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Theorem ig1peu 20086
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 2435 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
2 ig1peu.u . . . . . . . . . . 11  |-  U  =  (LIdeal `  P )
31, 2lidlss 16272 . . . . . . . . . 10  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
433ad2ant2 979 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  C_  ( Base `  P ) )
54ssdifd 3475 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( ( Base `  P
)  \  {  .0.  } ) )
6 imass2 5232 . . . . . . . 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 15834 . . . . . . . . 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 20004 . . . . . . . 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 3350 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  NN0 )
16 nn0uz 10512 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
1715, 16syl6sseq 3386 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  ( ZZ>= `  0 )
)
1811ply1rng 16634 . . . . . . . . . 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 16275 . . . . . . . . 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 3935 . . . . . . 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 2681 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  {  .0.  }  =/=  I )
26 pssdifn0 3681 . . . . . . 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 19996 . . . . . . . . . 10  |-  D :
( Base `  P ) --> RR*
29 ffn 5583 . . . . . . . . . 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 3477 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( Base `  P )
)
33 fnimaeq0 5558 . . . . . . . 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 2633 . . . . . 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 10551 . . . . 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 5774 . . . . 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 2435 . . . . . . . . . . 11  |-  (algSc `  P )  =  (algSc `  P )
46 eqid 2435 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4711, 45, 46, 1ply1sclf 16669 . . . . . . . . . 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 3340 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  ( Base `  P
) )
51 eldifsni 3920 . . . . . . . . . . . . . 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 2435 . . . . . . . . . . . . . 14  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
5411, 1, 12, 53drnguc1p 20085 . . . . . . . . . . . . 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 2435 . . . . . . . . . . . . 13  |-  (Unit `  R )  =  (Unit `  R )
5710, 56, 53uc1pldg 20063 . . . . . . . . . . . 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 2435 . . . . . . . . . . . 12  |-  ( invr `  R )  =  (
invr `  R )
6056, 59unitinvcl 15771 . . . . . . . . . . 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 15756 . . . . . . . . . 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 5863 . . . . . . . 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 3461 . . . . . . . . 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 2435 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
682, 1, 67lidlmcl 16280 . . . . . . . 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 20066 . . . . . . . 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 3522 . . . . . . 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 2435 . . . . . . . . . 10  |-  (RLReg `  R )  =  (RLReg `  R )
7675, 56unitrrg 16345 . . . . . . . . 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 3341 . . . . . . 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 20030 . . . . . . 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 5720 . . . . . . . 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 2443 . . . . . . 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 3044 . . . . . 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 2444 . . . . . 6  |-  ( ( D `  h )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  )  -> 
( ( D `  g )  =  ( D `  h )  <-> 
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
8685rexbidv 2718 . . . . 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 2822 . . 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 2435 . . . . . . 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 3554 . . . . . . . . 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 3338 . . . . . . . 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 3338 . . . . . . . 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 20067 . . . . . 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 3553 . . . . . . . . . . . . . 14  |-  ( I  i^i  M )  C_  I
109108, 93sseldi 3338 . . . . . . . . . . . . 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 3338 . . . . . . . . . . . . 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 16279 . . . . . . . . . . . . 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 3919 . . . . . . . . . . 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 5968 . . . . . . . . . 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 10550 . . . . . . . . 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 5208 . . . . . . . . . . 11  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  ran  D
123 frn 5589 . . . . . . . . . . . 12  |-  ( D : ( Base `  P
) --> RR*  ->  ran  D  C_  RR* )
12428, 123ax-mp 8 . . . . . . . . . . 11  |-  ran  D  C_ 
RR*
125122, 124sstri 3349 . . . . . . . . . 10  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  RR*
126125, 38sseldi 3338 . . . . . . . . 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 15661 . . . . . . . . . . . 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 3351 . . . . . . . . . . . 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 3341 . . . . . . . . . 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 3341 . . . . . . . . . 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 14861 . . . . . . . . . 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 19997 . . . . . . . . 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 9135 . . . . . . . 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 2659 . . . . 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 14867 . . . . 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 2790 . 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 5720 . . . 4  |-  ( g  =  h  ->  ( D `  g )  =  ( D `  h ) )
149148eqeq1d 2443 . . 3  |-  ( g  =  h  ->  (
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <->  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
150149reu4 3120 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   E!wreu 2699    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204   `'ccnv 4869   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   RRcr 8981   0cc0 8982   RR*cxr 9111    < clt 9112    <_ cle 9113   NN0cn0 10213   ZZ>=cuz 10480   Basecbs 13461   .rcmulr 13522   0gc0g 13715   Grpcgrp 14677   -gcsg 14680   Ringcrg 15652  Unitcui 15736   invrcinvr 15768   DivRingcdr 15827  LIdealclidl 16234  RLRegcrlreg 16331  algSccascl 16363  Poly1cpl1 16563  coe1cco1 16566   deg1 cdg1 19969  Monic1pcmn1 20040  Unic1pcuc1p 20041
This theorem is referenced by:  ig1pval3  20089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-subrg 15858  df-lmod 15944  df-lss 16001  df-sra 16236  df-rgmod 16237  df-lidl 16238  df-rlreg 16335  df-ascl 16366  df-psr 16409  df-mvr 16410  df-mpl 16411  df-opsr 16417  df-psr1 16568  df-vr1 16569  df-ply1 16570  df-coe1 16573  df-cnfld 16696  df-mdeg 19970  df-deg1 19971  df-mon1 20045  df-uc1p 20046
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