MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ig1peu Structured version   Unicode version

Theorem ig1peu 21665
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 2443 . . . . . . . . . . 11  |-  ( Base `  P )  =  (
Base `  P )
2 ig1peu.u . . . . . . . . . . 11  |-  U  =  (LIdeal `  P )
31, 2lidlss 17313 . . . . . . . . . 10  |-  ( I  e.  U  ->  I  C_  ( Base `  P
) )
433ad2ant2 1010 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  C_  ( Base `  P ) )
54ssdifd 3513 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( ( Base `  P
)  \  {  .0.  } ) )
6 imass2 5225 . . . . . . . 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 16861 . . . . . . . . 9  |-  ( R  e.  DivRing  ->  R  e.  Ring )
983ad2ant1 1009 . . . . . . . 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 21582 . . . . . . . 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 3387 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  NN0 )
16 nn0uz 10916 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
1715, 16syl6sseq 3423 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  C_  ( ZZ>= `  0 )
)
1811ply1rng 17725 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  P  e. 
Ring )
199, 18syl 16 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  P  e.  Ring )
20 simp2 989 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  e.  U
)
212, 12lidl0cl 17316 . . . . . . . . 9  |-  ( ( P  e.  Ring  /\  I  e.  U )  ->  .0.  e.  I )
2219, 20, 21syl2anc 661 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  .0.  e.  I
)
2322snssd 4039 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  {  .0.  }  C_  I )
24 simp3 990 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  I  =/=  {  .0.  } )
2524necomd 2640 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  {  .0.  }  =/=  I )
26 pssdifn0 3761 . . . . . . 7  |-  ( ( {  .0.  }  C_  I  /\  {  .0.  }  =/=  I )  ->  (
I  \  {  .0.  } )  =/=  (/) )
2723, 25, 26syl2anc 661 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  =/=  (/) )
2810, 11, 1deg1xrf 21574 . . . . . . . . . 10  |-  D :
( Base `  P ) --> RR*
29 ffn 5580 . . . . . . . . . 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 3515 . . . . . . . 8  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  \  {  .0.  } )  C_  ( Base `  P )
)
33 fnimaeq0 5553 . . . . . . . 8  |-  ( ( D  Fn  ( Base `  P )  /\  (
I  \  {  .0.  } )  C_  ( Base `  P ) )  -> 
( ( D "
( I  \  {  .0.  } ) )  =  (/) 
<->  ( I  \  {  .0.  } )  =  (/) ) )
3431, 32, 33syl2anc 661 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( ( D
" ( I  \  {  .0.  } ) )  =  (/)  <->  ( I  \  {  .0.  } )  =  (/) ) )
3534necon3bid 2637 . . . . . 6  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( ( D
" ( I  \  {  .0.  } ) )  =/=  (/)  <->  ( I  \  {  .0.  } )  =/=  (/) ) )
3627, 35mpbird 232 . . . . 5  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( D "
( I  \  {  .0.  } ) )  =/=  (/) )
37 infmssuzcl 10959 . . . . 5  |-  ( ( ( D " (
I  \  {  .0.  } ) )  C_  ( ZZ>=
`  0 )  /\  ( D " ( I 
\  {  .0.  }
) )  =/=  (/) )  ->  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  ( D "
( I  \  {  .0.  } ) ) )
3817, 36, 37syl2anc 661 . . . 4  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  ( D
" ( I  \  {  .0.  } ) ) )
39 fvelimab 5768 . . . . 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 661 . . . 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 210 . . 3  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  E. h  e.  ( I  \  {  .0.  } ) ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) )
4219adantr 465 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  P  e.  Ring )
43 simpl2 992 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  I  e.  U )
449adantr 465 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  R  e.  Ring )
45 eqid 2443 . . . . . . . . . . 11  |-  (algSc `  P )  =  (algSc `  P )
46 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4711, 45, 46, 1ply1sclf 17760 . . . . . . . . . 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 991 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  R  e.  DivRing )
5032sselda 3377 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  ( Base `  P
) )
51 eldifsni 4022 . . . . . . . . . . . . . 14  |-  ( h  e.  ( I  \  {  .0.  } )  ->  h  =/=  .0.  )
5251adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  =/=  .0.  )
53 eqid 2443 . . . . . . . . . . . . . 14  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
5411, 1, 12, 53drnguc1p 21664 . . . . . . . . . . . . 13  |-  ( ( R  e.  DivRing  /\  h  e.  ( Base `  P
)  /\  h  =/=  .0.  )  ->  h  e.  (Unic1p `  R ) )
5549, 50, 52, 54syl3anc 1218 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  (Unic1p `  R ) )
56 eqid 2443 . . . . . . . . . . . . 13  |-  (Unit `  R )  =  (Unit `  R )
5710, 56, 53uc1pldg 21642 . . . . . . . . . . . 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 2443 . . . . . . . . . . . 12  |-  ( invr `  R )  =  (
invr `  R )
6056, 59unitinvcl 16788 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  (
(coe1 `  h ) `  ( D `  h ) )  e.  (Unit `  R ) )  -> 
( ( invr `  R
) `  ( (coe1 `  h ) `  ( D `  h )
) )  e.  (Unit `  R ) )
6144, 58, 60syl2anc 661 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) )  e.  (Unit `  R ) )
