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Theorem deg1mhm 27505
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 20048 . . . . 5  |-  ( R  e. Domn  ->  P  e. Domn )
3 deg1mhm.b . . . . . . 7  |-  B  =  ( Base `  P
)
4 deg1mhm.z . . . . . . 7  |-  .0.  =  ( 0g `  P )
5 eqid 2438 . . . . . . 7  |-  (mulGrp `  P )  =  (mulGrp `  P )
63, 4, 5isdomn3 27502 . . . . . 6  |-  ( P  e. Domn 
<->  ( P  e.  Ring  /\  ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P )
) ) )
76simprbi 452 . . . . 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 14756 . . . 4  |-  ( ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P ) )  ->  Y  e.  Mnd )
118, 10syl 16 . . 3  |-  ( R  e. Domn  ->  Y  e.  Mnd )
12 nn0subm 16756 . . . 4  |-  NN0  e.  (SubMnd ` fld )
13 deg1mhm.n . . . . 5  |-  N  =  (flds  NN0 )
1413submmnd 14756 . . . 4  |-  ( NN0 
e.  (SubMnd ` fld )  ->  N  e. 
Mnd )
1512, 14mp1i 12 . . 3  |-  ( R  e. Domn  ->  N  e.  Mnd )
1611, 15jca 520 . 2  |-  ( R  e. Domn  ->  ( Y  e. 
Mnd  /\  N  e.  Mnd ) )
17 deg1mhm.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
1817, 1, 3deg1xrf 20006 . . . . . . 7  |-  D : B
--> RR*
19 ffn 5593 . . . . . . 7  |-  ( D : B --> RR*  ->  D  Fn  B )
2018, 19ax-mp 8 . . . . . 6  |-  D  Fn  B
21 difss 3476 . . . . . 6  |-  ( B 
\  {  .0.  }
)  C_  B
22 fnssres 5560 . . . . . 6  |-  ( ( D  Fn  B  /\  ( B  \  {  .0.  } )  C_  B )  ->  ( D  |`  ( B  \  {  .0.  }
) )  Fn  ( B  \  {  .0.  }
) )
2320, 21, 22mp2an 655 . . . . 5  |-  ( D  |`  ( B  \  {  .0.  } ) )  Fn  ( B  \  {  .0.  } )
2423a1i 11 . . . 4  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) )  Fn  ( B  \  {  .0.  }
) )
25 fvres 5747 . . . . . . 7  |-  ( x  e.  ( B  \  {  .0.  } )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  =  ( D `
 x ) )
2625adantl 454 . . . . . 6  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  =  ( D `
 x ) )
27 domnrng 16358 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e.  Ring )
2827adantr 453 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  R  e.  Ring )
29 eldifi 3471 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
3029adantl 454 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  x  e.  B )
31 eldifsni 3930 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  =/=  .0.  )
3231adantl 454 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  x  =/=  .0.  )
3317, 1, 4, 3deg1nn0cl 20013 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  B  /\  x  =/=  .0.  )  ->  ( D `  x )  e.  NN0 )
3428, 30, 32, 33syl3anc 1185 . . . . . 6  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( D `  x
)  e.  NN0 )
3526, 34eqeltrd 2512 . . . . 5  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  e.  NN0 )
3635ralrimiva 2791 . . . 4  |-  ( R  e. Domn  ->  A. x  e.  ( B  \  {  .0.  } ) ( ( D  |`  ( B  \  {  .0.  } ) ) `  x )  e.  NN0 )
37 ffnfv 5896 . . . 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 647 . . 3  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) ) : ( B  \  {  .0.  } ) --> NN0 )
39 eqid 2438 . . . . . 6  |-  (RLReg `  R )  =  (RLReg `  R )
40 eqid 2438 . . . . . 6  |-  ( .r
`  P )  =  ( .r `  P
)
4127adantr 453 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  R  e.  Ring )
4229ad2antrl 710 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  x  e.  B )
4331ad2antrl 710 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  x  =/=  .0.  )
44 simpl 445 . . . . . . 7  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  R  e. Domn )
45 eqid 2438 . . . . . . . 8  |-  (coe1 `  x
)  =  (coe1 `  x
)
