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Theorem ig1peu 19557
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 2283 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
2 ig1peu.u . . . . . . . . . . 11  |-  U  =  (LIdeal `  P )
31, 2lidlss 15961 . . . . . . . . . 10  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
433ad2ant2 977 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  C_  ( Base `  P ) )
5 ssdif 3311 . . . . . . . . 9  |-  ( I 
C_  ( Base `  P
)  ->  ( I  \  {  .0.  } ) 
C_  ( ( Base `  P )  \  {  .0.  } ) )
64, 5syl 15 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( ( Base `  P
)  \  {  .0.  } ) )
7 imass2 5049 . . . . . . . 8  |-  ( ( I  \  {  .0.  } )  C_  ( ( Base `  P )  \  {  .0.  } )  -> 
( D " (
I  \  {  .0.  } ) )  C_  ( D " ( ( Base `  P )  \  {  .0.  } ) ) )
86, 7syl 15 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  ( D " ( (
Base `  P )  \  {  .0.  } ) ) )
9 drngrng 15519 . . . . . . . . 9  |-  ( R  e.  DivRing  ->  R  e.  Ring )
1093ad2ant1 976 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  R  e.  Ring )
11 ig1peu.d . . . . . . . . 9  |-  D  =  ( deg1  `  R )
12 ig1peu.p . . . . . . . . 9  |-  P  =  (Poly1 `  R )
13 ig1peu.z . . . . . . . . 9  |-  .0.  =  ( 0g `  P )
1411, 12, 13, 1deg1n0ima 19475 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( D
" ( ( Base `  P )  \  {  .0.  } ) )  C_  NN0 )
1510, 14syl 15 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( ( Base `  P
)  \  {  .0.  } ) )  C_  NN0 )
168, 15sstrd 3189 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  NN0 )
17 nn0uz 10262 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
1816, 17syl6sseq 3224 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  ( ZZ>= `  0 )
)
1912ply1rng 16326 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
Ring )
2010, 19syl 15 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  P  e.  Ring )
21 simp2 956 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  e.  U
)
222, 13lidl0cl 15964 . . . . . . . . 9  |-  ( ( P  e.  Ring  /\  I  e.  U )  ->  .0.  e.  I )
2320, 21, 22syl2anc 642 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  .0.  e.  I
)
2423snssd 3760 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  {  .0.  }  C_  I )
25 simp3 957 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  =/=  {  .0.  } )
2625necomd 2529 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  {  .0.  }  =/=  I )
27 pssdifn0 3515 . . . . . . 7  |-  ( ( {  .0.  }  C_  I  /\  {  .0.  }  =/=  I )  ->  (
I  \  {  .0.  } )  =/=  (/) )
2824, 26, 27syl2anc 642 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  =/=  (/) )
2911, 12, 1deg1xrf 19467 . . . . . . . . . 10  |-  D :
( Base `  P ) --> RR*
30 ffn 5389 . . . . . . . . . 10  |-  ( D : ( Base `  P
) --> RR*  ->  D  Fn  ( Base `  P )
)
3129, 30ax-mp 8 . . . . . . . . 9  |-  D  Fn  ( Base `  P )
3231a1i 10 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  D  Fn  ( Base `  P ) )
33 difss 3303 . . . . . . . . 9  |-  ( I 
\  {  .0.  }
)  C_  I
3433, 4syl5ss 3190 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( Base `  P )
)
35 fnimaeq0 5365 . . . . . . . 8  |-  ( ( D  Fn  ( Base `  P )  /\  (
I  \  {  .0.  } )  C_  ( Base `  P ) )  -> 
( ( D "
( I  \  {  .0.  } ) )  =  (/) 
<->  ( I  \  {  .0.  } )  =  (/) ) )
3632, 34, 35syl2anc 642 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( ( D
" ( I  \  {  .0.  } ) )  =  (/)  <->  ( I  \  {  .0.  } )  =  (/) ) )
3736necon3bid 2481 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( ( D
" ( I  \  {  .0.  } ) )  =/=  (/)  <->  ( I  \  {  .0.  } )  =/=  (/) ) )
3828, 37mpbird 223 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  =/=  (/) )
39 infmssuzcl 10301 . . . . 5  |-  ( ( ( D " (
I  \  {  .0.  } ) )  C_  ( ZZ>=
`  0 )  /\  ( D " ( I 
\  {  .0.  }
) )  =/=  (/) )  ->  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  ( D "
( I  \  {  .0.  } ) ) )
4018, 38, 39syl2anc 642 . . . 4  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  ( D
" ( I  \  {  .0.  } ) ) )
41 fvelimab 5578 . . . . 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 ,  `'  <  ) ) )
4232, 34, 41syl2anc 642 . . . 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 ,  `'  <  ) ) )
4340, 42mpbid 201 . . 3  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  E. h  e.  ( I  \  {  .0.  } ) ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
4420adantr 451 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  P  e.  Ring )
