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Theorem cdlemk19u 31159
Description: Part of Lemma K of [Crawley] p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with  F,  N,  U. (Contributed by NM, 31-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 ) )
cdlemk5.u  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
Assertion
Ref Expression
cdlemk19u  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  F
)  =  N )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, z,  ./\    .<_ , b   
z, g,  .<_    .\/ , b,
z    A, b, g, z    B, b, z    F, b, g, z    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
Allowed substitution hints:    U( z, g, b)    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk19u
StepHypRef Expression
1 simp1l 979 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp1 955 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) ) )
3 simp2l 981 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
4 simp2r 982 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  N  e.  T )
5 simp3 957 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
6 cdlemk5.b . . . 4  |-  B  =  ( Base `  K
)
7 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
10 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
11 cdlemk5.h . . . 4  |-  H  =  ( LHyp `  K
)
12 cdlemk5.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
13 cdlemk5.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
14 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
15 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
16 cdlemk5.x . . . 4  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
17 cdlemk5.u . . . 4  |-  U  =  ( g  e.  T  |->  if ( F  =  N ,  g ,  X ) )
186, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemk35u 31153 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T  /\  F  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  F
)  e.  T )
192, 3, 4, 3, 5, 18syl131anc 1195 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  F
)  e.  T )
20 simpr 447 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  F  =  N )
21 simpl2l 1008 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  F  e.  T )
2216, 17cdlemk40t 31107 . . . . . 6  |-  ( ( F  =  N  /\  F  e.  T )  ->  ( U `  F
)  =  F )
2320, 21, 22syl2anc 642 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( U `  F
)  =  F )
2423fveq1d 5527 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( ( U `  F ) `  P
)  =  ( F `
 P ) )
25 fveq1 5524 . . . . 5  |-  ( F  =  N  ->  ( F `  P )  =  ( N `  P ) )
2625adantl 452 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( F `  P
)  =  ( N `
 P ) )
2724, 26eqtrd 2315 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =  N )  ->  ( ( U `  F ) `  P
)  =  ( N `
 P ) )
28 simpl1 958 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) ) )
29 simpl2l 1008 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  ->  F  e.  T )
30 simpr 447 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  ->  F  =/=  N )
31 simpl2r 1009 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  ->  N  e.  T )
32 simpl3 960 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
336, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemk19u1 31158 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  F  =/=  N  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( ( U `  F ) `  P
)  =  ( N `
 P ) )
3428, 29, 30, 31, 32, 33syl131anc 1195 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  F  =/=  N )  -> 
( ( U `  F ) `  P
)  =  ( N `
 P ) )
3527, 34pm2.61dane 2524 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( ( U `  F ) `  P
)  =  ( N `
 P ) )
367, 10, 11, 12cdlemd 30396 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  F )  e.  T  /\  N  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( U `
 F ) `  P )  =  ( N `  P ) )  ->  ( U `  F )  =  N )
371, 19, 4, 5, 35, 36syl311anc 1196 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  N  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( U `  F
)  =  N )
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   ifcif 3565   class class class wbr 4023    e. cmpt 4077    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   ` 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   trLctrl 30347
This theorem is referenced by:  cdlemk19w  31161
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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