Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme4 Structured version   Unicode version
Theorem cdleme4 36360
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 6281 . 2 |- ( R .\/ U ) = ( R .\/ ( ( P .\/ Q ) ./\ W ) )
3 simp1l 1018 . . . 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 1115 . . . 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 1027 . . . . 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 1028 . . . . 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 2454 . . . . . 6 |- ( Base ` K ) = ( Base ` K )
8 cdleme4.j . . . . . 6 |- .\/ = ( join ` K )
9 cdleme4.a . . . . . 6 |- A = ( Atoms ` K )
107, 8, 9hlatjcl 35488 . . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
113, 5, 6, 10syl3anc 1226 . . . 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 1019 . . . . 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 36119 . . . . 5 |- ( W e. H -> W e. ( Base ` K ) )
1512, 14syl 16 . . . 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 996 . . . 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 35985 . . . 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 1239 . . 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 994 . . . . 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 1029 . . . . 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 2454 . . . . . 6 |- ( 1. ` K ) = ( 1. ` K )
2417, 8, 23, 9, 13lhpjat2 36142 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R .\/ W ) = ( 1. ` K ) )
2521, 22, 24syl2anc 659 . . . 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 6286 . . 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 35483 . . . . 5 |- ( K e. HL -> K e. OL )
283, 27syl 16 . . . 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 35349 . . . 4 |- ( ( K e. OL /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
3028, 11, 29syl2anc 659 . . 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 2499 . 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 2508 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   1.cp1 15867   OLcol 35296   Atomscatm 35385   HLchlt 35472   LHypclh 36105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109
This theorem is referenced by:  cdleme5  36362  cdleme7aa  36364  cdleme7c  36367  cdleme7e  36369  cdleme20i  36440  cdleme36a  36583  cdleme37m  36585  cdleme39a  36588
  Copyright terms: Public domain W3C validator