Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  igenval2 Structured version   Unicode version

Theorem igenval2 30358
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 30355 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R
) )
41, 2igenss 30354 . . . 4  |-  ( ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )
5 igenmin 30356 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  j  e.  ( Idl `  R
)  /\  S  C_  j
)  ->  ( R  IdlGen  S )  C_  j
)
653expia 1198 . . . . . 6  |-  ( ( R  e.  RingOps  /\  j  e.  ( Idl `  R
) )  ->  ( S  C_  j  ->  ( R  IdlGen  S )  C_  j ) )
76ralrimiva 2881 . . . . 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 1176 . . 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 2539 . . . 4  |-  ( ( R  IdlGen  S )  =  I  ->  ( ( R  IdlGen  S )  e.  ( Idl `  R
)  <->  I  e.  ( Idl `  R ) ) )
11 sseq2 3531 . . . 4  |-  ( ( R  IdlGen  S )  =  I  ->  ( S  C_  ( R  IdlGen  S )  <-> 
S  C_  I )
)
12 sseq1 3530 . . . . . 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 2906 . . . 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 1299 . . 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 30356 . . . . . 6  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  S  C_  I
)  ->  ( R  IdlGen  S )  C_  I
)
18173adant3r3 1207 . . . . 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 3531 . . . . . . . . . 10  |-  ( i  =  j  ->  ( S  C_  i  <->  S  C_  j
) )
2120ralrab 3270 . . . . . . . . 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 4303 . . . . . . . 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 1019 . . . . . 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 30353 . . . . . 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 3546 . . . 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 3526 . . 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 973    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821    C_ wss 3481   |^|cint 4287   ran crn 5005   ` cfv 5593  (class class class)co 6294   1stc1st 6792   RingOpscrngo 25168   Idlcidl 30299    IdlGen cigen 30351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fo 5599  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-1st 6794  df-2nd 6795  df-grpo 24984  df-gid 24985  df-ablo 25075  df-rngo 25169  df-idl 30302  df-igen 30352
This theorem is referenced by:  prnc  30359
  Copyright terms: Public domain W3C validator