ( B  \  {  .0.  }
) )
2320, 21, 22mp2an 654 . . . . 5  |-  ( D  |`  ( B  \  {  .0.  } ) )  Fn  ( B  \  {  .0.  } )
2423a1i 11 . . . 4  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) )  Fn  ( B  \  {  .0.  }
) )
25 fvres 5678 . . . . . . 7  |-  ( x  e.  ( B  \  {  .0.  } )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  =  ( D `
 x ) )
2625adantl 453 . . . . . 6  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  =  ( D `
 x ) )
27 domnrng 16276 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e.  Ring )
2827adantr 452 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  R  e.  Ring )
29 eldifi 3405 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
3029adantl 453 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  x  e.  B )
31 eldifsni 3864 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  =/=  .0.  )
3231adantl 453 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  x  =/=  .0.  )
3317, 1, 4, 3deg1nn0cl 19871 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  B  /\  x  =/=  .0.  )  ->  ( D `  x )  e.  NN0 )
3428, 30, 32, 33syl3anc 1184 . . . . . 6  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( D `  x
)  e.  NN0 )
3526, 34eqeltrd 2454 . . . . 5  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  e.  NN0 )
3635ralrimiva 2725 . . . 4  |-  ( R  e. Domn  ->  A. x  e.  ( B  \  {  .0.  } ) ( ( D  |`  ( B  \  {  .0.  } ) ) `  x )  e.  NN0 )
37 ffnfv 5826 . . . 4  |-  ( ( D  |`  ( B  \  {  .0.  } ) ) : ( B 
\  {  .0.  }
) --> NN0  <->  ( ( D  |`  ( B  \  {  .0.  } ) )  Fn  ( B  \  {  .0.  } )  /\  A. x  e.  ( B  \  {  .0.  } ) ( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  e.  NN0 )
)
3824, 36, 37sylanbrc 646 . . 3  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) ) : ( B  \  {  .0.  } ) --> NN0 )
39 eqid 2380 . . . . . 6  |-  (RLReg `  R )  =  (RLReg `  R )
40 eqid 2380 . . . . . 6  |-  ( .r
`  P )  =  ( .r `  P
)
4127adantr 452 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  R  e.  Ring )
4229ad2antrl 709 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  x  e.  B )
4331ad2antrl 709 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  x  =/=  .0.  )
44 simpl 444 . . . . . . 7  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  R  e. Domn )
45 eqid 2380 . . . . . . . 8  |-  (coe1 `  x
)  =  (coe1 `  x
)
4617, 1, 4, 3, 39, 45deg1ldgdomn 19877 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  B  /\  x  =/=  .0.  )  ->  (
(coe1 `  x ) `  ( D `  x ) )  e.  (RLReg `  R ) )
4744, 42, 43, 46syl3anc 1184 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( (coe1 `  x ) `  ( D `  x ) )  e.  (RLReg `  R ) )
48 eldifi 3405 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  e.  B )
4948ad2antll 710 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
y  e.  B )
50 eldifsni 3864 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  =/=  .0.  )
5150ad2antll 710 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
y  =/=  .0.  )
5217, 1, 39, 3, 40, 4, 41, 42, 43, 47, 49, 51deg1mul2 19897 . . . . 5  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( D `  (
x ( .r `  P ) y ) )  =  ( ( D `  x )  +  ( D `  y ) ) )
53 domnrng 16276 . . . . . . . . . 10  |-  ( P  e. Domn  ->  P  e.  Ring )
542, 53syl 16 . . . . . . . . 9  |-  ( R  e. Domn  ->  P  e.  Ring )
5554adantr 452 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  P  e.  Ring )
563, 40rngcl 15597 . . . . . . . 8  |-  ( ( P  e.  Ring  /\  x  e.  B  /\  y  e.  B )  ->  (
x ( .r `  P ) y )  e.  B )
5755, 42, 49, 56syl3anc 1184 . . . . . . 7  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( x ( .r
`  P ) y )  e.  B )
582adantr 452 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  P  e. Domn )
593, 40, 4domnmuln0 16278 . . . . . . . 8  |-  ( ( P  e. Domn  /\  (
x  e.  B  /\  x  =/=  .0.  )  /\  ( y  e.  B  /\  y  =/=  .0.  ) )  ->  (
x ( .r `  P ) y )  =/=  .0.  )
6058, 42, 43, 49, 51, 59syl122anc 1193 . . . . . . 7  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( x ( .r
`  P ) y )  =/=  .0.  )
61 eldifsn 3863 . . . . . . 7  |-  ( ( x ( .r `  P ) y )  e.  ( B  \  {  .0.  } )  <->  ( (
x ( .r `  P ) y )  e.  B  /\  (
x ( .r `  P ) y )  =/=  .0.  ) )
6257, 60, 61sylanbrc 646 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( x ( .r
`  P ) y )  e.  ( B 
