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Theorem ltrniotacnvN 30769
Description: Version of cdleme51finvtrN 30747 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
ltrniotacnvN  |-  ( ( ( 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 ltrniotacnvN
Dummy variables  s 
t  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 ltrniotaval.l . 2  |-  .<_  =  ( le `  K )
3 eqid 2283 . 2  |-  ( join `  K )  =  (
join `  K )
4 eqid 2283 . 2  |-  ( meet `  K )  =  (
meet `  K )
5 ltrniotaval.a . 2  |-  A  =  ( Atoms `  K )
6 ltrniotaval.h . 2  |-  H  =  ( LHyp `  K
)
7 eqid 2283 . 2  |-  ( ( P ( join `  K
) Q ) (
meet `  K ) W )  =  ( ( P ( join `  K ) Q ) ( meet `  K
) W )
8 eqid 2283 . 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 2283 . 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 2283 . 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, 12cdlemg1finvtrlemN 30764 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  `' F  e.  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   [_csb 3081   ifcif 3565   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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