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Theorem ig1peu 19961
Description: There is a unique monic polynomial of minimal degree in any nonzero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
ig1peu.p  |-  P  =  (Poly1 `  R )
ig1peu.u  |-  U  =  (LIdeal `  P )
ig1peu.z  |-  .0.  =  ( 0g `  P )
ig1peu.m  |-  M  =  (Monic1p `  R )
ig1peu.d  |-  D  =  ( deg1  `  R )
Assertion
Ref Expression
ig1peu  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  E! g  e.  ( I  i^i  M
) ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
Distinct variable groups:    D, g    g, I    g, M    P, g    R, g    U, g    .0. , g

Proof of Theorem ig1peu
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
2 ig1peu.u . . . . . . . . . . 11  |-  U  =  (LIdeal `  P )
31, 2lidlss 16207 . . . . . . . . . 10  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
433ad2ant2 979 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  C_  ( Base `  P ) )
54ssdifd 3426 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( ( Base `  P
)  \  {  .0.  } ) )
6 imass2 5180 . . . . . . . 8  |-  ( ( I  \  {  .0.  } )  C_  ( ( Base `  P )  \  {  .0.  } )  -> 
( D " (
I  \  {  .0.  } ) )  C_  ( D " ( ( Base `  P )  \  {  .0.  } ) ) )
75, 6syl 16 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  ( D " ( (
Base `  P )  \  {  .0.  } ) ) )
8 drngrng 15769 . . . . . . . . 9  |-  ( R  e.  DivRing  ->  R  e.  Ring )
983ad2ant1 978 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  R  e.  Ring )
10 ig1peu.d . . . . . . . . 9  |-  D  =  ( deg1  `  R )
11 ig1peu.p . . . . . . . . 9  |-  P  =  (Poly1 `  R )
12 ig1peu.z . . . . . . . . 9  |-  .0.  =  ( 0g `  P )
1310, 11, 12, 1deg1n0ima 19879 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( D
" ( ( Base `  P )  \  {  .0.  } ) )  C_  NN0 )
149, 13syl 16 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( ( Base `  P
)  \  {  .0.  } ) )  C_  NN0 )
157, 14sstrd 3301 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  NN0 )
16 nn0uz 10452 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
1715, 16syl6sseq 3337 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  ( ZZ>= `  0 )
)
1811ply1rng 16569 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
Ring )
199, 18syl 16 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  P  e.  Ring )
20 simp2 958 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  e.  U
)
212, 12lidl0cl 16210 . . . . . . . . 9  |-  ( ( P  e.  Ring  /\  I  e.  U )  ->  .0.  e.  I )
2219, 20, 21syl2anc 643 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  .0.  e.  I
)
2322snssd 3886 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  {  .0.  }  C_  I )
24 simp3 959 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  =/=  {  .0.  } )
2524necomd 2633 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  {  .0.  }  =/=  I )
26 pssdifn0 3632 . . . . . . 7  |-  ( ( {  .0.  }  C_  I  /\  {  .0.  }  =/=  I )  ->  (
I  \  {  .0.  } )  =/=  (/) )
2723, 25, 26syl2anc 643 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  =/=  (/) )
2810, 11, 1deg1xrf 19871 . . . . . . . . . 10  |-  D :
( Base `  P ) --> RR*
29 ffn 5531 . . . . . . . . . 10  |-  ( D : ( Base `  P
) --> RR*  ->  D  Fn  ( Base `  P )
)
3028, 29ax-mp 8 . . . . . . . . 9  |-  D  Fn  ( Base `  P )
3130a1i 11 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  D  Fn  ( Base `  P ) )
324ssdifssd 3428 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( Base `  P )
)
33 fnimaeq0 5506 . . . . . . . 8  |-  ( ( D  Fn  ( Base `  P )  /\  (
I  \  {  .0.  } )  C_  ( Base `  P ) )  -> 
( ( D "
( I  \  {  .0.  } ) )  =  (/) 
<->  ( I  \  {  .0.  } )  =  (/) ) )
3431, 32, 33syl2anc 643 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( ( D
" ( I  \  {  .0.  } ) )  =  (/)  <->  ( I  \  {  .0.  } )  =  (/) ) )
3534necon3bid 2585 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( ( D
" ( I  \  {  .0.  } ) )  =/=  (/)  <->  ( I  \  {  .0.  } )  =/=  (/) ) )
3627, 35mpbird 224 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  =/=  (/) )
37 infmssuzcl 10491 . . . . 5  |-  ( ( ( D " (
I  \  {  .0.  } ) )  C_  ( ZZ>=
`  0 )  /\  ( D " ( I 
\  {  .0.  }
) )  =/=  (/) )  ->  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  ( D "
( I  \  {  .0.  } ) ) )
3817, 36, 37syl2anc 643 . . . 4  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  ( D
" ( I  \  {  .0.  } ) ) )
39 fvelimab 5721 . . . . 5  |-  ( ( D  Fn  ( Base `  P )  /\  (