6246, 56unitcl 16773 . . . . . . . . . 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 5865 . . . . . . . 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 3499 . . . . . . . . 9  |-  ( h  e.  ( I  \  {  .0.  } )  ->  h  e.  I )
6665adantl 466 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  h  e.  I )
67 eqid 2443 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
682, 1, 67lidlmcl 17321 . . . . . . . 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 1219 . . . . . . 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 21645 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  h  e.  (Unic1p `  R ) )  ->  ( ( (algSc `  P ) `  (
( invr `  R ) `  ( (coe1 `  h ) `  ( D `  h ) ) ) ) ( .r `  P ) h )  e.  M
)
7244, 55, 71syl2anc 661 . . . . . . 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 3561 . . . . . 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 2443 . . . . . . . . . 10  |-  (RLReg `  R )  =  (RLReg `  R )
7574, 56unitrrg 17387 . . . . . . . . 9  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
7644, 75syl 16 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/=  {  .0.  }
)  /\  h  e.  ( I  \  {  .0.  } ) )  ->  (Unit `  R )  C_  (RLReg `  R ) )
7776, 61sseldd 3378 . . . . . . 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 21609 . . . . . . 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 1218 . . . . . 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 5712 . . . . . . . 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 2451 . . . . . . 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 3094 . . . . . 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 661 . . . . 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 2452 . . . . . 6  |-  ( ( D `  h )  =  sup ( ( D " ( I 
\  {  .0.  }
) ) ,  RR ,  `'  <  )  -> 
( ( D `  g )  =  ( D `  h )  <-> 
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
8584rexbidv 2757 . . . . 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 220 . . . 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 2862 . . 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 2443 . . . . . . 7  |-  ( -g `  P )  =  (
-g `  P )
909ad2antrr 725 . . . . . . 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 3592 . . . . . . . . 9  |-  ( I  i^i  M )  C_  M
92 simprl 755 . . . . . . . . 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 3375 . . . . . . . 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 465 . . . . . . 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 755 . . . . . . 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 756 . . . . . . . . 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 3375 . . . . . . . 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 465 . . . . . . 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 756 . . . . . . 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 21646 . . . . . 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 434 . . . . 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 725 . . . . . . . . 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 725 . . . . . . . . . 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 465 . . . . . . . . . . . . 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 992 . . . . . . . . . . . . 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 3591 . . . . . . . . . . . . . 14  |-  ( I  i^i  M )  C_  I
108107, 92sseldi 3375 . . . . . . . . . . . . 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 3375 . . . . . . . . . . . . 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 17320 . . . . . . . . . . . . 13  |-  ( ( ( P  e.  Ring  /\  I  e.  U )  /\  ( g  e.  I  /\  h  e.  I ) )  -> 
( g ( -g `  P ) h )  e.  I )
111105, 106, 108, 109, 110syl22anc 1219 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 461 . . . . . . . . . . 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 4021 . . . . . . . . . . 11  |-  ( ( g ( -g `  P
) h )  e.  ( I  \  {  .0.  } )  <->  ( (
g ( -g `  P
) h )  e.  I  /\  ( g ( -g `  P
) h )  =/= 
.0.  ) )
115112, 113, 114sylanbrc 664 . . . . . . . . . 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 5976 . . . . . . . . . 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 1218 . . . . . . . . 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 infmssuzle 10958 . . . . . . . . 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 661 . . . . . . . 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 434 . . . . . . 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 5201 . . . . . . . . . . 11  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  ran  D
122 frn 5586 . . . . . . . . . . . 12  |-  ( D : ( Base `  P
) --> RR*  ->  ran  D  C_  RR* )
12328, 122ax-mp 5 . . . . . . . . . . 11  |-  ran  D  C_ 
RR*
124121, 123sstri 3386 . . . . . . . . . 10  |-  ( D
" ( I  \  {  .0.  } ) ) 
C_  RR*
125124, 38sseldi 3375 . . . . . . . . 9  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  sup ( ( D
" ( I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  e.  RR* )
126125adantr 465 . . . . . . . 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 rnggrp 16672 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  P  e. 
