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Theorem cdlemk17-2N 29871
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119.  Q,  C are k2, f2. (Contributed by NM, 1-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
)
Assertion
Ref Expression
cdlemk17-2N  |-  ( ( ( 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 )  =  ( ( P  .\/  ( R `  F )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( F  o.  `' C
) ) ) ) )
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
Allowed substitution hints:    A( f)    B( f, i)    Q( f, i)    S( f, i)    H( f)    K( f)    .<_ ( f)

Proof of Theorem cdlemk17-2N
StepHypRef Expression
1 simp11 990 . . 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 991 . . 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 520 . 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 993 . 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 994 . 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 995 . 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 998 . 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 992 . 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 1085 . 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 1086 . 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 996 . 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
)
2212, 13, 14, 15, 16, 17, 18, 19, 20, 21cdlemk17 29848 . 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 )  =  ( ( P  .\/  ( R `  F )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( F  o.  `' C
) ) ) ) )
233, 4, 5, 6, 7, 8, 9, 10, 11, 22syl333anc 1219 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 )  =  ( ( P  .\/  ( R `  F )
)  ./\  ( ( Q `  P )  .\/  ( R `  ( F  o.  `' C
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920    e. cmpt 3974    _I cid 4197   `'ccnv 4579    |` cres 4582    o. ccom 4584   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28254   HLchlt 28341   LHypclh 28974   LTrncltrn 29091   trLctrl 29148
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-llines 28488  df-lplanes 28489  df-lvols 28490  df-lines 28491  df-psubsp 28493  df-pmap 28494  df-padd 28786  df-lhyp 28978  df-laut 28979  df-ldil 29094  df-ltrn 29095  df-trl 29149
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