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Theorem igenval2 26676
Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
igenval2.1  |-  G  =  ( 1st `  R
)
igenval2.2  |-  X  =  ran  G
Assertion
Ref Expression
igenval2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  (
( R  IdlGen  S )  =  I  <->  ( I  e.  ( Idl `  R
)  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  I  C_  j )
) ) )
Distinct variable groups:    R, j    S, j    j, I
Allowed substitution hints:    G( j)    X( j)

Proof of Theorem igenval2
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 igenval2.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 igenval2.2 . . . . 5  |-  X  =  ran  G
31, 2igenidl 26673 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R
) )
41, 2igenss 26672 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )
5 igenmin 26674 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  j  e.  ( Idl `  R
)  /\  S  C_  j
)  ->  ( R  IdlGen  S )  C_  j
)
653expia 1155 . . . . . 6  |-  ( ( R  e.  RingOps  /\  j  e.  ( Idl `  R
) )  ->  ( S  C_  j  ->  ( R  IdlGen  S )  C_  j ) )
76ralrimiva 2789 . . . . 5  |-  ( R  e.  RingOps  ->  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  ( R  IdlGen  S ) 
C_  j ) )
87adantr 452 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  A. j  e.  ( Idl `  R
) ( S  C_  j  ->  ( R  IdlGen  S )  C_  j )
)
93, 4, 83jca 1134 . . 3  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  (
( R  IdlGen  S )  e.  ( Idl `  R
)  /\  S  C_  ( R  IdlGen  S )  /\  A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  ( R  IdlGen  S )  C_  j ) ) )
10 eleq1 2496 . . . 4  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( R  IdlGen  S )  e.  ( Idl `  R
)  <->  I  e.  ( Idl `  R ) ) )
11 sseq2 3370 . . . 4  |-  ( ( R  IdlGen  S )  =  I  ->  ( S  C_  ( R  IdlGen  S )  <-> 
S  C_  I )
)
12 sseq1 3369 . . . . . 6  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( R  IdlGen  S )  C_  j 
<->  I  C_  j )
)
1312imbi2d 308 . . . . 5  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( S  C_  j  ->  ( R  IdlGen  S )  C_  j )  <->  ( S  C_  j  ->  I  C_  j
) ) )
1413ralbidv 2725 . . . 4  |-  ( ( R  IdlGen  S )  =  I  ->  ( A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  ( R  IdlGen  S )  C_  j )  <->  A. j  e.  ( Idl `  R
) ( S  C_  j  ->  I  C_  j
) ) )
1510, 11, 143anbi123d 1254 . . 3  |-  ( ( R  IdlGen  S )  =  I  ->  ( (
( R  IdlGen  S )  e.  ( Idl `  R
)  /\  S  C_  ( R  IdlGen  S )  /\  A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  ( R  IdlGen  S )  C_  j ) )  <->  ( I  e.  ( Idl `  R
)  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  I  C_  j )
) ) )
169, 15syl5ibcom 212 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  (
( R  IdlGen  S )  =  I  ->  (
I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R
) ( S  C_  j  ->  I  C_  j
) ) ) )
17 igenmin 26674 . . . . . 6  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  C_  I
)
18173adant3r3 1164 . . . . 5  |-  ( ( R  e.  RingOps  /\  (
I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R
) ( S  C_  j  ->  I  C_  j
) ) )  -> 
( R  IdlGen  S ) 
C_  I )
1918adantlr 696 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  C_  X )  /\  ( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j ) ) )  ->  ( R  IdlGen  S )  C_  I )
20 sseq2 3370 . . . . . . . . . 10  |-  ( i  =  j  ->  ( S  C_  i  <->  S  C_  j
) )
2120ralrab 3096 . . . . . . . . 9  |-  ( A. j  e.  { i  e.  ( Idl `  R
)  |  S  C_  i } I  C_  j  <->  A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j ) )
2221biimpri 198 . . . . . . . 8  |-  ( A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j )  ->  A. j  e.  { i  e.  ( Idl `  R )  |  S  C_  i } I  C_  j )
23 ssint 4066 . . . . . . . 8  |-  ( I 
C_  |^| { i  e.  ( Idl `  R
)  |  S  C_  i }  <->  A. j  e.  {
i  e.  ( Idl `  R )  |  S  C_  i } I  C_  j )
2422, 23sylibr 204 . . . . . . 7  |-  ( A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j )  ->  I  C_ 
|^| { i  e.  ( Idl `  R )  |  S  C_  i } )
25243ad2ant3 980 . . . . . 6  |-  ( ( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R
) ( S  C_  j  ->  I  C_  j
) )  ->  I  C_ 
|^| { i  e.  ( Idl `  R )  |  S  C_  i } )
2625adantl 453 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  C_  X )  /\  ( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j ) ) )  ->  I  C_  |^| { i  e.  ( Idl `  R
)  |  S  C_  i } )
271, 2igenval 26671 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { i  e.  ( Idl `  R )  |  S  C_  i } )
2827adantr 452 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  C_  X )  /\  ( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j ) ) )  ->  ( R  IdlGen  S )  =  |^| { i  e.  ( Idl `  R
)  |  S  C_  i } )
2926, 28sseqtr4d 3385 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  C_  X )  /\  ( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j ) ) )  ->  I  C_  ( R  IdlGen  S ) )
3019, 29eqssd 3365 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  C_  X )  /\  ( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j ) ) )  ->  ( R  IdlGen  S )  =  I )
3130ex 424 . 2  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  (
( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  I  C_  j )
)  ->  ( R  IdlGen  S )  =  I ) )
3216, 31impbid 184 1  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  (
( R  IdlGen  S )  =  I  <->  ( I  e.  ( Idl `  R
)  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  I  C_  j )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709    C_ wss 3320   |^|cint 4050   ran crn 4879   ` cfv 5454  (class class class)co 6081   1stc1st 6347   RingOpscrngo 21963   Idlcidl 26617    IdlGen cigen 26669
This theorem is referenced by:  prnc  26677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ablo 21870  df-rngo 21964  df-idl 26620  df-igen 26670
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