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Theorem cdlemk20-2N 34503
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be fi. Our  D,  C,  O,  Q,  U,  V represent their f1, f2, k1, k2, sigma1, sigma2. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemk2.b  |-  B  =  ( Base `  K
)
cdlemk2.l  |-  .<_  =  ( le `  K )
cdlemk2.j  |-  .\/  =  ( join `  K )
cdlemk2.m  |-  ./\  =  ( meet `  K )
cdlemk2.a  |-  A  =  ( Atoms `  K )
cdlemk2.h  |-  H  =  ( LHyp `  K
)
cdlemk2.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk2.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk2.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk2.q  |-  Q  =  ( S `  C
)
cdlemk2.v  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T  ( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
cdlemk2a.o  |-  O  =  ( S `  D
)
Assertion
Ref Expression
cdlemk20-2N  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( V `  D ) `  P )  =  ( O `  P ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    C, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i    ./\ , d    .\/ , d    C, d, k    Q, d    P, d    R, d    T, d    W, d    ./\ , k    .<_ , k    .\/ , k    A, k    C, k   
k, F    k, H    k, K    k, N    Q, k    P, k    R, k    T, k    k, W    F, d    i, k, f, D, d
Allowed substitution hints:    A( f, d)    B( f, i, k, d)    Q( f, i)    S( f, i, k, d)    H( f, d)    K( f, d)    .<_ ( f, d)    N( d)    O( f, i, k, d)    V( f, i, k, d)

Proof of Theorem cdlemk20-2N
StepHypRef Expression
1 simp11 1044 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
2 simp12 1045 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H )
31, 2jca 539 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp211 1152 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T )
5 simp212 1153 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  e.  T )
6 simp213 1154 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  N  e.  T )
7 simp22l 1133 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  D  e.  T )
86, 7jca 539 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( N  e.  T  /\  D  e.  T ) )
9 simp33 1052 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
10 simp13 1046 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  F )  =  ( R `  N ) )
11 simp32l 1139 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
12 simp32r 1140 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  =/=  (  _I  |`  B ) )
13 simp22r 1134 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  D  =/=  (  _I  |`  B ) )
1411, 12, 133jca 1194 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
15 simp31 1050 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( R `  C )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 C ) ) )
16 cdlemk2.b . . 3  |-  B  =  ( Base `  K
)
17 cdlemk2.l . . 3  |-  .<_  =  ( le `  K )
18 cdlemk2.j . . 3  |-  .\/  =  ( join `  K )
19 cdlemk2.m . . 3  |-  ./\  =  ( meet `  K )
20 cdlemk2.a . . 3  |-  A  =  ( Atoms `  K )
21 cdlemk2.h . . 3  |-  H  =  ( LHyp `  K
)
22 cdlemk2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
23 cdlemk2.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
24 cdlemk2.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
25 cdlemk2.q . . 3  |-  Q  =  ( S `  C
)
26 cdlemk2.v . . 3  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T  ( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
27 cdlemk2a.o . . 3  |-  O  =  ( S `  D
)
2816, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27cdlemk20 34485 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  C  e.  T )  /\  (
( N  e.  T  /\  D  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  C )
) ) )  -> 
( ( V `  D ) `  P
)  =  ( O `
 P ) )
293, 4, 5, 8, 9, 10, 14, 15, 28syl332anc 1307 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  C  e.  T  /\  N  e.  T )  /\  ( D  e.  T  /\  D  =/=  (  _I  |`  B ) )  /\  ( C  e.  T  /\  C  =/=  (  _I  |`  B ) ) )  /\  (
( ( R `  C )  =/=  ( R `  F )  /\  ( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  C ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( V `  D ) `  P )  =  ( O `  P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   class class class wbr 4415    |-> cmpt 4474    _I cid 4762   `'ccnv 4851    |` cres 4854    o. ccom 4856   ` cfv 5600   iota_crio 6275  (class class class)co 6314   Basecbs 15169   lecple 15245   joincjn 16237   meetcmee 16238   Atomscatm 32873   HLchlt 32960   LHypclh 33593   LTrncltrn 33710   trLctrl 33768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-riotaBAD 32569
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-1st 6819  df-2nd 6820  df-undef 7045  df-map 7499  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-p1 16334  df-lat 16340  df-clat 16402  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lplanes 33108  df-lvols 33109  df-lines 33110  df-psubsp 33112  df-pmap 33113  df-padd 33405  df-lhyp 33597  df-laut 33598  df-ldil 33713  df-ltrn 33714  df-trl 33769
This theorem is referenced by: (None)
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