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Theorem cdleme4 34191
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 6204 . 2 |- ( R .\/ U ) = ( R .\/ ( ( P .\/ Q ) ./\ W ) )
3 simp1l 1012 . . . 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 1109 . . . 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 1021 . . . . 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 1022 . . . . 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 2451 . . . . . 6 |- ( Base ` K ) = ( Base ` K )
8 cdleme4.j . . . . . 6 |- .\/ = ( join ` K )
9 cdleme4.a . . . . . 6 |- A = ( Atoms ` K )
107, 8, 9hlatjcl 33320 . . . . 5 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
113, 5, 6, 10syl3anc 1219 . . . 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 1013 . . . . 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 33951 . . . . 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 990 . . . 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 33817 . . . 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 1232 . . 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 988 . . . . 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 1023 . . . . 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 2451 . . . . . 6 |- ( 1. ` K ) = ( 1. ` K )
2417, 8, 23, 9, 13lhpjat2 33974 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R .\/ W ) = ( 1. ` K ) )
2521, 22, 24syl2anc 661 . . . 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 6209 . . 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 33315 . . . . 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 33181 . . . 4 |- ( ( K e. OL /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
3028, 11, 29syl2anc 661 . . 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 2496 . 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 2505 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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   lecple 14356   joincjn 15225   meetcmee 15226   1.cp1 15319   OLcol 33128   Atomscatm 33217   HLchlt 33304   LHypclh 33937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-psubsp 33456  df-pmap 33457  df-padd 33749  df-lhyp 33941
This theorem is referenced by:  cdleme5  34193  cdleme7aa  34195  cdleme7c  34198  cdleme7e  34200  cdleme20i  34270  cdleme36a  34413  cdleme37m  34415  cdleme39a  34418
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