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Theorem pridlc 26799
Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
Hypotheses
Ref Expression
ispridlc.1  |-  G  =  ( 1st `  R
)
ispridlc.2  |-  H  =  ( 2nd `  R
)
ispridlc.3  |-  X  =  ran  G
Assertion
Ref Expression
pridlc  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  e.  P ) )  -> 
( A  e.  P  \/  B  e.  P
) )

Proof of Theorem pridlc
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispridlc.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 ispridlc.2 . . . . 5  |-  H  =  ( 2nd `  R
)
3 ispridlc.3 . . . . 5  |-  X  =  ran  G
41, 2, 3ispridlc 26798 . . . 4  |-  ( R  e. CRingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( (
a H b )  e.  P  ->  (
a  e.  P  \/  b  e.  P )
) ) ) )
54biimpa 470 . . 3  |-  ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  ->  ( P  e.  ( Idl `  R
)  /\  P  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P ) ) ) )
65simp3d 969 . 2  |-  ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  ->  A. a  e.  X  A. b  e.  X  ( (
a H b )  e.  P  ->  (
a  e.  P  \/  b  e.  P )
) )
7 oveq1 5881 . . . . . . . 8  |-  ( a  =  A  ->  (
a H b )  =  ( A H b ) )
87eleq1d 2362 . . . . . . 7  |-  ( a  =  A  ->  (
( a H b )  e.  P  <->  ( A H b )  e.  P ) )
9 eleq1 2356 . . . . . . . 8  |-  ( a  =  A  ->  (
a  e.  P  <->  A  e.  P ) )
109orbi1d 683 . . . . . . 7  |-  ( a  =  A  ->  (
( a  e.  P  \/  b  e.  P
)  <->  ( A  e.  P  \/  b  e.  P ) ) )
118, 10imbi12d 311 . . . . . 6  |-  ( a  =  A  ->  (
( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P
) )  <->  ( ( A H b )  e.  P  ->  ( A  e.  P  \/  b  e.  P ) ) ) )
12 oveq2 5882 . . . . . . . 8  |-  ( b  =  B  ->  ( A H b )  =  ( A H B ) )
1312eleq1d 2362 . . . . . . 7  |-  ( b  =  B  ->  (
( A H b )  e.  P  <->  ( A H B )  e.  P
) )
14 eleq1 2356 . . . . . . . 8  |-  ( b  =  B  ->  (
b  e.  P  <->  B  e.  P ) )
1514orbi2d 682 . . . . . . 7  |-  ( b  =  B  ->  (
( A  e.  P  \/  b  e.  P
)  <->  ( A  e.  P  \/  B  e.  P ) ) )
1613, 15imbi12d 311 . . . . . 6  |-  ( b  =  B  ->  (
( ( A H b )  e.  P  ->  ( A  e.  P  \/  b  e.  P
) )  <->  ( ( A H B )  e.  P  ->  ( A  e.  P  \/  B  e.  P ) ) ) )
1711, 16rspc2v 2903 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P ) )  -> 
( ( A H B )  e.  P  ->  ( A  e.  P  \/  B  e.  P
) ) ) )
1817com12 27 . . . 4  |-  ( A. a  e.  X  A. b  e.  X  (
( a H b )  e.  P  -> 
( a  e.  P  \/  b  e.  P
) )  ->  (
( A  e.  X  /\  B  e.  X
)  ->  ( ( A H B )  e.  P  ->  ( A  e.  P  \/  B  e.  P ) ) ) )
1918exp3a 425 . . 3  |-  ( A. a  e.  X  A. b  e.  X  (
( a H b )  e.  P  -> 
( a  e.  P  \/  b  e.  P
) )  ->  ( A  e.  X  ->  ( B  e.  X  -> 
( ( A H B )  e.  P  ->  ( A  e.  P  \/  B  e.  P
) ) ) ) )
20193imp2 1166 . 2  |-  ( ( A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P
) )  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  e.  P ) )  ->  ( A  e.  P  \/  B  e.  P ) )
216, 20sylan 457 1  |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R )
)  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  e.  P ) )  -> 
( A  e.  P  \/  B  e.  P
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  CRingOpsccring 26723   Idlcidl 26735   PrIdlcpridl 26736
This theorem is referenced by:  pridlc2  26800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-com2 21094  df-crngo 26724  df-idl 26738  df-pridl 26739  df-igen 26788
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