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Theorem cdlemg2idN 31330
Description: Version of cdleme31id 31128 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 21-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg2id.l  |-  .<_  =  ( le `  K )
cdlemg2id.a  |-  A  =  ( Atoms `  K )
cdlemg2id.h  |-  H  =  ( LHyp `  K
)
cdlemg2id.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg2id.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
cdlemg2idN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  ( F `  X )  =  X )

Proof of Theorem cdlemg2idN
Dummy variables  s 
t  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp111 1086 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  K  e.  HL )
2 simp112 1087 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  W  e.  H )
3 simp12 988 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp13 989 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
5 simp113 1088 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  F  e.  T )
6 simp2l 983 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  ( F `  P )  =  Q )
7 cdlemg2id.b . . . . 5  |-  B  =  ( Base `  K
)
8 cdlemg2id.l . . . . 5  |-  .<_  =  ( le `  K )
9 eqid 2435 . . . . 5  |-  ( join `  K )  =  (
join `  K )
10 eqid 2435 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
11 cdlemg2id.a . . . . 5  |-  A  =  ( Atoms `  K )
12 cdlemg2id.h . . . . 5  |-  H  =  ( LHyp `  K
)
13 cdlemg2id.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
14 eqid 2435 . . . . 5  |-  ( ( P ( join `  K
) Q ) (
meet `  K ) W )  =  ( ( P ( join `  K ) Q ) ( meet `  K
) W )
15 eqid 2435 . . . . 5  |-  ( ( t ( join `  K
) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) )  =  ( ( t ( join `  K
) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) )
16 eqid 2435 . . . . 5  |-  ( ( P ( join `  K
) Q ) (
meet `  K )
( ( ( t ( join `  K
) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ( join `  K
) ( ( s ( join `  K
) t ) (
meet `  K ) W ) ) )  =  ( ( P ( join `  K
) Q ) (
meet `  K )
( ( ( t ( join `  K
) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ( join `  K
) ( ( s ( join `  K
) t ) (
meet `  K ) W ) ) )
17 eqid 2435 . . . . 5  |-  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s ( join `  K ) ( x ( meet `  K
) W ) )  =  x )  -> 
z  =  ( if ( s  .<_  ( P ( join `  K
) Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P ( join `  K ) Q ) )  ->  y  =  ( ( P (
join `  K ) Q ) ( meet `  K ) ( ( ( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ( join `  K
) ( ( s ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ) ,  [_ s  /  t ]_ (
( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ( join `  K
) ( x (
meet `  K ) W ) ) ) ) ,  x ) )  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s ( join `  K ) ( x ( meet `  K
) W ) )  =  x )  -> 
z  =  ( if ( s  .<_  ( P ( join `  K
) Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P ( join `  K ) Q ) )  ->  y  =  ( ( P (
join `  K ) Q ) ( meet `  K ) ( ( ( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ( join `  K
) ( ( s ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ) ,  [_ s  /  t ]_ (
( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ( join `  K
) ( x (
meet `  K ) W ) ) ) ) ,  x ) )
187, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemg2dN 31324 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  ( F `  P )  =  Q ) )  ->  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s ( join `  K ) ( x ( meet `  K
) W ) )  =  x )  -> 
z  =  ( if ( s  .<_  ( P ( join `  K
) Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P ( join `  K ) Q ) )  ->  y  =  ( ( P (
join `  K ) Q ) ( meet `  K ) ( ( ( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ( join `  K
) ( ( s ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ) ,  [_ s  /  t ]_ (
( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ( join `  K
) ( x (
meet `  K ) W ) ) ) ) ,  x ) ) )
191, 2, 3, 4, 5, 6, 18syl222anc 1200 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s ( join `  K ) ( x ( meet `  K
) W ) )  =  x )  -> 
z  =  ( if ( s  .<_  ( P ( join `  K
) Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P ( join `  K ) Q ) )  ->  y  =  ( ( P (
join `  K ) Q ) ( meet `  K ) ( ( ( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ( join `  K
) ( ( s ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ) ,  [_ s  /  t ]_ (
( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ( join `  K
) ( x (
meet `  K ) W ) ) ) ) ,  x ) ) )
2019fveq1d 5722 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  ( F `  X )  =  ( ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s ( join `  K ) ( x ( meet `  K
) W ) )  =  x )  -> 
z  =  ( if ( s  .<_  ( P ( join `  K
) Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P ( join `  K ) Q ) )  ->  y  =  ( ( P (
join `  K ) Q ) ( meet `  K ) ( ( ( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ( join `  K
) ( ( s ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ) ,  [_ s  /  t ]_ (
( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ( join `  K
) ( x (
meet `  K ) W ) ) ) ) ,  x ) ) `  X ) )
21 simp2r 984 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  X  e.  B )
22 simp3 959 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  P  =  Q )
2317cdleme31id 31128 . . 3  |-  ( ( X  e.  B  /\  P  =  Q )  ->  ( ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s ( join `  K ) ( x ( meet `  K
) W ) )  =  x )  -> 
z  =  ( if ( s  .<_  ( P ( join `  K
) Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P ( join `  K ) Q ) )  ->  y  =  ( ( P (
join `  K ) Q ) ( meet `  K ) ( ( ( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ( join `  K
) ( ( s ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ) ,  [_ s  /  t ]_ (
( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ( join `  K
) ( x (
meet `  K ) W ) ) ) ) ,  x ) ) `  X )  =  X )
2421, 22, 23syl2anc 643 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  (
( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P
( join `  K ) Q ) ,  (
iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P ( join `  K ) Q ) )  ->  y  =  ( ( P (
join `  K ) Q ) ( meet `  K ) ( ( ( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ( join `  K
) ( ( s ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ) ,  [_ s  /  t ]_ (
( t ( join `  K ) ( ( P ( join `  K
) Q ) (
meet `  K ) W ) ) (
meet `  K )
( Q ( join `  K ) ( ( P ( join `  K
) t ) (
meet `  K ) W ) ) ) ) ( join `  K
) ( x (
meet `  K ) W ) ) ) ) ,  x ) ) `  X )  =  X )
2520, 24eqtrd 2467 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F `  P )  =  Q  /\  X  e.  B
)  /\  P  =  Q )  ->  ( F `  X )  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   [_csb 3243   ifcif 3731   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   iota_crio 6534   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893
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