Grp )
12819, 127syl 16 . . . . . . . . . . 11  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  P  e.  Grp )
129128adantr 465 . . . . . . . . . 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 3388 . . . . . . . . . . . 12  |-  ( ( R  e.  DivRing  /\  I  e.  U  /\  I  =/= 
{  .0.  } )  ->  ( I  i^i 
M )  C_  ( Base `  P ) )
131130adantr 465 . . . . . . . . . . 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 3378 . . . . . . . . . 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 3378 . . . . . . . . . 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 15627 . . . . . . . . . 10  |-  ( ( P  e.  Grp  /\  g  e.  ( Base `  P )  /\  h  e.  ( Base `  P
) )  ->  (
g ( -g `  P
) h )  e.  ( Base `  P
) )
135129, 132, 133, 134syl3anc 1218 . . . . . . . . 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 21575 . . . . . . . . 9  |-  ( ( g ( -g `  P
) h )  e.  ( Base `  P
)  ->  ( D `  ( g ( -g `  P ) h ) )  e.  RR* )
137135, 136syl 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* )
138 xrlenlt 9463 . . . . . . . 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 661 . . . . . . 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 214 . . . . . 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 2696 . . . . 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 44 . . . 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 15633 . . . . 5  |-  ( ( P  e.  Grp  /\  g  e.  ( Base `  P )  /\  h  e.  ( Base `  P
) )  ->  (
( g ( -g `  P ) h )  =  .0.  <->  g  =  h ) )
144129, 132, 133, 143syl3anc 1218 . . . 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 214 . . 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 2829 . 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 5712 . . . 4  |-  ( g  =  h  ->  ( D `  g )  =  ( D `  h ) )
148147eqeq1d 2451 . . 3  |-  ( g  =  h  ->  (
( D `  g
)  =  sup (
( D " (
I  \  {  .0.  } ) ) ,  RR ,  `'  <  )  <->  ( D `  h )  =  sup ( ( D "
( I  \  {  .0.  } ) ) ,  RR ,  `'  <  ) ) )
149148reu4 3174 . 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 664 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737   E!wreu 2738    \ cdif 3346    i^i cin 3348    C_ wss 3349   (/)c0 3658   {csn 3898   class class class wbr 4313   `'ccnv 4860   ran crn 4862   "cima 4864    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   supcsup 7711   RRcr 9302   0cc0 9303   RR*cxr 9438    < clt 9439    <_ cle 9440   NN0cn0 10600   ZZ>=cuz 10882   Basecbs 14195   .rcmulr 14260   0gc0g 14399   Grpcgrp 15431   -gcsg 15434   Ringcrg 16667  Unitcui 16753   invrcinvr 16785   DivRingcdr 16854  LIdealclidl 17273  RLRegcrlreg 17372  algSccascl 17405  Poly1cpl1 17655  coe1cco1 17656   deg1 cdg1 21545  Monic1pcmn1 21619  Unic1pcuc1p 21620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-ofr 6342  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-fz 11459  df-fzo 11570  df-seq 11828  df-hash 12125  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-0g 14401  df-gsum 14402  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-mhm 15485  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mulg 15569  df-subg 15699  df-ghm 15766  df-cntz 15856  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-cring 16670  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-drng 16856  df-subrg 16885  df-lmod 16972  df-lss 17036  df-sra 17275  df-rgmod 17276  df-lidl 17277  df-rlreg 17376  df-ascl 17408  df-psr 17445  df-mvr 17446  df-mpl 17447  df-opsr 17449  df-psr1 17658  df-vr1 17659  df-ply1 17660  df-coe1 17661  df-cnfld 17841  df-mdeg 21546  df-deg1 21547  df-mon1 21624  df-uc1p 21625
This theorem is referenced by:  ig1pval3  21668
  Copyright terms: Public domain W3C validator