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Theorem cdlemefs27cl 35701
Description: Part of proof of Lemma E in [Crawley] p. 113. Closure of N. TODO FIX COMMENT This is the start of a re-proof of cdleme27cl 35654 etc. with the s .<_ ( P .\/ Q ) condition (so as to not have the C hypothesis). (Contributed by NM, 24-Mar-2013.)
Hypotheses
Ref Expression
cdlemefs26.b |- B = ( Base ` K )
cdlemefs26.l |- .<_ = ( le ` K )
cdlemefs26.j |- .\/ = ( join ` K )
cdlemefs26.m |- ./\ = ( meet ` K )
cdlemefs26.a |- A = ( Atoms ` K )
cdlemefs26.h |- H = ( LHyp ` K )
cdlemefs27.u |- U = ( ( P .\/ Q ) ./\ W )
cdlemefs27.d |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs27.e |- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemefs27.i |- I = ( iota_ u e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> u = E ) )
cdlemefs27.n |- N = if ( s .<_ ( P .\/ Q ) , I , C )
Assertion
Ref Expression
cdlemefs27cl |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
Distinct variable groups:   u, t, A   t, B, u   u, E   t, H   t, .\/ , u   t, K   t, .<_ , u   t, ./\ , u   t, P, u   t, Q, u   t, U, u   t, W, u   t, s, u
Allowed substitution hints:   A( s)   B( s)   C( u, t, s)   D( u, t, s)   P( s)   Q( s)   U( s)   E( t, s)   H( u, s)   I( u, t, s)   .\/ ( s)   K( u, s)   .<_ ( s)   ./\ ( s)   N( u, t, s)   W( s)
Proof of Theorem cdlemefs27cl
StepHypRef Expression
1 cdlemefs27.n . 2 |- N = if ( s .<_ ( P .\/ Q ) , I , C )
2 simpr2 1068 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> s .<_ ( P .\/ Q ) )
32iftrued 4094 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> if ( s .<_ ( P .\/ Q ) , I , C ) = I )
4 simpl1 1064 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> ( K e. HL /\ W e. H ) )
5 simpl2 1065 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> ( P e. A /\ -. P .<_ W ) )
6 simpl3 1066 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> ( Q e. A /\ -. Q .<_ W ) )
7 simpr1 1067 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> ( s e. A /\ -. s .<_ W ) )
8 simpr3 1069 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> P =/= Q )
9 cdlemefs26.b . . . . 5 |- B = ( Base ` K )
10 cdlemefs26.l . . . . 5 |- .<_ = ( le ` K )
11 cdlemefs26.j . . . . 5 |- .\/ = ( join ` K )
12 cdlemefs26.m . . . . 5 |- ./\ = ( meet ` K )
13 cdlemefs26.a . . . . 5 |- A = ( Atoms ` K )
14 cdlemefs26.h . . . . 5 |- H = ( LHyp ` K )
15 cdlemefs27.u . . . . 5 |- U = ( ( P .\/ Q ) ./\ W )
16 cdlemefs27.d . . . . 5 |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
17 cdlemefs27.e . . . . 5 |- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
18 cdlemefs27.i . . . . 5 |- I = ( iota_ u e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> u = E ) )
199, 10, 11, 12, 13, 14, 15, 16, 17, 18cdleme25cl 35645 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( s e. A /\ -. s .<_ W ) /\ ( P =/= Q /\ s .<_ ( P .\/ Q ) ) ) -> I e. B )
204, 5, 6, 7, 8, 2, 19syl312anc 1347 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> I e. B )
213, 20eqeltrd 2701 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> if ( s .<_ ( P .\/ Q ) , I , C ) e. B )
221, 21syl5eqel 2705 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ifcif 4086   class class class wbr 4653   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270
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
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-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-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
This theorem is referenced by:  cdlemefs29bpre0N  35704  cdlemefs29bpre1N  35705  cdlemefs29cpre1N  35706  cdlemefs29clN  35707  cdlemefs32fvaN  35710  cdlemefs32fva1  35711
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