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Theorem cdlemk19u 36436
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 1021 . 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 997 . . 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 1023 . . 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 1024 . . 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 999 . . 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 36430 . . 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 1242 . 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 461 . . . . . 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 1050 . . . . . 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 36384 . . . . . 6  |-  ( ( F  =  N  /\  F  e.  T )  ->  ( U `  F
)  =  F )
2320, 21, 22syl2anc 661 . . . . 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 5858 . . . 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 5855 . . . . 5  |-  ( F  =  N  ->  ( F `  P )  =  ( N `  P ) )
2625adantl 466 . . . 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 2484 . . 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 1000 . . . 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 1050 . . . 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 461 . . . 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 1051 . . . 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 1002 . . . 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 36435 . . . 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 1242 . . 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 2761 . 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 35672 . 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 1243 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   ifcif 3926   class class class wbr 4437    |-> cmpt 4495    _I cid 4780   `'ccnv 4988    |` cres 4991    o. ccom 4993   ` cfv 5578   iota_crio 6241  (class class class)co 6281   Basecbs 14509   lecple 14581   joincjn 15447   meetcmee 15448   Atomscatm 34728   HLchlt 34815   LHypclh 35448   LTrncltrn 35565   trLctrl 35623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-riotaBAD 34424
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-undef 7004  df-map 7424  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-llines 34962  df-lplanes 34963  df-lvols 34964  df-lines 34965  df-psubsp 34967  df-pmap 34968  df-padd 35260  df-lhyp 35452  df-laut 35453  df-ldil 35568  df-ltrn 35569  df-trl 35624
This theorem is referenced by:  cdlemk19w  36438
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