I  \  {  .0.  } )  C_  ( Base `  P ) )  -> 
( sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  )  e.  ( D " (
I  \  {  .0.  } ) )  <->  E. h  e.  ( I  \  {  .0.  } ) ( D `
 h )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
4031, 32, 39syl2anc 643 . . . 4  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  ( D " (
I  \  {  .0.  } ) )  <->  E. h  e.  ( I  \  {  .0.  } ) ( D `
 h )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
4138, 40mpbid 202 . . 3  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  E. h  e.  ( I  \  {  .0.  } ) ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
4219adantr 452 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  P  e.  Ring )
43 simpl2 961 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  I  e.  U )
449adantr 452 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  R  e.  Ring )
45 eqid 2387 . . . . . . . . . . 11  |-  (algSc `  P )  =  (algSc `  P )
46 eqid 2387 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4711, 45, 46, 1ply1sclf 16604 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (algSc `  P ) : (
Base `  R ) --> ( Base `  P )
)
4844, 47syl 16 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (algSc `  P ) : (
Base `  R ) --> ( Base `  P )
)
49 simpl1 960 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  R  e.  DivRing )
5032sselda 3291 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  ( Base `  P
) )
51 eldifsni 3871 . . . . . . . . . . . . . 14  |-  ( h  e.  ( I  \  {  .0.  } )  ->  h  =/=  .0.  )
5251adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  =/=  .0.  )
53 eqid 2387 . . . . . . . . . . . . . 14  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
5411, 1, 12, 53drnguc1p 19960 . . . . . . . . . . . . 13  |-  ( ( R  e.  DivRing  /\  h  e.  ( Base `  P
)  /\  h  =/=  .0.  )  ->  h  e.  (Unic1p `  R ) )
5549, 50, 52, 54syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  (Unic1p `  R ) )
56 eqid 2387 . . . . . . . . . . . . 13  |-  (Unit `  R )  =  (Unit `  R )
5710, 56, 53uc1pldg 19938 . . . . . . . . . . . 12  |-  ( h  e.  (Unic1p `  R )  -> 
( (coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )
5855, 57syl 16 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
(coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )
59 eqid 2387 . . . . . . . . . . . 12  |-  ( invr `  R )  =  (
invr `  R )
6056, 59unitinvcl 15706 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
(coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )  -> 
( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) )  e.  (Unit `  R ) )
6144, 58, 60syl2anc 643 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (Unit `  R ) )
6246, 56unitcl 15691 . . . . . . . . . 10  |-  ( ( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) )  e.  (Unit `  R )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (
Base `  R )
)
6361, 62syl 16 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (
Base `  R )
)
6448, 63ffvelrnd 5810 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
(algSc `  P ) `  ( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) ) )  e.  ( Base `  P
) )
65 eldifi 3412 . . . . . . . . 9  |-  ( h  e.  ( I  \  {  .0.  } )  ->  h  e.  I )
6665adantl 453 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  I )
67 eqid 2387 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
682, 1, 67lidlmcl 16215 . . . . . . . 8  |-  ( ( ( P  e.  Ring  /\  I  e.  U )  /\  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) )  e.  ( Base `  P
)  /\  h  e.  I ) )  -> 
( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  I
)
6942, 43, 64, 66, 68syl22anc 1185 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( (algSc `  P
) `  ( ( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  I
)
70 ig1peu.m . . . . . . . . 9  |-  M  =  (Monic1p `  R )
7153, 70, 11, 67, 45, 10, 59uc1pmon1p 19941 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  h  e.  (Unic1p `  R ) )  ->  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  M
)
7244, 55, 71syl2anc 643 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( (algSc `  P
) `  ( ( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  M
)
73 elin 3473 . . . . . . 7  |-  ( ( ( (algSc `  P
) `  ( ( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  ( I  i^i  M )  <-> 
( ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  I  /\  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  M
) )
7469, 72, 73sylanbrc 646 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( (algSc `  P
