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Theorem cdleme4 33875
Description: Part of proof of Lemma E in [Crawley] p. 113. F and G represent f(s) and fs(r). Here show p \/ q = r \/ u at the top of p. 114. (Contributed by NM, 7-Jun-2012.)
Hypotheses
Ref Expression
cdleme4.l |- .<_ = ( le ` K )
cdleme4.j |- .\/ = ( join ` K )
cdleme4.m |- ./\ = ( meet ` K )
cdleme4.a |- A = ( Atoms ` K )
cdleme4.h |- H = ( LHyp ` K )
cdleme4.u |- U = ( ( P .\/ Q ) ./\ W )
Assertion
Ref Expression
cdleme4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) = ( R .\/ U ) )
Proof of Theorem cdleme4
StepHypRef Expression
1 cdleme4.u . . 3 |- U = ( ( P .\/ Q ) ./\ W )
21oveq2i 6319 . 2 |- ( R .\/ U ) = ( R .\/ ( ( P .\/ Q ) ./\ W ) )
3 simp1l 1054 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> K e. HL )
4 simp23l 1151 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> R e. A )
5 simp21 1063 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> P e. A )
6 simp22 1064 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> Q e. A )
7 eqid 2471 . . . . . 6 |- ( Base ` K ) = ( Base ` K )
8 cdleme4.j . . . . . 6 |- .\/ = ( join ` K )
9 cdleme4.a . . . . . 6 |- A = ( Atoms ` K )
107, 8, 9hlatjcl 33003 . . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
113, 5, 6, 10syl3anc 1292 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
12 simp1r 1055 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> W e. H )
13 cdleme4.h . . . . . 6 |- H = ( LHyp ` K )
147, 13lhpbase 33634 . . . . 5 |- ( W e. H -> W e. ( Base ` K ) )
1512, 14syl 17 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> W e. ( Base ` K ) )
16 simp3 1032 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> R .<_ ( P .\/ Q ) )
17 cdleme4.l . . . . 5 |- .<_ = ( le ` K )
18 cdleme4.m . . . . 5 |- ./\ = ( meet ` K )
197, 17, 8, 18, 9atmod3i1 33500 . . . 4 |- ( ( K e. HL /\ ( R e. A /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ R .<_ ( P .\/ Q ) ) -> ( R .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( R .\/ W ) ) )
203, 4, 11, 15, 16, 19syl131anc 1305 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( R .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( R .\/ W ) ) )
21 simp1 1030 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( K e. HL /\ W e. H ) )
22 simp23 1065 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( R e. A /\ -. R .<_ W ) )
23 eqid 2471 . . . . . 6 |- ( 1. ` K ) = ( 1. ` K )
2417, 8, 23, 9, 13lhpjat2 33657 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R .\/ W ) = ( 1. ` K ) )
2521, 22, 24syl2anc 673 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( R .\/ W ) = ( 1. ` K ) )
2625oveq2d 6324 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( R .\/ W ) ) = ( ( P .\/ Q ) ./\ ( 1. ` K ) ) )
27 hlol 32998 . . . . 5 |- ( K e. HL -> K e. OL )
283, 27syl 17 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> K e. OL )
297, 18, 23olm11 32864 . . . 4 |- ( ( K e. OL /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
3028, 11, 29syl2anc 673 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
3120, 26, 303eqtrd 2509 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( R .\/ ( ( P .\/ Q ) ./\ W ) ) = ( P .\/ Q ) )
322, 31syl5req 2518 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) = ( R .\/ U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   1.cp1 16362   OLcol 32811   Atomscatm 32900   HLchlt 32987   LHypclh 33620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624
This theorem is referenced by:  cdleme5  33877  cdleme7aa  33879  cdleme7c  33882  cdleme7e  33884  cdleme20i  33955  cdleme36a  34098  cdleme37m  34100  cdleme39a  34103
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