\  {  .0.  }
) )
63 fvres 5678 . . . . . 6  |-  ( ( x ( .r `  P ) y )  e.  ( B  \  {  .0.  } )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  (
x ( .r `  P ) y ) )  =  ( D `
 ( x ( .r `  P ) y ) ) )
6462, 63syl 16 . . . . 5  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  (
x ( .r `  P ) y ) )  =  ( D `
 ( x ( .r `  P ) y ) ) )
65 fvres 5678 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  y
)  =  ( D `
 y ) )
6625, 65oveqan12d 6032 . . . . . 6  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) )  ->  (
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  +  ( ( D  |`  ( B  \  {  .0.  } ) ) `  y ) )  =  ( ( D `  x )  +  ( D `  y ) ) )
6766adantl 453 . . . . 5  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( ( ( D  |`  ( B  \  {  .0.  } ) ) `  x )  +  ( ( D  |`  ( B  \  {  .0.  }
) ) `  y
) )  =  ( ( D `  x
)  +  ( D `
 y ) ) )
6852, 64, 673eqtr4d 2422 . . . 4  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  (
x ( .r `  P ) y ) )  =  ( ( ( D  |`  ( B  \  {  .0.  }
) ) `  x
)  +  ( ( D  |`  ( B  \  {  .0.  } ) ) `  y ) ) )
6968ralrimivva 2734 . . 3  |-  ( R  e. Domn  ->  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( x ( .r `  P
) y ) )  =  ( ( ( D  |`  ( B  \  {  .0.  } ) ) `  x )  +  ( ( D  |`  ( B  \  {  .0.  } ) ) `  y ) ) )
70 eqid 2380 . . . . . . . 8  |-  ( 1r
`  P )  =  ( 1r `  P
)
713, 70rngidcl 15604 . . . . . . 7  |-  ( P  e.  Ring  ->  ( 1r
`  P )  e.  B )
7254, 71syl 16 . . . . . 6  |-  ( R  e. Domn  ->  ( 1r `  P )  e.  B
)
73 domnnzr 16275 . . . . . . 7  |-  ( P  e. Domn  ->  P  e. NzRing )
7470, 4nzrnz 16251 . . . . . . 7  |-  ( P  e. NzRing  ->  ( 1r `  P )  =/=  .0.  )
752, 73, 743syl 19 . . . . . 6  |-  ( R  e. Domn  ->  ( 1r `  P )  =/=  .0.  )
76 eldifsn 3863 . . . . . 6  |-  ( ( 1r `  P )  e.  ( B  \  {  .0.  } )  <->  ( ( 1r `  P )  e.  B  /\  ( 1r
`  P )  =/= 
.0.  ) )
7772, 75, 76sylanbrc 646 . . . . 5  |-  ( R  e. Domn  ->  ( 1r `  P )  e.  ( B  \  {  .0.  } ) )
78 fvres 5678 . . . . 5  |-  ( ( 1r `  P )  e.  ( B  \  {  .0.  } )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 1r `  P ) )  =  ( D `  ( 1r `  P ) ) )
7977, 78syl 16 . . . 4  |-  ( R  e. Domn  ->  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 1r `  P ) )  =  ( D `
 ( 1r `  P ) ) )
805, 70rngidval 15586 . . . . . . 7  |-  ( 1r
`  P )  =  ( 0g `  (mulGrp `  P ) )
819, 80subm0 14676 . . . . . 6  |-  ( ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P ) )  ->  ( 1r `  P )  =  ( 0g `  Y ) )
828, 81syl 16 . . . . 5  |-  ( R  e. Domn  ->  ( 1r `  P )  =  ( 0g `  Y ) )
8382fveq2d 5665 . . . 4  |-  ( R  e. Domn  ->  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 1r `  P ) )  =  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 0g
`  Y ) ) )
84 domnnzr 16275 . . . . 5  |-  ( R  e. Domn  ->  R  e. NzRing )
85 eqid 2380 . . . . . . 7  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
861, 70, 85, 17mon1pid 27186 . . . . . 6  |-  ( R  e. NzRing  ->  ( ( 1r
`  P )  e.  (Monic1p `  R )  /\  ( D `  ( 1r
`  P ) )  =  0 ) )
8786simprd 450 . . . . 5  |-  ( R  e. NzRing  ->  ( D `  ( 1r `  P ) )  =  0 )
8884, 87syl 16 . . . 4  |-  ( R  e. Domn  ->  ( D `  ( 1r `  P ) )  =  0 )
8979, 83, 883eqtr3d 2420 . . 3  |-  ( R  e. Domn  ->  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 0g `  Y ) )  =  0 )
9038, 69, 893jca 1134 . 2  |-  ( R  e. Domn  ->  ( ( D  |`  ( B  \  {  .0.  } ) ) : ( B  \  {  .0.  } ) --> NN0  /\  A. x  e.  ( B 
\  {  .0.  }
) A. y  e.  ( B  \  {  .0.  } ) ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( x ( .r `  P
) y ) )  =  ( ( ( D  |`  ( B  \  {  .0.  } ) ) `  x )  +  ( ( D  |`  ( B  \  {  .0.  } ) ) `  y ) )  /\  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 0g `  Y ) )  =  0 ) )
915, 3mgpbas 15574 . . . . 5  |-  B  =  ( Base `  (mulGrp `  P ) )
929, 91ressbas2 13440 . . . 4  |-  ( ( B  \  {  .0.  } )  C_  B  ->  ( B  \  {  .0.  } )  =  ( Base `  Y ) )
9321, 92ax-mp 8 . . 3  |-  ( B 
\  {  .0.  }
)  =  ( Base `  Y )
94 nn0sscn 10151 . . . 4  |-  NN0  C_  CC
95 cnfldbas 16623 . . . . 5  |-  CC  =  ( Base ` fld )
9613, 95ressbas2 13440 . . . 4  |-  ( NN0  C_  CC  ->  NN0  =  (
Base `  N )
)
9794, 96ax-mp 8 . . 3  |-  NN0  =  ( Base `  N )
98 fvex 5675 . . . . . 6  |-  ( Base `  P )  e.  _V
993, 98eqeltri 2450 . . . . 5  |-  B  e. 