) `  ( ( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  ( I  i^i  M ) )
75 eqid 2387 . . . . . . . . . 10  |-  (RLReg `  R )  =  (RLReg `  R )
7675, 56unitrrg 16280 . . . . . . . . 9  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
7744, 76syl 16 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (Unit `  R )  C_  (RLReg `  R ) )
7877, 61sseldd 3292 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (RLReg `  R ) )
7910, 11, 75, 1, 67, 45deg1mul3 19905 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (RLReg `  R )  /\  h  e.  ( Base `  P
) )  ->  ( D `  ( (
(algSc `  P ) `  ( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) ) ) ( .r `  P ) h ) )  =  ( D `  h
) )
8044, 78, 50, 79syl3anc 1184 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  ( D `  ( (
(algSc `  P ) `  ( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) ) ) ( .r `  P ) h ) )  =  ( D `  h
) )
81 fveq2 5668 . . . . . . . 8  |-  ( g  =  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  ->  ( D `  g )  =  ( D `  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h ) ) )
8281eqeq1d 2395 . . . . . . 7  |-  ( g  =  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  ->  (
( D `  g
)  =  ( D `
 h )  <->  ( D `  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h ) )  =  ( D `  h
) ) )
8382rspcev 2995 . . . . . 6  |-  ( ( ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  ( I  i^i  M )  /\  ( D `  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h ) )  =  ( D `  h
) )  ->  E. g  e.  ( I  i^i  M
) ( D `  g )  =  ( D `  h ) )
8474, 80, 83syl2anc 643 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  E. g  e.  ( I  i^i  M
) ( D `  g )  =  ( D `  h ) )
85 eqeq2 2396 . . . . . 6  |-  ( ( D `  h )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  )  -> 
( ( D `  g )  =  ( D `  h )  <-> 
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
8685rexbidv 2670 . . . . 5  |-  ( ( D `  h )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  )  -> 
( E. g  e.  ( I  i^i  M
) ( D `  g )  =  ( D `  h )  <->  E. g  e.  (
I  i^i  M )
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
8784, 86syl5ibcom 212 . . . 4  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( D `  h
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  ->  E. g  e.  (
I  i^i  M )
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
8887rexlimdva 2773 . . 3  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( E. h  e.  ( I  \  {  .0.  } ) ( D `
 h )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  ->  E. g  e.  ( I  i^i  M
) ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
8941, 88mpd 15 . 2  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  E. g  e.  ( I  i^i  M ) ( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
90 eqid 2387 . . . . . . 7  |-  ( -g `  P )  =  (
-g `  P )
919ad2antrr 707 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  ->  R  e.  Ring )
92 inss2 3505 . . . . . . . . 9  |-  ( I  i^i  M )  C_  M
93 simprl 733 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  g  e.  ( I  i^i  M
) )
9492, 93sseldi 3289 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  g  e.  M )
9594adantr 452 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  ->  g  e.  M )
96 simprl 733 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  ->  ( D `  g )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
97 simprr 734 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  h  e.  ( I  i^i  M
) )
9892, 97sseldi 3289 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  h  e.  M )
9998adantr 452 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  ->  h  e.  M )
100 simprr 734 . . . . . . 7  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  ->  ( D `  h )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
10110, 70, 11, 90, 91, 95, 96, 99, 100deg1submon1p 19942 . . . . . 6  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )  ->  ( D `  ( g
( -g `  P ) h ) )  <  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
102101ex 424 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
( ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )  ->  ( D `  ( g ( -g `  P ) h ) )  <  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
10317ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( g ( -g `  P ) h )  =/=  .0.  )  -> 
( D " (
I  \  {  .0.  } ) )  C_  ( ZZ>=
`  0 ) )