45 simpl2 959 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  I  e.  U )
4610adantr 451 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  R  e.  Ring )
47 eqid 2283 . . . . . . . . . . 11  |-  (algSc `  P )  =  (algSc `  P )
48 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4912, 47, 48, 1ply1sclf 16361 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (algSc `  P ) : (
Base `  R ) --> ( Base `  P )
)
5046, 49syl 15 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (algSc `  P ) : (
Base `  R ) --> ( Base `  P )
)
51 simpl1 958 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  R  e.  DivRing )
5234sselda 3180 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  ( Base `  P
) )
53 eldifsni 3750 . . . . . . . . . . . . . 14  |-  ( h  e.  ( I  \  {  .0.  } )  ->  h  =/=  .0.  )
5453adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  =/=  .0.  )
55 eqid 2283 . . . . . . . . . . . . . 14  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
5612, 1, 13, 55drnguc1p 19556 . . . . . . . . . . . . 13  |-  ( ( R  e.  DivRing  /\  h  e.  ( Base `  P
)  /\  h  =/=  .0.  )  ->  h  e.  (Unic1p `  R ) )
5751, 52, 54, 56syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  (Unic1p `  R ) )
58 eqid 2283 . . . . . . . . . . . . 13  |-  (Unit `  R )  =  (Unit `  R )
5911, 58, 55uc1pldg 19534 . . . . . . . . . . . 12  |-  ( h  e.  (Unic1p `  R )  -> 
( (coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )
6057, 59syl 15 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
(coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )
61 eqid 2283 . . . . . . . . . . . 12  |-  ( invr `  R )  =  (
invr `  R )
6258, 61unitinvcl 15456 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
(coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )  -> 
( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) )  e.  (Unit `  R ) )
6346, 60, 62syl2anc 642 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (Unit `  R ) )
6448, 58unitcl 15441 . . . . . . . . . 10  |-  ( ( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) )  e.  (Unit `  R )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (
Base `  R )
)
6563, 64syl 15 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (
Base `  R )
)
66 ffvelrn 5663 . . . . . . . . 9  |-  ( ( (algSc `  P ) : ( Base `  R
) --> ( Base `  P
)  /\  ( ( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (
Base `  R )
)  ->  ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) )  e.  ( Base `  P
) )
6750, 65, 66syl2anc 642 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
(algSc `  P ) `  ( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) ) )  e.  ( Base `  P
) )
68 eldifi 3298 . . . . . . . . 9  |-  ( h  e.  ( I  \  {  .0.  } )  ->  h  e.  I )
6968adantl 452 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  I )
70 eqid 2283 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
712, 1, 70lidlmcl 15969 . . . . . . . 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
)
7244, 45, 67, 69, 71syl22anc 1183 . . . . . . 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
)
73 ig1peu.m . . . . . . . . 9  |-  M  =  (Monic1p `  R )
7455, 73, 12, 70, 47, 11, 61uc1pmon1p 19537 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  h  e.  (Unic1p `  R ) )  ->  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  M
)
7546, 57, 74syl2anc 642 . . . . . . 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
)
76 elin 3358 . . . . . . 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
) )
7772, 75, 76sylanbrc 645 . . . . . 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 ) )
78 eqid 2283 . . . . . . . . . 10  |-  (RLReg `  R )  =  (RLReg `  R )
7978, 58unitrrg 16034 . . . . . . . . 9  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
8046, 79syl 15 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (Unit `  R )  C_  (RLReg `  R ) )
8180, 63sseldd 3181 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (RLReg `  R ) )
8211, 12, 78, 1, 70, 47deg1mul3 19501 . . . . . . 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
) )
8346, 81, 52, 82syl3anc 1182 . . . . . 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
) )
84 fveq2 5525 . . . . . . . 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 ) ) )
8584eqeq1d 2291 . . . . . . 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
) ) )
8685rspcev 2884 . . . . . 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 ) )
