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Theorem ig1peuOLD 23123
Description: There is a unique monic polynomial of minimal degree in any nonzero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) Obsolete version of ig1peu 23122 as of 25-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
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
ig1peuOLD  |-  ( ( 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 ig1peuOLD
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 eqid 2451 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
2 ig1peu.u . . . . . . . . . . 11  |-  U  =  (LIdeal `  P )
31, 2lidlss 18433 . . . . . . . . . 10  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
433ad2ant2 1030 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  C_  ( Base `  P ) )
54ssdifd 3569 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( ( Base `  P
)  \  {  .0.  } ) )
6 imass2 5204 . . . . . . . 8  |-  ( ( I  \  {  .0.  } )  C_  ( ( Base `  P )  \  {  .0.  } )  -> 
( D " (
I  \  {  .0.  } ) )  C_  ( D " ( ( Base `  P )  \  {  .0.  } ) ) )
75, 6syl 17 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  ( D " ( (
Base `  P )  \  {  .0.  } ) ) )
8 drngring 17982 . . . . . . . . 9  |-  ( R  e.  DivRing  ->  R  e.  Ring )
983ad2ant1 1029 . . . . . . . 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 23038 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( D
" ( ( Base `  P )  \  {  .0.  } ) )  C_  NN0 )
149, 13syl 17 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( ( Base `  P
)  \  {  .0.  } ) )  C_  NN0 )
157, 14sstrd 3442 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  NN0 )
16 nn0uz 11193 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
1715, 16syl6sseq 3478 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  ( ZZ>= `  0 )
)
1811ply1ring 18841 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
Ring )
199, 18syl 17 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  P  e.  Ring )
20 simp2 1009 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  e.  U
)
212, 12lidl0cl 18435 . . . . . . . . 9  |-  ( ( P  e.  Ring  /\  I  e.  U )  ->  .0.  e.  I )
2219, 20, 21syl2anc 667 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  .0.  e.  I
)
2322snssd 4117 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  {  .0.  }  C_  I )
24 simp3 1010 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  =/=  {  .0.  } )
2524necomd 2679 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  {  .0.  }  =/=  I )
26 pssdifn0 3827 . . . . . . 7  |-  ( ( {  .0.  }  C_  I  /\  {  .0.  }  =/=  I )  ->  (
I  \  {  .0.  } )  =/=  (/) )
2723, 25, 26syl2anc 667 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  =/=  (/) )
2810, 11, 1deg1xrf 23030 . . . . . . . . . 10  |-  D :
( Base `  P ) --> RR*
29 ffn 5728 . . . . . . . . . 10  |-  ( D : ( Base `  P
) --> RR*  ->  D  Fn  ( Base `  P )
)
3028, 29ax-mp 5 . . . . . . . . 9  |-  D  Fn  ( Base `  P )
3130a1i 11 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  D  Fn  ( Base `  P ) )
324ssdifssd 3571 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( Base `  P )
)
33 fnimaeq0 5697 . . . . . . . 8  |-  ( ( D  Fn  ( Base `  P )  /\  (
I  \  {  .0.  } )  C_  ( Base `  P ) )  -> 
( ( D "
( I  \  {  .0.  } ) )  =  (/) 
<->  ( I  \  {  .0.  } )  =  (/) ) )
3431, 32, 33syl2anc 667 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( ( D
" ( I  \  {  .0.  } ) )  =  (/)  <->  ( I  \  {  .0.  } )  =  (/) ) )
3534necon3bid 2668 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( ( D
" ( I  \  {  .0.  } ) )  =/=  (/)  <->  ( I  \  {  .0.  } )  =/=  (/) ) )
3627, 35mpbird 236 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  =/=  (/) )
37 infmssuzclOLD 11247 . . . . 5  |-  ( ( ( D " (
I  \  {  .0.  } ) )  C_  ( ZZ>=
`  0 )  /\  ( D " ( I 
\  {  .0.  }
) )  =/=  (/) )  ->  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  ( D "
( I  \  {  .0.  } ) ) )
3817, 36, 37syl2anc 667 . . . 4  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  ( D
" ( I  \  {  .0.  } ) ) )
39 fvelimab 5921 . . . . 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 667 . . . 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 214 . . 3  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  E. h  e.  ( I  \  {  .0.  } ) ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
4219adantr 467 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  P  e.  Ring )
43 simpl2 1012 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  I  e.  U )
449adantr 467 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  R  e.  Ring )
45 eqid 2451 . . . . . . . . . . 11  |-  (algSc `  P )  =  (algSc `  P )
46 eqid 2451 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4711, 45, 46, 1ply1sclf 18878 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (algSc `  P ) : (
Base `  R ) --> ( Base `  P )
)
4844, 47syl 17 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (algSc `  P ) : (
Base `  R ) --> ( Base `  P )
)
49 simpl1 1011 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  R  e.  DivRing )
5032sselda 3432 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  ( Base `  P
) )
51 eldifsni 4098 . . . . . . . . . . . . . 14  |-  ( h  e.  ( I  \  {  .0.  } )  ->  h  =/=  .0.  )
5251adantl 468 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  =/=  .0.  )
53 eqid 2451 . . . . . . . . . . . . . 14  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
5411, 1, 12, 53drnguc1p 23121 . . . . . . . . . . . . 13  |-  ( ( R  e.  DivRing  /\  h  e.  ( Base `  P
)  /\  h  =/=  .0.  )  ->  h  e.  (Unic1p `  R ) )
5549, 50, 52, 54syl3anc 1268 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  (Unic1p `  R ) )
56 eqid 2451 . . . . . . . . . . . . 13  |-  (Unit `  R )  =  (Unit `  R )
5710, 56, 53uc1pldg 23099 . . . . . . . . . . . 12  |-  ( h  e.  (Unic1p `  R )  -> 
( (coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )
5855, 57syl 17 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
(coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )
59 eqid 2451 . . . . . . . . . . . 12  |-  ( invr `  R )  =  (
invr `  R )
6056, 59unitinvcl 17902 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
(coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )  -> 
( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) )  e.  (Unit `  R ) )
6144, 58, 60syl2anc 667 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (Unit `  R ) )
