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Theorem cdlemk18-2N 34869
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119.  N,  V,  Q,  C are k, sigma2 (p), k2, f2. (Contributed by NM, 2-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 ) ) ) ) ) )
Assertion
Ref Expression
cdlemk18-2N  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( N `  P )  =  ( ( V `  F
) `  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
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)    V( f, i, k, d)

Proof of Theorem cdlemk18-2N
StepHypRef Expression
1 simp11 1018 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
2 simp12 1019 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H
)
31, 2jca 532 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp21 1021 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  e.  T
)
5 simp22 1022 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  e.  T
)
6 simp23 1023 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  N  e.  T
)
7 simp33 1026 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp13 1020 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  F )  =  ( R `  N ) )
9 simp32l 1113 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
10 simp32r 1114 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  C  =/=  (  _I  |`  B ) )
11 simp31 1024 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( R `  C )  =/=  ( R `  F )
)
12 cdlemk2.b . . 3  |-  B  =  ( Base `  K
)
13 cdlemk2.l . . 3  |-  .<_  =  ( le `  K )
14 cdlemk2.j . . 3  |-  .\/  =  ( join `  K )
15 cdlemk2.m . . 3  |-  ./\  =  ( meet `  K )
16 cdlemk2.a . . 3  |-  A  =  ( Atoms `  K )
17 cdlemk2.h . . 3  |-  H  =  ( LHyp `  K
)
18 cdlemk2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
19 cdlemk2.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
20 cdlemk2.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
21 cdlemk2.q . . 3  |-  Q  =  ( S `  C
)
22 cdlemk2.v . . 3  |-  V  =  ( d  e.  T  |->  ( iota_ k  e.  T  ( k `  P
)  =  ( ( P  .\/  ( R `
 d ) ) 
./\  ( ( Q `
 P )  .\/  ( R `  ( d  o.  `' C ) ) ) ) ) )
2312, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22cdlemk18 34851 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  C  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B )  /\  ( R `  C )  =/=  ( R `  F ) ) )  ->  ( N `  P )  =  ( ( V `  F
) `  P )
)
243, 4, 5, 6, 7, 8, 9, 10, 11, 23syl333anc 1251 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  e.  T  /\  C  e.  T  /\  N  e.  T
)  /\  ( ( R `  C )  =/=  ( R `  F
)  /\  ( F  =/=  (  _I  |`  B )  /\  C  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( N `  P )  =  ( ( V `  F
) `  P )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401    |-> cmpt 4459    _I cid 4740   `'ccnv 4948    |` cres 4951    o. ccom 4953   ` cfv 5527   iota_crio 6161  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   Atomscatm 33247   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   trLctrl 34141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-riotaBAD 32943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-undef 6903  df-map 7327  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482  df-lvols 33483  df-lines 33484  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142
This theorem is referenced by:  cdlemk18-3N  34883
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