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Theorem ltrniotafvawN 30694
Description: Version of cdleme46fvaw 30617 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ltrniotaval.l  |-  .<_  =  ( le `  K )
ltrniotaval.a  |-  A  =  ( Atoms `  K )
ltrniotaval.h  |-  H  =  ( LHyp `  K
)
ltrniotaval.t  |-  T  =  ( ( LTrn `  K
) `  W )
ltrniotaval.f  |-  F  =  ( iota_ f  e.  T
( f `  P
)  =  Q )
Assertion
Ref Expression
ltrniotafvawN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( F `  R )  e.  A  /\  -.  ( F `  R )  .<_  W ) )
Distinct variable groups:    A, f    f, H    f, K    .<_ , f    P, f    Q, f    T, f   
f, W
Allowed substitution hints:    R( f)    F( f)

Proof of Theorem ltrniotafvawN
Dummy variables  s 
t  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2389 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 ltrniotaval.l . 2  |-  .<_  =  ( le `  K )
3 eqid 2389 . 2  |-  ( join `  K )  =  (
join `  K )
4 eqid 2389 . 2  |-  ( meet `  K )  =  (
meet `  K )
5 ltrniotaval.a . 2  |-  A  =  ( Atoms `  K )
6 ltrniotaval.h . 2  |-  H  =  ( LHyp `  K
)
7 eqid 2389 . 2  |-  ( ( P ( join `  K
) Q ) (
meet `  K ) W )  =  ( ( P ( join `  K ) Q ) ( meet `  K
) W )
8 eqid 2389 . 2  |-  ( ( 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 ) ) )
9 eqid 2389 . 2  |-  ( ( 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 ) ) )
10 eqid 2389 . 2  |-  ( x  e.  ( Base `  K
)  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  ( Base `  K
) A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s ( join `  K ) ( x ( meet `  K
) W ) )  =  x )  -> 
z  =  ( if ( s  .<_  ( P ( join `  K
) Q ) ,  ( iota_ y  e.  (
Base `  K ) 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.  ( Base `  K
)  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  ( Base `  K
) A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s ( join `  K ) ( x ( meet `  K
) W ) )  =  x )  -> 
z  =  ( if ( s  .<_  ( P ( join `  K
) Q ) ,  ( iota_ y  e.  (
Base `  K ) 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 ) )
11 ltrniotaval.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
12 ltrniotaval.f . 2  |-  F  =  ( iota_ f  e.  T
( f `  P
)  =  Q )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemg1fvawlemN 30689 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( ( F `  R )  e.  A  /\  -.  ( F `  R )  .<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   [_csb 3196   ifcif 3684   class class class wbr 4155    e. cmpt 4209   ` cfv 5396  (class class class)co 6022   iota_crio 6480   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   Atomscatm 29380   HLchlt 29467   LHypclh 30100   LTrncltrn 30217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-map 6958  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275
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