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Theorem cdlemg2idN 34126
Description: Version of cdleme31id 33924 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 1135 . . . 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 1136 . . . 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 1037 . . . 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 1038 . . . 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 1137 . . . 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 1032 . . . 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 2423 . . . . 5  |-  ( join `  K )  =  (
join `  K )
10 eqid 2423 . . . . 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 2423 . . . . 5  |-  ( ( P ( join `  K
) Q ) (
meet `  K ) W )  =  ( ( P ( join `  K ) Q ) ( meet `  K
) W )
15 eqid 2423 . . . . 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 2423 . . . . 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 2423 . . . . 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 34120 . . . 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 1281 . . 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 5881 . 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 1033 . . 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 1008 . . 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 33924 . . 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 666 . 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 2464 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   [_csb 3396   ifcif 3910   class class class wbr 4421    |-> cmpt 4480   ` cfv 5599   iota_crio 6264  (class class class)co 6303   Basecbs 15114   lecple 15190   joincjn 16182   meetcmee 16183   Atomscatm 32792   HLchlt 32879   LHypclh 33512   LTrncltrn 33629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-riotaBAD 32488
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-undef 7026  df-map 7480  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-oposet 32705  df-ol 32707  df-oml 32708  df-covers 32795  df-ats 32796  df-atl 32827  df-cvlat 32851  df-hlat 32880  df-llines 33026  df-lplanes 33027  df-lvols 33028  df-lines 33029  df-psubsp 33031  df-pmap 33032  df-padd 33324  df-lhyp 33516  df-laut 33517  df-ldil 33632  df-ltrn 33633  df-trl 33688
This theorem is referenced by: (None)
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