4617, 1, 4, 3, 39, 45deg1ldgdomn 20019 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  B  /\  x  =/=  .0.  )  ->  (
(coe1 `  x ) `  ( D `  x ) )  e.  (RLReg `  R ) )
4744, 42, 43, 46syl3anc 1185 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( (coe1 `  x ) `  ( D `  x ) )  e.  (RLReg `  R ) )
48 eldifi 3471 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  e.  B )
4948ad2antll 711 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
y  e.  B )
50 eldifsni 3930 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  =/=  .0.  )
5150ad2antll 711 . . . . . 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 20039 . . . . 5  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( D `  (
x ( .r `  P ) y ) )  =  ( ( D `  x )  +  ( D `  y ) ) )
53 domnrng 16358 . . . . . . . . . 10  |-  ( P  e. Domn  ->  P  e.  Ring )
542, 53syl 16 . . . . . . . . 9  |-  ( R  e. Domn  ->  P  e.  Ring )
5554adantr 453 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  P  e.  Ring )
563, 40rngcl 15679 . . . . . . . 8  |-  ( ( P  e.  Ring  /\  x  e.  B  /\  y  e.  B )  ->  (
x ( .r `  P ) y )  e.  B )
5755, 42, 49, 56syl3anc 1185 . . . . . . 7  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( x ( .r
`  P ) y )  e.  B )
582adantr 453 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  P  e. Domn )
593, 40, 4domnmuln0 16360 . . . . . . . 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 1194 . . . . . . 7  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( x ( .r
`  P ) y )  =/=  .0.  )
61 eldifsn 3929 . . . . . . 7  |-  ( ( x ( .r `  P ) y )  e.  ( B  \  {  .0.  } )  <->  ( (
x ( .r `  P ) y )  e.  B  /\  (
x ( .r `  P ) y )  =/=  .0.  ) )
6257, 60, 61sylanbrc 647 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( x ( .r
`  P ) y )  e.  ( B 
\  {  .0.  }
) )
63 fvres 5747 . . . . . 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 5747 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  y
)  =  ( D `
 y ) )
6625, 65oveqan12d 6102 . . . . . 6  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) )  ->  (
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  +  ( ( D  |`  ( B  \  {  .0.  } ) ) `  y ) )  =  ( ( D `  x )  +  ( D `  y ) ) )
6766adantl 454 . . . . 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 2480 . . . 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 2800 . . 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 2438 . . . . . . . 8  |-  ( 1r
`  P )  =  ( 1r `  P
)
713, 70rngidcl 15686 . . . . . . 7  |-  ( P  e.  Ring  ->  ( 1r
`  P )  e.  B )
7254, 71syl 16 . . . . . 6  |-  ( R  e. Domn  ->  ( 1r `  P )  e.  B
)
73 domnnzr 16357 . . . . . . 7  |-  ( P  e. Domn  ->  P  e. NzRing )
7470, 4nzrnz 16333 . . . . . . 7  |-  ( P  e. NzRing  ->  ( 1r `  P )  =/=  .0.  )
752, 73, 743syl 19 . . . . . 6  |-  ( R  e. Domn  ->  ( 1r `  P )  =/=  .0.  )
76 eldifsn 3929 . . . . . 6  |-  ( ( 1r `  P )  e.  ( B  \  {  .0.  } )  <->  ( ( 1r `  P )  e.  B  /\  ( 1r
`  P )  =/= 
.0.  ) )
7772, 75, 76sylanbrc 647 . . . . 5  |-  ( R  e. Domn  ->  ( 1r `  P )  e.  ( B  \  {  .0.  } ) )
78 fvres 5747 . . . . 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 15668 . . . . . . 7  |-  ( 1r
`  P )  =  ( 0g `  (mulGrp `  P ) )
819, 80subm0 14758 . . . . . 6  |-  ( ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P ) )  ->  ( 1r `  P )  =  ( 0g `  Y ) )
828, 81syl 16 . . . . 5  |-  ( R  e. Domn  ->  ( 1r `  P )  =  ( 0g `  Y ) )
8382fveq2d 5734 . . . 4  |-  ( R  e. Domn  ->  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 1r `  P ) )  =  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 0g
