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Theorem cdleme20l1 34273
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. D, F, Y, G represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
Hypotheses
Ref Expression
cdleme19.l |- .<_ = ( le ` K )
cdleme19.j |- .\/ = ( join ` K )
cdleme19.m |- ./\ = ( meet ` K )
cdleme19.a |- A = ( Atoms ` K )
cdleme19.h |- H = ( LHyp ` K )
cdleme19.u |- U = ( ( P .\/ Q ) ./\ W )
cdleme19.f |- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme19.g |- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme19.d |- D = ( ( R .\/ S ) ./\ W )
cdleme19.y |- Y = ( ( R .\/ T ) ./\ W )
cdleme20.v |- V = ( ( S .\/ T ) ./\ W )
Assertion
Ref Expression
cdleme20l1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( F .\/ D ) e. ( LLines ` K ) )
Proof of Theorem cdleme20l1
StepHypRef Expression
1 simp11l 1099 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. HL )
2 simp11 1018 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
3 simp12 1019 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
4 simp13 1020 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
5 simp22 1022 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. A )
6 simp23 1023 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ W )
75, 6jca 532 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
8 simp31 1024 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> P =/= Q )
9 simp32 1025 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
10 cdleme19.l . . . 4 |- .<_ = ( le ` K )
11 cdleme19.j . . . 4 |- .\/ = ( join ` K )
12 cdleme19.m . . . 4 |- ./\ = ( meet ` K )
13 cdleme19.a . . . 4 |- A = ( Atoms ` K )
14 cdleme19.h . . . 4 |- H = ( LHyp ` K )
15 cdleme19.u . . . 4 |- U = ( ( P .\/ Q ) ./\ W )
16 cdleme19.f . . . 4 |- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
1710, 11, 12, 13, 14, 15, 16cdleme3fa 34189 . . 3 |- ( ( ( 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 ) ) ) -> F e. A )
182, 3, 4, 7, 8, 9, 17syl132anc 1237 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> F e. A )
19 simp11r 1100 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. H )
20 simp21 1021 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R e. A )
21 simp33 1026 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
22 cdleme19.d . . . 4 |- D = ( ( R .\/ S ) ./\ W )
2310, 11, 12, 13, 14, 22cdlemeda 34251 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> D e. A )
241, 19, 5, 6, 20, 21, 9, 23syl223anc 1245 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> D e. A )
2510, 11, 12, 13, 14, 15, 16, 16, 22, 22cdleme19c 34258 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= D )
261, 19, 3, 4, 7, 20, 8, 9, 25syl233anc 1248 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> F =/= D )
27 eqid 2451 . . 3 |- ( LLines ` K ) = ( LLines ` K )
2811, 13, 27llni2 33465 . 2 |- ( ( ( K e. HL /\ F e. A /\ D e. A ) /\ F =/= D ) -> ( F .\/ D ) e. ( LLines ` K ) )
291, 18, 24, 26, 28syl31anc 1222 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( F .\/ D ) e. ( LLines ` K ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   lecple 14356   joincjn 15225   meetcmee 15226   Atomscatm 33217   HLchlt 33304   LLinesclln 33444   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-llines 33451  df-lines 33454  df-psubsp 33456  df-pmap 33457  df-padd 33749  df-lhyp 33941
This theorem is referenced by:  cdleme20l2  34274  cdleme20l  34275
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