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Theorem deg1mhm 27402
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 20007 . . . . 5  |-  ( R  e. Domn  ->  P  e. Domn )
3 deg1mhm.b . . . . . . 7  |-  B  =  ( Base `  P
)
4 deg1mhm.z . . . . . . 7  |-  .0.  =  ( 0g `  P )
5 eqid 2412 . . . . . . 7  |-  (mulGrp `  P )  =  (mulGrp `  P )
63, 4, 5isdomn3 27399 . . . . . 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 14717 . . . 4  |-  ( ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P ) )  ->  Y  e.  Mnd )
118, 10syl 16 . . 3  |-  ( R  e. Domn  ->  Y  e.  Mnd )
12 nn0subm 16717 . . . 4  |-  NN0  e.  (SubMnd ` fld )
13 deg1mhm.n . . . . 5  |-  N  =  (flds  NN0 )
1413submmnd 14717 . . . 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 19965 . . . . . . 7  |-  D : B
--> RR*
19 ffn 5558 . . . . . . 7  |-  ( D : B --> RR*  ->  D  Fn  B )
2018, 19ax-mp 8 . . . . . 6  |-  D  Fn  B
21 difss 3442 . . . . . 6  |-  ( B 
\  {  .0.  }
)  C_  B
22 fnssres 5525 . . . . . 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 5712 . . . . . . 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 16319 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e.  Ring )
2827adantr 452 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  R  e.  Ring )
29 eldifi 3437 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
3029adantl 453 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  x  e.  B )
31 eldifsni 3896 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  =/=  .0.  )
3231adantl 453 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  x  =/=  .0.  )
3317, 1, 4, 3deg1nn0cl 19972 . . . . . . 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 2486 . . . . 5  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  e.  NN0 )
3635ralrimiva 2757 . . . 4  |-  ( R  e. Domn  ->  A. x  e.  ( B  \  {  .0.  } ) ( ( D  |`  ( B  \  {  .0.  } ) ) `  x )  e.  NN0 )
37 ffnfv 5861 . . . 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 2412 . . . . . 6  |-  (RLReg `  R )  =  (RLReg `  R )
40 eqid 2412 . . . . . 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 2412 . . . . . . . 8  |-  (coe1 `  x
)  =  (coe1 `  x
)
4617, 1, 4, 3, 39, 45deg1ldgdomn 19978 . . . . . . 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 3437 . . . . . . 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 3896 . . . . . . 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 19998 . . . . 5  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( D `  (
x ( .r `  P ) y ) )  =  ( ( D `  x )  +  ( D `  y ) ) )
53 domnrng 16319 . . . . . . . . . 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 15640 . . . . . . . 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 16321 . . . . . . . 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 3895 . . . . . . 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 5712 . . . . . 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 5712 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  y
)  =  ( D `
 y ) )
6625, 65oveqan12d 6067 . . . . . 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 2454 . . . 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 2766 . . 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 2412 . . . . . . . 8  |-  ( 1r
`  P )  =  ( 1r `  P
)
713, 70rngidcl 15647 . . . . . . 7  |-  ( P  e.  Ring  ->  ( 1r
`  P )  e.  B )
7254, 71syl 16 . . . . . 6  |-  ( R  e. Domn  ->  ( 1r `  P )  e.  B
)
73 domnnzr 16318 . . . . . . 7  |-  ( P  e. Domn  ->  P  e. NzRing )
7470, 4nzrnz 16294 . . . . . . 7  |-  ( P  e. NzRing  ->  ( 1r `  P )  =/=  .0.  )
752, 73, 743syl 19 . . . . . 6  |-  ( R  e. Domn  ->  ( 1r `  P )  =/=  .0.  )
