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Theorem deg1mhm 31143
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 22502 . . . . 5  |-  ( R  e. Domn  ->  P  e. Domn )
3 deg1mhm.b . . . . . . 7  |-  B  =  ( Base `  P
)
4 deg1mhm.z . . . . . . 7  |-  .0.  =  ( 0g `  P )
5 eqid 2443 . . . . . . 7  |-  (mulGrp `  P )  =  (mulGrp `  P )
63, 4, 5isdomn3 31140 . . . . . 6  |-  ( P  e. Domn 
<->  ( P  e.  Ring  /\  ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P )
) ) )
76simprbi 464 . . . . 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 15964 . . . 4  |-  ( ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P ) )  ->  Y  e.  Mnd )
118, 10syl 16 . . 3  |-  ( R  e. Domn  ->  Y  e.  Mnd )
12 nn0subm 18452 . . . 4  |-  NN0  e.  (SubMnd ` fld )
13 deg1mhm.n . . . . 5  |-  N  =  (flds  NN0 )
1413submmnd 15964 . . . 4  |-  ( NN0 
e.  (SubMnd ` fld )  ->  N  e. 
Mnd )
1512, 14mp1i 12 . . 3  |-  ( R  e. Domn  ->  N  e.  Mnd )
1611, 15jca 532 . 2  |-  ( R  e. Domn  ->  ( Y  e. 
Mnd  /\  N  e.  Mnd ) )
17 deg1mhm.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
1817, 1, 3deg1xrf 22459 . . . . . . 7  |-  D : B
--> RR*
19 ffn 5721 . . . . . . 7  |-  ( D : B --> RR*  ->  D  Fn  B )
2018, 19ax-mp 5 . . . . . 6  |-  D  Fn  B
21 difss 3616 . . . . . 6  |-  ( B 
\  {  .0.  }
)  C_  B
22 fnssres 5684 . . . . . 6  |-  ( ( D  Fn  B  /\  ( B  \  {  .0.  } )  C_  B )  ->  ( D  |`  ( B  \  {  .0.  }
) )  Fn  ( B  \  {  .0.  }
) )
2320, 21, 22mp2an 672 . . . . 5  |-  ( D  |`  ( B  \  {  .0.  } ) )  Fn  ( B  \  {  .0.  } )
2423a1i 11 . . . 4  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) )  Fn  ( B  \  {  .0.  }
) )
25 fvres 5870 . . . . . . 7  |-  ( x  e.  ( B  \  {  .0.  } )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  =  ( D `
 x ) )
2625adantl 466 . . . . . 6  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  =  ( D `
 x ) )
27 domnring 17924 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e.  Ring )
2827adantr 465 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  R  e.  Ring )
29 eldifi 3611 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
3029adantl 466 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  x  e.  B )
31 eldifsni 4141 . . . . . . . 8  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  =/=  .0.  )
3231adantl 466 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  ->  x  =/=  .0.  )
3317, 1, 4, 3deg1nn0cl 22466 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  B  /\  x  =/=  .0.  )  ->  ( D `  x )  e.  NN0 )
3428, 30, 32, 33syl3anc 1229 . . . . . 6  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( D `  x
)  e.  NN0 )
3526, 34eqeltrd 2531 . . . . 5  |-  ( ( R  e. Domn  /\  x  e.  ( B  \  {  .0.  } ) )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  e.  NN0 )
3635ralrimiva 2857 . . . 4  |-  ( R  e. Domn  ->  A. x  e.  ( B  \  {  .0.  } ) ( ( D  |`  ( B  \  {  .0.  } ) ) `  x )  e.  NN0 )
37 ffnfv 6042 . . . 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 664 . . 3  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) ) : ( B  \  {  .0.  } ) --> NN0 )
39 eqid 2443 . . . . . 6  |-  (RLReg `  R )  =  (RLReg `  R )
40 eqid 2443 . . . . . 6  |-  ( .r
`  P )  =  ( .r `  P
)
4127adantr 465 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  R  e.  Ring )
4229ad2antrl 727 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  x  e.  B )
4331ad2antrl 727 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  x  =/=  .0.  )
44 simpl 457 . . . . . . 7  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  R  e. Domn )
45 eqid 2443 . . . . . . . 8  |-  (coe1 `  x
)  =  (coe1 `  x
)
