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Theorem igenval2 26691
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 26688 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R
) )
41, 2igenss 26687 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )
5 igenmin 26689 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  j  e.  ( Idl `  R
)  /\  S  C_  j
)  ->  ( R  IdlGen  S )  C_  j
)
653expia 1153 . . . . . 6  |-  ( ( R  e.  RingOps  /\  j  e.  ( Idl `  R
) )  ->  ( S  C_  j  ->  ( R  IdlGen  S )  C_  j ) )
76ralrimiva 2626 . . . . 5  |-  ( R  e.  RingOps  ->  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  ( R  IdlGen  S ) 
C_  j ) )
87adantr 451 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  A. j  e.  ( Idl `  R
) ( S  C_  j  ->  ( R  IdlGen  S )  C_  j )
)
93, 4, 83jca 1132 . . 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 2343 . . . 4  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( R  IdlGen  S )  e.  ( Idl `  R
)  <->  I  e.  ( Idl `  R ) ) )
11 sseq2 3200 . . . 4  |-  ( ( R  IdlGen  S )  =  I  ->  ( S  C_  ( R  IdlGen  S )  <-> 
S  C_  I )
)
12 sseq1 3199 . . . . . 6  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( R  IdlGen  S )  C_  j 
<->  I  C_  j )
)
1312imbi2d 307 . . . . 5  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( S  C_  j  ->  ( R  IdlGen  S )  C_  j )  <->  ( S  C_  j  ->  I  C_  j
) ) )
1413ralbidv 2563 . . . 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 1252 . . 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 211 . 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 26689 . . . . . 6  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  C_  I
)
18173adant3r3 1162 . . . . 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 695 . . . 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 3200 . . . . . . . . . 10  |-  ( i  =  j  ->  ( S  C_  i  <->  S  C_  j
) )
2120ralrab 2927 . . . . . . . . 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 197 . . . . . . . 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 3878 . . . . . . . 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 203 . . . . . . 7  |-  ( A. j  e.  ( Idl `  R ) ( S 
C_  j  ->  I  C_  j )  ->  I  C_ 
|^| { i  e.  ( Idl `  R )  |  S  C_  i } )
25243ad2ant3 978 . . . . . 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 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 ) ) )  ->  I  C_  |^| { i  e.  ( Idl `  R
)  |  S  C_  i } )
271, 2igenval 26686 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  = 
|^| { i  e.  ( Idl `  R )  |  S  C_  i } )
2827adantr 451 . . . . 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 3215 . . . 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 3196 . . 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 423 . 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 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   |^|cint 3862   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   RingOpscrngo 21042   Idlcidl 26632    IdlGen cigen 26684
This theorem is referenced by:  prnc  26692
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-reu 2550  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ablo 20949  df-rngo 21043  df-idl 26635  df-igen 26685
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