_V
100 difexg 4285 . . . . 5  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
10199, 100ax-mp 8 . . . 4  |-  ( B 
\  {  .0.  }
)  e.  _V
1025, 40mgpplusg 15572 . . . . 5  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  P ) )
1039, 102ressplusg 13491 . . . 4  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  P )  =  ( +g  `  Y
) )
104101, 103ax-mp 8 . . 3  |-  ( .r
`  P )  =  ( +g  `  Y
)
105 nn0ex 10152 . . . 4  |-  NN0  e.  _V
106 cnfldadd 16624 . . . . 5  |-  +  =  ( +g  ` fld )
10713, 106ressplusg 13491 . . . 4  |-  ( NN0 
e.  _V  ->  +  =  ( +g  `  N ) )
108105, 107ax-mp 8 . . 3  |-  +  =  ( +g  `  N )
109 eqid 2380 . . 3  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
110 cnfld0 16641 . . . . 5  |-  0  =  ( 0g ` fld )
11113, 110subm0 14676 . . . 4  |-  ( NN0 
e.  (SubMnd ` fld )  ->  0  =  ( 0g `  N
) )
11212, 111ax-mp 8 . . 3  |-  0  =  ( 0g `  N )
11393, 97, 104, 108, 109, 112ismhm 14660 . 2  |-  ( ( D  |`  ( B  \  {  .0.  } ) )  e.  ( Y MndHom  N )  <->  ( ( Y  e.  Mnd  /\  N  e.  Mnd )  /\  (
( D  |`  ( B  \  {  .0.  }
) ) : ( B  \  {  .0.  } ) --> NN0  /\  A. x  e.  ( B  \  {  .0.  } ) A. y  e.  ( B  \  {  .0.  } ) ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( x ( .r `  P
) y ) )  =  ( ( ( D  |`  ( B  \  {  .0.  } ) ) `  x )  +  ( ( D  |`  ( B  \  {  .0.  } ) ) `  y ) )  /\  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 0g `  Y ) )  =  0 ) ) )
11416, 90, 113sylanbrc 646 1  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) )  e.  ( Y MndHom  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642   _Vcvv 2892    \ cdif 3253    C_ wss 3256   {csn 3750    |` cres 4813    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916    + caddc 8919   RR*cxr 9045   NN0cn0 10146   Basecbs 13389   ↾s cress 13390   +g cplusg 13449   .rcmulr 13450   0gc0g 13643   Mndcmnd 14604   MndHom cmhm 14656  SubMndcsubmnd 14657  mulGrpcmgp 15568   Ringcrg 15580   1rcur 15582  NzRingcnzr 16248  RLRegcrlreg 16259  Domncdomn 16260  Poly1cpl1 16491  coe1cco1 16494  ℂfldccnfld 16619   deg1 cdg1 19837  Monic1pcmn1 19908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-ofr 6238  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-fzo 11059  df-seq 11244  df-hash 11539  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-0g 13647  df-gsum 13648  df-mre 13731  df-mrc 13732  df-acs 13734  df-mnd 14610  df-mhm 14658  df-submnd 14659  df-grp 14732  df-minusg 14733  df-sbg 14734  df-mulg 14735  df-subg 14861  df-ghm 14924  df-cntz 15036  df-cmn 15334  df-abl 15335  df-mgp 15569  df-rng 15583  df-cring 15584  df-ur 15585  df-subrg 15786  df-lmod 15872  df-lss 15929  df-nzr 16249  df-rlreg 16263  df-domn 16264  df-ascl 16294  df-psr 16337  df-mvr 16338  df-mpl 16339  df-opsr 16345  df-psr1 16496  df-vr1 16497  df-ply1 16498  df-coe1 16501  df-cnfld 16620  df-mdeg 19838  df-deg1 19839  df-mon1 19913
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