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Theorem cdleme19f 35596
Description: Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, line 3. D, F, N, Y, G, O represent s2, f(s), fs(r), t2, f(t), ft(r). We prove that if r <_ s \/ t, then ft(r) = ft(r). (Contributed by NM, 14-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 )
cdleme19.n |- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
cdleme19.o |- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
Assertion
Ref Expression
cdleme19f |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> N = O )
Proof of Theorem cdleme19f
StepHypRef Expression
1 cdleme19.l . . . 4 |- .<_ = ( le ` K )
2 cdleme19.j . . . 4 |- .\/ = ( join ` K )
3 cdleme19.m . . . 4 |- ./\ = ( meet ` K )
4 cdleme19.a . . . 4 |- A = ( Atoms ` K )
5 cdleme19.h . . . 4 |- H = ( LHyp ` K )
6 cdleme19.u . . . 4 |- U = ( ( P .\/ Q ) ./\ W )
7 cdleme19.f . . . 4 |- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme19.g . . . 4 |- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 cdleme19.d . . . 4 |- D = ( ( R .\/ S ) ./\ W )
10 cdleme19.y . . . 4 |- Y = ( ( R .\/ T ) ./\ W )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10cdleme19e 35595 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( F .\/ D ) = ( G .\/ Y ) )
1211oveq2d 6666 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ D ) ) = ( ( P .\/ Q ) ./\ ( G .\/ Y ) ) )
13 cdleme19.n . 2 |- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
14 cdleme19.o . 2 |- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
1512, 13, 143eqtr4g 2681 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> N = O )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   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-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:  cdleme20  35612
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