6246, 56unitcl 17887 . . . . . . . . . 10  |-  ( ( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) )  e.  (Unit `  R )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (
Base `  R )
)
6361, 62syl 17 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (
Base `  R )
)
6448, 63ffvelrnd 6023 . . . . . . . 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 3555 . . . . . . . . 9  |-  ( h  e.  ( I  \  {  .0.  } )  ->  h  e.  I )
6665adantl 468 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  I )
67 eqid 2451 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
682, 1, 67lidlmcl 18441 . . . . . . . 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 1269 . . . . . . 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 23102 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  h  e.  (Unic1p `  R ) )  ->  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  M
)
7244, 55, 71syl2anc 667 . . . . . . 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
)
7369, 72elind 3618 . . . . . 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 ) )
74 eqid 2451 . . . . . . . . . 10  |-  (RLReg `  R )  =  (RLReg `  R )
7574, 56unitrrg 18517 . . . . . . . . 9  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
7644, 75syl 17 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (Unit `  R )  C_  (RLReg `  R ) )
7776, 61sseldd 3433 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (RLReg `  R ) )
7810, 11, 74, 1, 67, 45deg1mul3 23064 . . . . . . 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
) )
7944, 77, 50, 78syl3anc 1268 . . . . . 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
) )
80 fveq2 5865 . . . . . . . 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 ) ) )
8180eqeq1d 2453 . . . . . . 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
) ) )
8281rspcev 3150 . . . . . 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 ) )
8373, 79, 82syl2anc 667 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  E. g  e.  ( I  i^i  M
) ( D `  g )  =  ( D `  h ) )
84 eqeq2 2462 . . . . . 6  |-  ( ( D `  h )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  )  -> 
( ( D `  g )  =  ( D `  h )  <-> 
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
8584rexbidv 2901 . . . . 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 ,  `'  <  ) ) )
8683, 85syl5ibcom 224 . . . 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 ,  `'  <  ) ) )
8786rexlimdva 2879 . . 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 ,  `'  <  ) ) )
8841, 87mpd 15 . 2  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  E. g  e.  ( I  i^i  M ) ( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
89 eqid 2451 . . . . . . 7  |-  ( -g `  P )  =  (
-g `  P )
909ad2antrr 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 ,  `'  <  ) ) )  ->  R  e.  Ring )
91 inss2 3653 . . . . . . . . 9  |-  ( I  i^i  M )  C_  M
92 simprl 764 . . . . . . . . 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
) )
9391, 92sseldi 3430 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  g  e.  M )
9493adantr 467 . . . . . . 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 )
95 simprl 764 . . . . . . 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 ,  `'  <  ) )
96 simprr 766 . . . . . . . . 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
) )
9791, 96sseldi 3430 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  h  e.  M )
9897adantr 467 . . . . . . 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 )
99 simprr 766 . . . . . . 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 ,  `'  <  ) )
10010, 70, 11, 89, 90, 94, 95, 98, 99deg1submon1p 23103 . . . . . 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 ,  `'  <  ) )
101100ex 436 . . . . 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 ,  `'  <  ) ) )
10217ad2antrr 732 . . . . . . . . 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 ) )
10330a1i 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 ) )
10432ad2antrr 732 . . . . . . . . . 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 ) )
10519adantr 467 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  P  e.  Ring )
106 simpl2 1012 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  I  e.  U )
107 inss1 3652 . . . . . . . . . . . . . 14  |-  ( I  i^i  M )  C_  I
108107, 92sseldi 3430 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  g  e.  I )
109107, 96sseldi 3430 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  h  e.  I )
1102, 89lidlsubcl 18439 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Ring  /\  I  e.  U )  /\  ( g  e.  I  /\  h  e.  I ) )  -> 
( g ( -g `  P ) h )  e.  I )
111105, 106, 108, 109, 110syl22anc 1269 . . . . . . . . . . . 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 )
112111adantr 467 . . . . . . . . . . 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 )
113 simpr 463 . . . . . . . . . . 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.  )
114 eldifsn 4097 . . . . . . . . . . 11  |-  ( ( g ( -g `  P
) h )  e.  ( I  \  {  .0.  } )  <->  ( (
g ( -g `  P
) h )  e.  I  /\  ( g ( -g `  P
) h )  =/= 
.0.  ) )
115112, 113, 114sylanbrc 670 . . . . . . . . . 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.  } ) )
116 fnfvima 6143 . . . . . . . . . 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.  } ) ) )
117103, 104, 115, 116syl3anc 1268 . . . . . . . . 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.  } ) ) )
118 infmssuzleOLD 11246 . . . . . . . . 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 ) ) )
119102, 117, 118syl2anc 667 . . . . . . . 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 ) ) )
120119ex 436 . . . . . . 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 ) ) ) )
121 imassrn 5179 . . . . . . . . . . 11  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  ran  D
122 frn 5735 . . . . . . . . . . . 12  |-  ( D : ( Base `  P
) --> RR*  ->  ran  D  C_  RR* )
12328, 122ax-mp 5 . . . . . . . . . . 11  |-  ran  D  C_ 
RR*
124121, 123sstri 3441 . . . . . . . . . 10  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  RR*
125124, 38sseldi 3430 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  RR* )
126125adantr 467 . . . . . . . 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* )
127 ringgrp 17785 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  P  e. 
