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Theorem igenval2 28864
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 28861 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R
) )
41, 2igenss 28860 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )
5 igenmin 28862 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  j  e.  ( Idl `  R
)  /\  S  C_  j
)  ->  ( R  IdlGen  S )  C_  j
)
653expia 1189 . . . . . 6  |-  ( ( R  e.  RingOps  /\  j  e.  ( Idl `  R
) )  ->  ( S  C_  j  ->  ( R  IdlGen  S )  C_  j ) )
76ralrimiva 2798 . . . . 5  |-  ( R  e.  RingOps  ->  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  ( R  IdlGen  S ) 
C_  j ) )
87adantr 465 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  A. j  e.  ( Idl `  R
) ( S  C_  j  ->  ( R  IdlGen  S )  C_  j )
)
93, 4, 83jca 1168 . . 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 2502 . . . 4  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( R  IdlGen  S )  e.  ( Idl `  R
)  <->  I  e.  ( Idl `  R ) ) )
11 sseq2 3377 . . . 4  |-  ( ( R  IdlGen  S )  =  I  ->  ( S  C_  ( R  IdlGen  S )  <-> 
S  C_  I )
)
12 sseq1 3376 . . . . . 6  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( R  IdlGen  S )  C_  j 
<->  I  C_  j )
)
1312imbi2d 316 . . . . 5  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( S  C_  j  ->  ( R  IdlGen  S )  C_  j )  <->  ( S  C_  j  ->  I  C_  j
) ) )
1413ralbidv 2734 . . . 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 1289 . . 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 220 . 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 28862 . . . . . 6  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  C_  I
)
18173adant3r3 1198 . . . . 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 714 . . . 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 3377 . . . . . . . . . 10  |-  ( i  =  j  ->  ( S  C_  i  <->  S  C_  j
) )
2120ralrab 3120 . . . . . . . . 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 206 . . . . . . . 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 4143 . . . . . . . 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 212 . . . . . . 7  |-  ( A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j )  ->  I  C_ 
|^| { i  e.  ( Idl `  R )  |  S  C_  i } )
25243ad2ant3 1011 . . . . . 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 466 . . . . 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 28859 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { i  e.  ( Idl `  R )  |  S  C_  i } )
2827adantr 465 . . . . 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 3392 . . . 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 3372 . . 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 434 . 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 191 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   {crab 2718    C_ wss 3327   |^|cint 4127   ran crn 4840   ` cfv 5417  (class class class)co 6090   1stc1st 6574   RingOpscrngo 23861   Idlcidl 28805    IdlGen cigen 28857
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-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fo 5423  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-grpo 23677  df-gid 23678  df-ablo 23768  df-rngo 23862  df-idl 28808  df-igen 28858
This theorem is referenced by:  prnc  28865
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