4617, 1, 4, 3, 39, 45deg1ldgdomn 22472 . . . . . . 7  |-  ( ( R  e. Domn  /\  x  e.  B  /\  x  =/=  .0.  )  ->  (
(coe1 `  x ) `  ( D `  x ) )  e.  (RLReg `  R ) )
4744, 42, 43, 46syl3anc 1229 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( (coe1 `  x ) `  ( D `  x ) )  e.  (RLReg `  R ) )
48 eldifi 3611 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  e.  B )
4948ad2antll 728 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
y  e.  B )
50 eldifsni 4141 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  =/=  .0.  )
5150ad2antll 728 . . . . . 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 22493 . . . . 5  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( D `  (
x ( .r `  P ) y ) )  =  ( ( D `  x )  +  ( D `  y ) ) )
53 domnring 17924 . . . . . . . . . 10  |-  ( P  e. Domn  ->  P  e.  Ring )
542, 53syl 16 . . . . . . . . 9  |-  ( R  e. Domn  ->  P  e.  Ring )
5554adantr 465 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  P  e.  Ring )
563, 40ringcl 17191 . . . . . . . 8  |-  ( ( P  e.  Ring  /\  x  e.  B  /\  y  e.  B )  ->  (
x ( .r `  P ) y )  e.  B )
5755, 42, 49, 56syl3anc 1229 . . . . . . 7  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( x ( .r
`  P ) y )  e.  B )
582adantr 465 . . . . . . . 8  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  ->  P  e. Domn )
593, 40, 4domnmuln0 17926 . . . . . . . 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 1238 . . . . . . 7  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( x ( .r
`  P ) y )  =/=  .0.  )
61 eldifsn 4140 . . . . . . 7  |-  ( ( x ( .r `  P ) y )  e.  ( B  \  {  .0.  } )  <->  ( (
x ( .r `  P ) y )  e.  B  /\  (
x ( .r `  P ) y )  =/=  .0.  ) )
6257, 60, 61sylanbrc 664 . . . . . 6  |-  ( ( R  e. Domn  /\  (
x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) ) )  -> 
( x ( .r
`  P ) y )  e.  ( B 
\  {  .0.  }
) )
63 fvres 5870 . . . . . 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 5870 . . . . . . 7  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
( ( D  |`  ( B  \  {  .0.  } ) ) `  y
)  =  ( D `
 y ) )
6625, 65oveqan12d 6300 . . . . . 6  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } ) )  ->  (
( ( D  |`  ( B  \  {  .0.  } ) ) `  x
)  +  ( ( D  |`  ( B  \  {  .0.  } ) ) `  y ) )  =  ( ( D `  x )  +  ( D `  y ) ) )
6766adantl 466 . . . . 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 2494 . . . 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 2864 . . 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 2443 . . . . . . . 8  |-  ( 1r
`  P )  =  ( 1r `  P
)
713, 70ringidcl 17198 . . . . . . 7  |-  ( P  e.  Ring  ->  ( 1r
`  P )  e.  B )
7254, 71syl 16 . . . . . 6  |-  ( R  e. Domn  ->  ( 1r `  P )  e.  B
)
73 domnnzr 17923 . . . . . . 7  |-  ( P  e. Domn  ->  P  e. NzRing )
7470, 4nzrnz 17887 . . . . . . 7  |-  ( P  e. NzRing  ->  ( 1r `  P )  =/=  .0.  )
752, 73, 743syl 20 . . . . . 6  |-  ( R  e. Domn  ->  ( 1r `  P )  =/=  .0.  )
76 eldifsn 4140 . . . . . 6  |-  ( ( 1r `  P )  e.  ( B  \  {  .0.  } )  <->  ( ( 1r `  P )  e.  B  /\  ( 1r
`  P )  =/= 
.0.  ) )
7772, 75, 76sylanbrc 664 . . . . 5  |-  ( R  e. Domn  ->  ( 1r `  P )  e.  ( B  \  {  .0.  } ) )
78 fvres 5870 . . . . 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, 70ringidval 17134 . . . . . . 7  |-  ( 1r
`  P )  =  ( 0g `  (mulGrp `  P ) )
819, 80subm0 15966 . . . . . 6  |-  ( ( B  \  {  .0.  } )  e.  (SubMnd `  (mulGrp `  P ) )  ->  ( 1r `  P )  =  ( 0g `  Y ) )
828, 81syl 16 . . . . 5  |-  ( R  e. Domn  ->  ( 1r `  P )  =  ( 0g `  Y ) )
8382fveq2d 5860 . . . 4  |-  ( R  e. Domn  ->  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 1r `  P ) )  =  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 0g
