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Theorem cdlemkid 35607
Description: The value of the tau function (in Lemma K of [Crawley] p. 118) on the identity relation. (Contributed by NM, 25-Jul-2013.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk5.x  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
Assertion
Ref Expression
cdlemkid  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  [_ G  /  g ]_ X  =  (  _I  |`  B ) )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, G, z    ./\ , b, z    .<_ , b    z,
g,  .<_    .\/ , b, z    A, b, g, z    B, b, z    F, b, g, z   
z, G    H, b,
g, z    K, b,
g, z    N, b,
g, z    P, b,
z    R, b, z    T, b, z    W, b, g, z    z, Y    G, b
Allowed substitution hints:    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemkid
StepHypRef Expression
1 cdlemk5.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
2 fvex 5867 . . 3  |-  ( (
LTrn `  K ) `  W )  e.  _V
31, 2eqeltri 2544 . 2  |-  T  e. 
_V
4 nfv 1678 . . 3  |-  F/ b ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )
5 nfcv 2622 . . . . . 6  |-  F/_ b G
6 cdlemk5.x . . . . . . 7  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
7 nfra1 2838 . . . . . . . 8  |-  F/ b A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )
8 nfcv 2622 . . . . . . . 8  |-  F/_ b T
97, 8nfriota 6245 . . . . . . 7  |-  F/_ b
( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
106, 9nfcxfr 2620 . . . . . 6  |-  F/_ b X
115, 10nfcsb 3446 . . . . 5  |-  F/_ b [_ G  /  g ]_ X
1211nfeq1 2637 . . . 4  |-  F/ b
[_ G  /  g ]_ X  =  (  _I  |`  B )
1312a1i 11 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  F/ b [_ G  /  g ]_ X  =  (  _I  |`  B ) )
14 cdlemk5.b . . . 4  |-  B  =  ( Base `  K
)
15 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
16 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
17 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
18 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
19 cdlemk5.h . . . 4  |-  H  =  ( LHyp `  K
)
20 cdlemk5.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
21 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
22 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
2314, 15, 16, 17, 18, 19, 1, 20, 21, 22, 6cdlemkid4 35605 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  [_ G  /  g ]_ X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  (  _I  |`  B )
) ) )
24 eqeq1 2464 . . . 4  |-  ( (  _I  |`  B )  =  [_ G  /  g ]_ X  ->  ( (  _I  |`  B )  =  (  _I  |`  B )  <->  [_ G  /  g ]_ X  =  (  _I  |`  B ) ) )
2524adantl 466 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  (  _I  |`  B )  = 
[_ G  /  g ]_ X )  ->  (
(  _I  |`  B )  =  (  _I  |`  B )  <->  [_ G  /  g ]_ X  =  (  _I  |`  B ) ) )
26 eqidd 2461 . . . 4  |-  ( ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) )  -> 
(  _I  |`  B )  =  (  _I  |`  B ) )
2726a1i 11 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  (
( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) )  -> 
(  _I  |`  B )  =  (  _I  |`  B ) ) )
2814, 15, 16, 17, 18, 19, 1, 20, 21, 22, 6cdlemkid5 35606 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  [_ G  /  g ]_ X  e.  T )
2914, 19, 1, 20cdlemftr2 35237 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) )
30293ad2ant1 1012 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) )
314, 13, 23, 25, 27, 28, 30riotasv3d 33638 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B ) ) )  /\  T  e.  _V )  ->  [_ G  /  g ]_ X  =  (  _I  |`  B ) )
323, 31mpan2 671 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  G  =  (  _I  |`  B )
) )  ->  [_ G  /  g ]_ X  =  (  _I  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   F/wnf 1594    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   _Vcvv 3106   [_csb 3428   class class class wbr 4440    _I cid 4783   `'ccnv 4991    |` cres 4994    o. ccom 4996   ` cfv 5579   iota_crio 6235  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Atomscatm 33935   HLchlt 34022   LHypclh 34655   LTrncltrn 34772   trLctrl 34829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-undef 6992  df-map 7412  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830
This theorem is referenced by:  cdlemk35s-id  35609  cdlemk39s-id  35611  cdlemk53b  35627  cdlemk53  35628
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