10430a1i 11 . . . . . . . . . 10  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( g ( -g `  P ) h )  =/=  .0.  )  ->  D  Fn  ( Base `  P ) )
10532ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( g ( -g `  P ) h )  =/=  .0.  )  -> 
( I  \  {  .0.  } )  C_  ( Base `  P ) )
10619adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  P  e.  Ring )
107 simpl2 961 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  I  e.  U )
108 inss1 3504 . . . . . . . . . . . . . 14  |-  ( I  i^i  M )  C_  I
109108, 93sseldi 3289 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  g  e.  I )
110108, 97sseldi 3289 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  h  e.  I )
1112, 90lidlsubcl 16214 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Ring  /\  I  e.  U )  /\  ( g  e.  I  /\  h  e.  I ) )  -> 
( g ( -g `  P ) h )  e.  I )
112106, 107, 109, 110, 111syl22anc 1185 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
g ( -g `  P
) h )  e.  I )
113112adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( g ( -g `  P ) h )  =/=  .0.  )  -> 
( g ( -g `  P ) h )  e.  I )
114 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( g ( -g `  P ) h )  =/=  .0.  )  -> 
( g ( -g `  P ) h )  =/=  .0.  )
115 eldifsn 3870 . . . . . . . . . . 11  |-  ( ( g ( -g `  P
) h )  e.  ( I  \  {  .0.  } )  <->  ( (
g ( -g `  P
) h )  e.  I  /\  ( g ( -g `  P
) h )  =/= 
.0.  ) )
116113, 114, 115sylanbrc 646 . . . . . . . . . 10  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( g ( -g `  P ) h )  =/=  .0.  )  -> 
( g ( -g `  P ) h )  e.  ( I  \  {  .0.  } ) )
117 fnfvima 5915 . . . . . . . . . 10  |-  ( ( D  Fn  ( Base `  P )  /\  (
I  \  {  .0.  } )  C_  ( Base `  P )  /\  (
g ( -g `  P
) h )  e.  ( I  \  {  .0.  } ) )  -> 
( D `  (
g ( -g `  P
) h ) )  e.  ( D "
( I  \  {  .0.  } ) ) )
118104, 105, 116, 117syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( g ( -g `  P ) h )  =/=  .0.  )  -> 
( D `  (
g ( -g `  P
) h ) )  e.  ( D "
( I  \  {  .0.  } ) ) )
119 infmssuzle 10490 . . . . . . . . 9  |-  ( ( ( D " (
I  \  {  .0.  } ) )  C_  ( ZZ>=
`  0 )  /\  ( D `  ( g ( -g `  P
) h ) )  e.  ( D "
( I  \  {  .0.  } ) ) )  ->  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <_  ( D `  ( g ( -g `  P ) h ) ) )
120103, 118, 119syl2anc 643 . . . . . . . 8  |-  ( ( ( ( R  e.  DivRing 
/\  I  e.  U  /\  I  =/=  {  .0.  } )  /\  ( g  e.  ( I  i^i 
M )  /\  h  e.  ( I  i^i  M
) ) )  /\  ( g ( -g `  P ) h )  =/=  .0.  )  ->  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <_  ( D `  ( g ( -g `  P ) h ) ) )
121120ex 424 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
( g ( -g `  P ) h )  =/=  .0.  ->  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <_  ( D `  ( g ( -g `  P ) h ) ) ) )
122 imassrn 5156 . . . . . . . . . . 11  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  ran  D
123 frn 5537 . . . . . . . . . . . 12  |-  ( D : ( Base `  P
) --> RR*  ->  ran  D  C_  RR* )
12428, 123ax-mp 8 . . . . . . . . . . 11  |-  ran  D  C_ 
RR*
125122, 124sstri 3300 . . . . . . . . . 10  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  RR*
126125, 38sseldi 3289 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  RR* )
127126adantr 452 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  RR* )
128 rnggrp 15596 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  P  e. 
Grp )
12919, 128syl 16 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  P  e.  Grp )
130129adantr 452 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  P  e.  Grp )
131108, 4syl5ss 3302 . . . . . . . . . . . 12  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  i^i 
M )  C_  ( Base `  P ) )
132131adantr 452 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
I  i^i  M )  C_  ( Base `  P
) )
133132, 93sseldd 3292 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  g  e.  ( Base `  P
) )
134132, 97sseldd 3292 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  h  e.  ( Base `  P
) )
1351, 90grpsubcl 14796 . . . . . . . . . 10  |-  ( ( P  e.  Grp  /\  g  e.  ( Base `  P )  /\  h  e.  ( Base `  P
) )  ->  (
g ( -g `  P
) h )  e.  ( Base `  P
) )
136130, 133, 134, 135syl3anc 1184 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
g ( -g `  P
) h )  e.  ( Base `  P
) )
13710, 11, 1deg1xrcl 19872 . . . . . . . . 9  |-  ( ( g ( -g `  P
) h )  e.  ( Base `  P
)  ->  ( D `  ( g ( -g `  P ) h ) )  e.  RR* )
138136, 137syl 16 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  ( D `  ( g
( -g `  P ) h ) )  e. 
