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Theorem ltrniotacl 33855
Description: Version of cdleme50ltrn 33833 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 17-Apr-2013.)
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
ltrniotacl  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
Distinct variable groups:    A, f    f, H    f, K    .<_ , f    P, f    Q, f    T, f   
f, W
Allowed substitution hint:    F( f)

Proof of Theorem ltrniotacl
Dummy variables  s 
t  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 ltrniotaval.l . 2  |-  .<_  =  ( le `  K )
3 eqid 2429 . 2  |-  ( join `  K )  =  (
join `  K )
4 eqid 2429 . 2  |-  ( meet `  K )  =  (
meet `  K )
5 ltrniotaval.a . 2  |-  A  =  ( Atoms `  K )
6 ltrniotaval.h . 2  |-  H  =  ( LHyp `  K
)
7 eqid 2429 . 2  |-  ( ( P ( join `  K
) Q ) (
meet `  K ) W )  =  ( ( P ( join `  K ) Q ) ( meet `  K
) W )
8 eqid 2429 . 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 2429 . 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 2429 . 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, 12cdlemg1ltrnlem 33850 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F  e.  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   [_csb 3401   ifcif 3915   class class class wbr 4426    |-> cmpt 4484   ` cfv 5601   iota_crio 6266  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   Atomscatm 32538   HLchlt 32625   LHypclh 33258   LTrncltrn 33375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-riotaBAD 32234
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7028  df-map 7482  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773  df-lvols 32774  df-lines 32775  df-psubsp 32777  df-pmap 32778  df-padd 33070  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434
This theorem is referenced by:  ltrniotacnvval  33858  ltrniotaidvalN  33859  ltrniotavalbN  33860  cdlemg1ci2  33862  cdlemki  34117  cdlemkj  34139  cdlemm10N  34395  dicssdvh  34463  dicvaddcl  34467  dicvscacl  34468  dicn0  34469  diclspsn  34471  cdlemn2  34472  cdlemn2a  34473  cdlemn3  34474  cdlemn4  34475  cdlemn4a  34476  cdlemn6  34479  cdlemn8  34481  cdlemn9  34482  cdlemn11a  34484  dihordlem7b  34492  dihopelvalcpre  34525  dih1  34563  dihmeetlem1N  34567  dihglblem5apreN  34568  dihglbcpreN  34577  dihmeetlem4preN  34583  dihmeetlem13N  34596  dih1dimatlem0  34605  dihatlat  34611  dihatexv  34615  dihjatcclem3  34697  dihjatcclem4  34698
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