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Theorem cdlemk43N 30419
Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 31-Jul-2013.) (New usage is discouraged.)
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 ) )
cdlemk5.u  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
Assertion
Ref Expression
cdlemk43N  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
( U `  G
) `  P )  =  [_ G  /  g ]_ Y )
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:    U( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk43N
StepHypRef Expression
1 simp213 1100 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  F  =/=  N )
2 simp22l 1079 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  G  e.  T )
3 cdlemk5.x . . . . 5  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
4 cdlemk5.u . . . . 5  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
53, 4cdlemk40f 30375 . . . 4  |-  ( ( F  =/=  N  /\  G  e.  T )  ->  ( U `  G
)  =  [_ G  /  g ]_ X
)
61, 2, 5syl2anc 645 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( U `  G )  =  [_ G  /  g ]_ X )
76fveq1d 5487 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
( U `  G
) `  P )  =  ( [_ G  /  g ]_ X `  P ) )
8 simp1l 984 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
9 simp211 1098 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  F  e.  T )
10 simp212 1099 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  N  e.  T )
11 simp1r 985 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
12 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
13 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
14 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
15 cdlemk5.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
1612, 13, 14, 15trlnid 29635 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  N  e.  T )  /\  ( F  =/=  N  /\  ( R `  F )  =  ( R `  N ) ) )  ->  F  =/=  (  _I  |`  B ) )
178, 9, 10, 1, 11, 16syl122anc 1196 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  F  =/=  (  _I  |`  B ) )
189, 17jca 520 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )
19 simp22 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )
20 simp23 995 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
21 simp3 962 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )
22 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
23 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
24 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
25 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
26 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
27 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
2812, 22, 23, 24, 25, 13, 14, 15, 26, 27, 3cdlemk42 30397 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) )  -> 
( [_ G  /  g ]_ X `  P )  =  [_ G  / 
g ]_ Y )
298, 18, 19, 10, 20, 11, 21, 28syl331anc 1212 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( [_ G  /  g ]_ X `  P )  =  [_ G  / 
g ]_ Y )
307, 29eqtrd 2316 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  N  e.  T  /\  F  =/= 
N )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
( U `  G
) `  P )  =  [_ G  /  g ]_ Y )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    =/= wne 2447   A.wral 2544   [_csb 3082   ifcif 3566   class class class wbr 4024    e. cmpt 4078    _I cid 4303   `'ccnv 4687    |` cres 4690    o. ccom 4692   ` cfv 5221  (class class class)co 5819   iota_crio 6290   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615
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