RR* )
139 xrlenlt 9076 . . . . . . . 8  |-  ( ( sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  RR*  /\  ( D `  ( g
( -g `  P ) h ) )  e. 
RR* )  ->  ( sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <_  ( D `  ( g ( -g `  P ) h ) )  <->  -.  ( D `  ( g ( -g `  P ) h ) )  <  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
140127, 138, 139syl2anc 643 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  ( sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <_  ( D `  ( g ( -g `  P ) h ) )  <->  -.  ( D `  ( g ( -g `  P ) h ) )  <  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
141121, 140sylibd 206 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
( g ( -g `  P ) h )  =/=  .0.  ->  -.  ( D `  ( g ( -g `  P
) h ) )  <  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
142141necon4ad 2611 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
( D `  (
g ( -g `  P
) h ) )  <  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  ->  ( g
( -g `  P ) h )  =  .0.  ) )
143102, 142syld 42 . . . 4  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
( ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )  ->  ( g
( -g `  P ) h )  =  .0.  ) )
1441, 12, 90grpsubeq0 14802 . . . . 5  |-  ( ( P  e.  Grp  /\  g  e.  ( Base `  P )  /\  h  e.  ( Base `  P
) )  ->  (
( g ( -g `  P ) h )  =  .0.  <->  g  =  h ) )
145130, 133, 134, 144syl3anc 1184 . . . 4  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
( g ( -g `  P ) h )  =  .0.  <->  g  =  h ) )
146143, 145sylibd 206 . . 3  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  (
( ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )  ->  g  =  h ) )
147146ralrimivva 2741 . 2  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  A. g  e.  ( I  i^i  M ) A. h  e.  ( I  i^i  M ) ( ( ( D `
 g )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  ( D `
 h )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )  ->  g  =  h ) )
148 fveq2 5668 . . . 4  |-  ( g  =  h  ->  ( D `  g )  =  ( D `  h ) )
149148eqeq1d 2395 . . 3  |-  ( g  =  h  ->  (
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <->  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
150149reu4 3071 . 2  |-  ( E! g  e.  ( I  i^i  M ) ( D `  g )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  )  <->  ( E. g  e.  ( I  i^i  M ) ( D `
 g )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  /\  A. g  e.  ( I  i^i  M
) A. h  e.  ( I  i^i  M
) ( ( ( D `  g )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  )  /\  ( D `  h )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  ) )  ->  g  =  h ) ) )
15189, 147, 150sylanbrc 646 1  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  E! g  e.  ( I  i^i  M
) ( D `  g )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   E!wreu 2651    \ cdif 3260    i^i cin 3262    C_ wss 3263   (/)c0 3571   {csn 3757   class class class wbr 4153   `'ccnv 4817   ran crn 4819   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   supcsup 7380   RRcr 8922   0cc0 8923   RR*cxr 9052    < clt 9053    <_ cle 9054   NN0cn0 10153   ZZ>=cuz 10420   Basecbs 13396   .rcmulr 13457   0gc0g 13650   Grpcgrp 14612   -gcsg 14615   Ringcrg 15587  Unitcui 15671   invrcinvr 15703   DivRingcdr 15762  LIdealclidl 16169  RLRegcrlreg 16266  algSccascl 16298  Poly1cpl1 16498  coe1cco1 16501   deg1 cdg1 19844  Monic1pcmn1 19915  Unic1pcuc1p 19916
This theorem is referenced by:  ig1pval3  19964
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-ofr 6245  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-fzo 11066  df-seq 11251  df-hash 11546  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-0g 13654  df-gsum 13655  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-mhm 14665  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-ghm 14931  df-cntz 15043  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-cring 15591  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-drng 15764  df-subrg 15793  df-lmod 15879  df-lss 15936  df-sra 16171  df-rgmod 16172  df-lidl 16173  df-rlreg 16270  df-ascl 16301  df-psr 16344  df-mvr 16345  df-mpl 16346  df-opsr 16352  df-psr1 16503  df-vr1 16504  df-ply1 16505  df-coe1 16508  df-cnfld 16627  df-mdeg 19845  df-deg1 19846  df-mon1 19920  df-uc1p 19921
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