8777, 83, 86syl2anc 642 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  E. g  e.  ( I  i^i  M
) ( D `  g )  =  ( D `  h ) )
88 eqeq2 2292 . . . . . 6  |-  ( ( D `  h )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  )  -> 
( ( D `  g )  =  ( D `  h )  <-> 
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
8988rexbidv 2564 . . . . 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 ,  `'  <  ) ) )
9087, 89syl5ibcom 211 . . . 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 ,  `'  <  ) ) )
9190rexlimdva 2667 . . 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 ,  `'  <  ) ) )
9243, 91mpd 14 . 2  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  E. g  e.  ( I  i^i  M ) ( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
93 eqid 2283 . . . . . . 7  |-  ( -g `  P )  =  (
-g `  P )
9410ad2antrr 706 . . . . . . 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 )
95 inss2 3390 . . . . . . . . 9  |-  ( I  i^i  M )  C_  M
96 simprl 732 . . . . . . . . 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
) )
9795, 96sseldi 3178 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  g  e.  M )
9897adantr 451 . . . . . . 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 )
99 simprl 732 . . . . . . 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 ,  `'  <  ) )
100 simprr 733 . . . . . . . . 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
) )
10195, 100sseldi 3178 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  h  e.  M )
102101adantr 451 . . . . . . 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 )
103 simprr 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 `  h )  =  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
10411, 73, 12, 93, 94, 98, 99, 102, 103deg1submon1p 19538 . . . . . 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 ,  `'  <  ) )
105104ex 423 . . . . 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 ,  `'  <  ) ) )
10618ad2antrr 706 . . . . . . . . 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 ) )
10731a1i 10 . . . . . . . . . 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 ) )
10834ad2antrr 706 . . . . . . . . . 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 ) )
10920adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  P  e.  Ring )
110 simpl2 959 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  I  e.  U )
111 inss1 3389 . . . . . . . . . . . . . 14  |-  ( I  i^i  M )  C_  I
112111, 96sseldi 3178 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  g  e.  I )
113111, 100sseldi 3178 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  h  e.  I )
1142, 93lidlsubcl 15968 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Ring  /\  I  e.  U )  /\  ( g  e.  I  /\  h  e.  I ) )  -> 
( g ( -g `  P ) h )  e.  I )
115109, 110, 112, 113, 114syl22anc 1183 . . . . . . . . . . . 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 )
116115adantr 451 . . . . . . . . . . 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 )
117 simpr 447 . . . . . . . . . . 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.  )
118 eldifsn 3749 . . . . . . . . . . 11  |-  ( ( g ( -g `  P
) h )  e.  ( I  \  {  .0.  } )  <->  ( (
g ( -g `  P
) h )  e.  I  /\  ( g ( -g `  P
) h )  =/= 
.0.  ) )
119116, 117, 118sylanbrc 645 . . . . . . . . . 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.  } ) )
120 fnfvima 5756 . . . . . . . . . 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.  } ) ) )
121107, 108, 119, 120syl3anc 1182 . . . . . . . . 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.  } ) ) )
122 infmssuzle 10300 . . . . . . . . 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 ) ) )
123106, 121, 122syl2anc 642 . . . . . . . 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 ) ) )
124123ex 423 . . . . . . 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 ) ) ) )
125 imassrn 5025 . . . . . . . . . . 11  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  ran  D
126 frn 5395 . . . . . . . . . . . 12  |-  ( D : ( Base `  P
) --> RR*  ->  ran  D  C_  RR* )
12729, 126ax-mp 8 . . . . . . . . . . 11  |-  ran  D  C_ 
RR*
128125, 127sstri 3188 . . . . . . . . . 10  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  RR*
129128, 40sseldi 3178 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  RR* )
130129adantr 451 . . . . . . . 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* )
131 rnggrp 15346 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  P  e. 
