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Theorem cdlemk18-2N 36174
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 1091 . . 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 1092 . . 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 554 . 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 1094 . 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 1095 . 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 1096 . 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 1099 . 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 1093 . 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 1186 . 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 1187 . 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 1097 . 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 36156 . 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 1358 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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446
This theorem is referenced by:  cdlemk18-3N  36188
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