76 eldifsn 3895 . . . . . 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 5712 . . . . 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 15629 . . . . . . 7  |-  ( 1r
`  P )  =  ( 0g `  (mulGrp `  P ) )
819, 80subm0 14719 . . . . . 6  |-  ( ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P ) )  ->  ( 1r `  P )  =  ( 0g `  Y ) )
828, 81syl 16 . . . . 5  |-  ( R  e. Domn  ->  ( 1r `  P )  =  ( 0g `  Y ) )
8382fveq2d 5699 . . . 4  |-  ( R  e. Domn  ->  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 1r `  P ) )  =  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 0g
`  Y ) ) )
84 domnnzr 16318 . . . . 5  |-  ( R  e. Domn  ->  R  e. NzRing )
85 eqid 2412 . . . . . . 7  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
861, 70, 85, 17mon1pid 27400 . . . . . 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 2452 . . 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 15617 . . . . 5  |-  B  =  ( Base `  (mulGrp `  P ) )
929, 91ressbas2 13483 . . . 4  |-  ( ( B  \  {  .0.  } )  C_  B  ->  ( B  \  {  .0.  } )  =  ( Base `  Y ) )
9321, 92ax-mp 8 . . 3  |-  ( B 
\  {  .0.  }
)  =  ( Base `  Y )
94 nn0sscn 10190 . . . 4  |-  NN0  C_  CC
95 cnfldbas 16670 . . . . 5  |-  CC  =  ( Base ` fld )
9613, 95ressbas2 13483 . . . 4  |-  ( NN0  C_  CC  ->  NN0  =  (
Base `  N )
)
9794, 96ax-mp 8 . . 3  |-  NN0  =  ( Base `  N )
98 fvex 5709 . . . . . 6  |-  ( Base `  P )  e.  _V
993, 98eqeltri 2482 . . . . 5  |-  B  e. 
_V
100 difexg 4319 . . . . 5  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
10199, 100ax-mp 8 . . . 4  |-  ( B 
\  {  .0.  }
)  e.  _V
1025, 40mgpplusg 15615 . . . . 5  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  P ) )
1039, 102ressplusg 13534 . . . 4  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  P )  =  ( +g  `  Y
) )
104101, 103ax-mp 8 . . 3  |-  ( .r
`  P )  =  ( +g  `  Y
)
105 nn0ex 10191 . . . 4  |-  NN0  e.  _V
106 cnfldadd 16671 . . . . 5  |-  +  =  ( +g  ` fld )
10713, 106ressplusg 13534 . . . 4  |-  ( NN0 
e.  _V  ->  +  =  ( +g  `  N ) )
108105, 107ax-mp 8 . . 3  |-  +  =  ( +g  `  N )
109 eqid 2412 . . 3  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
110 cnfld0 16688 . . . . 5  |-  0  =  ( 0g ` fld )
11113, 110subm0 14719 . . . 4  |-  ( NN0 
e.  (SubMnd ` fld )  ->  0  =  ( 0g `  N
) )
11212, 111ax-mp 8 . . 3  |-  0  =  ( 0g `  N )
11393, 97, 104, 108, 109, 112ismhm 14703 . 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 1721    =/= wne 2575   A.wral 2674   _Vcvv 2924    \ cdif 3285    C_ wss 3288   {csn 3782    |` cres 4847    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954    + caddc 8957   RR*cxr 9083   NN0cn0 10185   Basecbs 13432   ↾s cress 13433   +g cplusg 13492   .rcmulr 13493   0gc0g 13686   Mndcmnd 14647   MndHom cmhm 14699  SubMndcsubmnd 14700  mulGrpcmgp 15611   Ringcrg 15623   1rcur 15625  NzRingcnzr 16291  RLRegcrlreg 16302  Domncdomn 16303  Poly1cpl1 16534  coe1cco1 16537  ℂfldccnfld 16666   deg1 cdg1 19938  Monic1pcmn1 20009
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-ofr 6273  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-fzo 11099  df-seq 11287  df-hash 11582  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-0g 13690  df-gsum 13691  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-mhm 14701  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-mulg 14778  df-subg 14904  df-ghm 14967  df-cntz 15079  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-cring 15627  df-ur 15628  df-subrg 15829  df-lmod 15915  df-lss 15972  df-nzr 16292  df-rlreg 16306  df-domn 16307  df-ascl 16337  df-psr 16380  df-mvr 16381  df-mpl 16382  df-opsr 16388  df-psr1 16539  df-vr1 16540  df-ply1 16541  df-coe1 16544  df-cnfld 16667  df-mdeg 19939  df-deg1 19940  df-mon1 20014
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