Grp )
13220, 131syl 15 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  P  e.  Grp )
133132adantr 451 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  P  e.  Grp )
134111, 4syl5ss 3190 . . . . . . . . . . . 12  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  i^i 
M )  C_  ( Base `  P ) )
135134adantr 451 . . . . . . . . . . 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
) )
136135, 96sseldd 3181 . . . . . . . . . 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
) )
137135, 100sseldd 3181 . . . . . . . . . 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
) )
1381, 93grpsubcl 14546 . . . . . . . . . 10  |-  ( ( P  e.  Grp  /\  g  e.  ( Base `  P )  /\  h  e.  ( Base `  P
) )  ->  (
g ( -g `  P
) h )  e.  ( Base `  P
) )
139133, 136, 137, 138syl3anc 1182 . . . . . . . . 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
) )
14011, 12, 1deg1xrcl 19468 . . . . . . . . 9  |-  ( ( g ( -g `  P
) h )  e.  ( Base `  P
)  ->  ( D `  ( g ( -g `  P ) h ) )  e.  RR* )
141139, 140syl 15 . . . . . . . 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* )
142 xrlenlt 8890 . . . . . . . 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 ,  `'  <  ) ) )
143130, 141, 142syl2anc 642 . . . . . . 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 ,  `'  <  ) ) )
144124, 143sylibd 205 . . . . . 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 ,  `'  <  ) ) )
145144necon4ad 2507 . . . . 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.  ) )
146105, 145syld 40 . . . 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.  ) )
1471, 13, 93grpsubeq0 14552 . . . . 5  |-  ( ( P  e.  Grp  /\  g  e.  ( Base `  P )  /\  h  e.  ( Base `  P
) )  ->  (
( g ( -g `  P ) h )  =  .0.  <->  g  =  h ) )
148133, 136, 137, 147syl3anc 1182 . . . 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 ) )
149146, 148sylibd 205 . . 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 ) )
150149ralrimivva 2635 . 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 ) )
151 fveq2 5525 . . . 4  |-  ( g  =  h  ->  ( D `  g )  =  ( D `  h ) )
152151eqeq1d 2291 . . 3  |-  ( g  =  h  ->  (
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <->  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
153152reu4 2959 . 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 ) ) )
15492, 150, 153sylanbrc 645 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   E!wreu 2545    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   RR*cxr 8866    < clt 8867    <_ cle 8868   NN0cn0 9965   ZZ>=cuz 10230   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Grpcgrp 14362   -gcsg 14365   Ringcrg 15337  Unitcui 15421   invrcinvr 15453   DivRingcdr 15512  LIdealclidl 15923  RLRegcrlreg 16020  algSccascl 16052  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440  Monic1pcmn1 19511  Unic1pcuc1p 19512
This theorem is referenced by:  ig1pval3  19560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-subrg 15543  df-lmod 15629  df-lss 15690  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rlreg 16024  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442  df-mon1 19516  df-uc1p 19517
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