`  Y ) ) )
84 domnnzr 16357 . . . . 5  |-  ( R  e. Domn  ->  R  e. NzRing )
85 eqid 2438 . . . . . . 7  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
861, 70, 85, 17mon1pid 27503 . . . . . 6  |-  ( R  e. NzRing  ->  ( ( 1r
`  P )  e.  (Monic1p `  R )  /\  ( D `  ( 1r
`  P ) )  =  0 ) )
8786simprd 451 . . . . 5  |-  ( R  e. NzRing  ->  ( D `  ( 1r `  P ) )  =  0 )
8884, 87syl 16 . . . 4  |-  ( R  e. Domn  ->  ( D `  ( 1r `  P ) )  =  0 )
8979, 83, 883eqtr3d 2478 . . 3  |-  ( R  e. Domn  ->  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 0g `  Y ) )  =  0 )
9038, 69, 893jca 1135 . 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 15656 . . . . 5  |-  B  =  ( Base `  (mulGrp `  P ) )
929, 91ressbas2 13522 . . . 4  |-  ( ( B  \  {  .0.  } )  C_  B  ->  ( B  \  {  .0.  } )  =  ( Base `  Y ) )
9321, 92ax-mp 8 . . 3  |-  ( B 
\  {  .0.  }
)  =  ( Base `  Y )
94 nn0sscn 10228 . . . 4  |-  NN0  C_  CC
95 cnfldbas 16709 . . . . 5  |-  CC  =  ( Base ` fld )
9613, 95ressbas2 13522 . . . 4  |-  ( NN0  C_  CC  ->  NN0  =  (
Base `  N )
)
9794, 96ax-mp 8 . . 3  |-  NN0  =  ( Base `  N )
98 fvex 5744 . . . . . 6  |-  ( Base `  P )  e.  _V
993, 98eqeltri 2508 . . . . 5  |-  B  e. 
_V
100 difexg 4353 . . . . 5  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
10199, 100ax-mp 8 . . . 4  |-  ( B 
\  {  .0.  }
)  e.  _V
1025, 40mgpplusg 15654 . . . . 5  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  P ) )
1039, 102ressplusg 13573 . . . 4  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  P )  =  ( +g  `  Y
) )
104101, 103ax-mp 8 . . 3  |-  ( .r
`  P )  =  ( +g  `  Y
)
105 nn0ex 10229 . . . 4  |-  NN0  e.  _V
106 cnfldadd 16710 . . . . 5  |-  +  =  ( +g  ` fld )
10713, 106ressplusg 13573 . . . 4  |-  ( NN0 
e.  _V  ->  +  =  ( +g  `  N ) )
108105, 107ax-mp 8 . . 3  |-  +  =  ( +g  `  N )
109 eqid 2438 . . 3  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
110 cnfld0 16727 . . . . 5  |-  0  =  ( 0g ` fld )
11113, 110subm0 14758 . . . 4  |-  ( NN0 
e.  (SubMnd ` fld )  ->  0  =  ( 0g `  N
) )
11212, 111ax-mp 8 . . 3  |-  0  =  ( 0g `  N )
11393, 97, 104, 108, 109, 112ismhm 14742 . 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 647 1  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) )  e.  ( Y MndHom  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   _Vcvv 2958    \ cdif 3319    C_ wss 3322   {csn 3816    |` cres 4882    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    + caddc 8995   RR*cxr 9121   NN0cn0 10223   Basecbs 13471   ↾s cress 13472   +g cplusg 13531   .rcmulr 13532   0gc0g 13725   Mndcmnd 14686   MndHom cmhm 14738  SubMndcsubmnd 14739  mulGrpcmgp 15650   Ringcrg 15662   1rcur 15664  NzRingcnzr 16330  RLRegcrlreg 16341  Domncdomn 16342  Poly1cpl1 16573  coe1cco1 16576  ℂfldccnfld 16705   deg1 cdg1 19979  Monic1pcmn1 20050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-ofr 6308  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-fzo 11138  df-seq 11326  df-hash 11621  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-0g 13729  df-gsum 13730  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-subg 14943  df-ghm 15006  df-cntz 15118  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-subrg 15868  df-lmod 15954  df-lss 16011  df-nzr 16331  df-rlreg 16345  df-domn 16346  df-ascl 16376  df-psr 16419  df-mvr 16420  df-mpl 16421  df-opsr 16427  df-psr1 16578  df-vr1 16579  df-ply1 16580  df-coe1 16583  df-cnfld 16706  df-mdeg 19980  df-deg1 19981  df-mon1 20055
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