Grp )
12819, 127syl 17 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  P  e.  Grp )
129128adantr 467 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  ( g  e.  ( I  i^i  M
)  /\  h  e.  ( I  i^i  M ) ) )  ->  P  e.  Grp )
130107, 4syl5ss 3443 . . . . . . . . . . . 12  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  i^i 
M )  C_  ( Base `  P ) )
131130adantr 467 . . . . . . . . . . 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
) )
132131, 92sseldd 3433 . . . . . . . . . 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
) )
133131, 96sseldd 3433 . . . . . . . . . 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
) )
1341, 89grpsubcl 16734 . . . . . . . . . 10  |-  ( ( P  e.  Grp  /\  g  e.  ( Base `  P )  /\  h  e.  ( Base `  P
) )  ->  (
g ( -g `  P
) h )  e.  ( Base `  P
) )
135129, 132, 133, 134syl3anc 1268 . . . . . . . . 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
) )
13610, 11, 1deg1xrcl 23031 . . . . . . . . 9  |-  ( ( g ( -g `  P
) h )  e.  ( Base `  P
)  ->  ( D `  ( g ( -g `  P ) h ) )  e.  RR* )
137135, 136syl 17 . . . . . . . 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* )
138 xrlenlt 9699 . . . . . . . 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 ,  `'  <  ) ) )
139126, 137, 138syl2anc 667 . . . . . . 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 ,  `'  <  ) ) )
140120, 139sylibd 218 . . . . . 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 ,  `'  <  ) ) )
141140necon4ad 2643 . . . . 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.  ) )
142101, 141syld 45 . . . 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.  ) )
1431, 12, 89grpsubeq0 16740 . . . . 5  |-  ( ( P  e.  Grp  /\  g  e.  ( Base `  P )  /\  h  e.  ( Base `  P
) )  ->  (
( g ( -g `  P ) h )  =  .0.  <->  g  =  h ) )
144129, 132, 133, 143syl3anc 1268 . . . 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 ) )
145142, 144sylibd 218 . . 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 ) )
146145ralrimivva 2809 . 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 ) )
147 fveq2 5865 . . . 4  |-  ( g  =  h  ->  ( D `  g )  =  ( D `  h ) )
148147eqeq1d 2453 . . 3  |-  ( g  =  h  ->  (
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <->  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
149148reu4 3232 . 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 ) ) )
15088, 146, 149sylanbrc 670 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   E!wreu 2739    \ cdif 3401    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402   `'ccnv 4833   ran crn 4835   "cima 4837    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   supcsup 7954   RRcr 9538   0cc0 9539   RR*cxr 9674    < clt 9675    <_ cle 9676   NN0cn0 10869   ZZ>=cuz 11159   Basecbs 15121   .rcmulr 15191   0gc0g 15338   Grpcgrp 16669   -gcsg 16671   Ringcrg 17780  Unitcui 17867   invrcinvr 17899   DivRingcdr 17975  LIdealclidl 18393  RLRegcrlreg 18503  algSccascl 18535  Poly1cpl1 18770  coe1cco1 18771   deg1 cdg1 23003  Monic1pcmn1 23074  Unic1pcuc1p 23075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-0g 15340  df-gsum 15341  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-ghm 16881  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-drng 17977  df-subrg 18006  df-lmod 18093  df-lss 18156  df-sra 18395  df-rgmod 18396  df-lidl 18397  df-rlreg 18507  df-ascl 18538  df-psr 18580  df-mvr 18581  df-mpl 18582  df-opsr 18584  df-psr1 18773  df-vr1 18774  df-ply1 18775  df-coe1 18776  df-cnfld 18971  df-mdeg 23004  df-deg1 23005  df-mon1 23080  df-uc1p 23081
This theorem is referenced by:  ig1pval3OLD  23132
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