`  Y ) ) )
84 domnnzr 17923 . . . . 5  |-  ( R  e. Domn  ->  R  e. NzRing )
85 eqid 2443 . . . . . . 7  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
861, 70, 85, 17mon1pid 31141 . . . . . 6  |-  ( R  e. NzRing  ->  ( ( 1r
`  P )  e.  (Monic1p `  R )  /\  ( D `  ( 1r
`  P ) )  =  0 ) )
8786simprd 463 . . . . 5  |-  ( R  e. NzRing  ->  ( D `  ( 1r `  P ) )  =  0 )
8884, 87syl 16 . . . 4  |-  ( R  e. Domn  ->  ( D `  ( 1r `  P ) )  =  0 )
8979, 83, 883eqtr3d 2492 . . 3  |-  ( R  e. Domn  ->  ( ( D  |`  ( B  \  {  .0.  } ) ) `  ( 0g `  Y ) )  =  0 )
9038, 69, 893jca 1177 . 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 17126 . . . . 5  |-  B  =  ( Base `  (mulGrp `  P ) )
929, 91ressbas2 14670 . . . 4  |-  ( ( B  \  {  .0.  } )  C_  B  ->  ( B  \  {  .0.  } )  =  ( Base `  Y ) )
9321, 92ax-mp 5 . . 3  |-  ( B 
\  {  .0.  }
)  =  ( Base `  Y )
94 nn0sscn 10807 . . . 4  |-  NN0  C_  CC
95 cnfldbas 18403 . . . . 5  |-  CC  =  ( Base ` fld )
9613, 95ressbas2 14670 . . . 4  |-  ( NN0  C_  CC  ->  NN0  =  (
Base `  N )
)
9794, 96ax-mp 5 . . 3  |-  NN0  =  ( Base `  N )
98 fvex 5866 . . . . . 6  |-  ( Base `  P )  e.  _V
993, 98eqeltri 2527 . . . . 5  |-  B  e. 
_V
100 difexg 4585 . . . . 5  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
10199, 100ax-mp 5 . . . 4  |-  ( B 
\  {  .0.  }
)  e.  _V
1025, 40mgpplusg 17124 . . . . 5  |-  ( .r
`  P )  =  ( +g  `  (mulGrp `  P ) )
1039, 102ressplusg 14721 . . . 4  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  P )  =  ( +g  `  Y
) )
104101, 103ax-mp 5 . . 3  |-  ( .r
`  P )  =  ( +g  `  Y
)
105 nn0ex 10808 . . . 4  |-  NN0  e.  _V
106 cnfldadd 18404 . . . . 5  |-  +  =  ( +g  ` fld )
10713, 106ressplusg 14721 . . . 4  |-  ( NN0 
e.  _V  ->  +  =  ( +g  `  N ) )
108105, 107ax-mp 5 . . 3  |-  +  =  ( +g  `  N )
109 eqid 2443 . . 3  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
110 cnfld0 18421 . . . . 5  |-  0  =  ( 0g ` fld )
11113, 110subm0 15966 . . . 4  |-  ( NN0 
e.  (SubMnd ` fld )  ->  0  =  ( 0g `  N
) )
11212, 111ax-mp 5 . . 3  |-  0  =  ( 0g `  N )
11393, 97, 104, 108, 109, 112ismhm 15947 . 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 664 1  |-  ( R  e. Domn  ->  ( D  |`  ( B  \  {  .0.  } ) )  e.  ( Y MndHom  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095    \ cdif 3458    C_ wss 3461   {csn 4014    |` cres 4991    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495    + caddc 9498   RR*cxr 9630   NN0cn0 10802   Basecbs 14614   ↾s cress 14615   +g cplusg 14679   .rcmulr 14680   0gc0g 14819   Mndcmnd 15898   MndHom cmhm 15943  SubMndcsubmnd 15944  mulGrpcmgp 17120   1rcur 17132   Ringcrg 17177  NzRingcnzr 17884  RLRegcrlreg 17906  Domncdomn 17907  Poly1cpl1 18195  coe1cco1 18196  ℂfldccnfld 18399   deg1 cdg1 22430  Monic1pcmn1 22504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-fz 11684  df-fzo 11807  df-seq 12090  df-hash 12388  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-0g 14821  df-gsum 14822  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-mhm 15945  df-submnd 15946  df-grp 16036  df-minusg 16037  df-sbg 16038  df-mulg 16039  df-subg 16177  df-ghm 16244  df-cntz 16334  df-cmn 16779  df-abl 16780  df-mgp 17121  df-ur 17133  df-ring 17179  df-cring 17180  df-subrg 17406  df-lmod 17493  df-lss 17558  df-nzr 17885  df-rlreg 17910  df-domn 17911  df-ascl 17942  df-psr 17984  df-mvr 17985  df-mpl 17986  df-opsr 17988  df-psr1 18198  df-vr1 18199  df-ply1 18200  df-coe1 18201  df-cnfld 18400  df-mdeg 22431  df-deg1 22432  df-mon1 22509
This theorem is referenced by: (None)
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