Theorem cdlemefs44 35714

Description: Value of f_s(r) when r is an atom under pq and s is any atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefs45 35717 instead TODO: FIX COMMENT. (Contributed by NM, 31-Mar-2013.)

Hypotheses

Ref	Expression
cdlemef44.b	|- B = ( Base ` K )
cdlemef44.l	|- .<_ = ( le ` K )
cdlemef44.j	|- .\/ = ( join ` K )
cdlemef44.m	|- ./\ = ( meet ` K )
cdlemef44.a	|- A = ( Atoms ` K )
cdlemef44.h	|- H = ( LHyp ` K )
cdlemef44.u	|- U = ( ( P .\/ Q ) ./\ W )
cdlemef44.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemef44.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) )
cdlemef44.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
cdlemefs44.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemefs44.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )

Assertion

Ref	Expression
cdlemefs44	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = [_ R / s ]_ [_ S / t ]_ E )

Distinct variable groups:   t, s, x, y, z, A   B, s, t, x, y, z   x, D, y, z   y, E   H, s, t, x, y, z   x, I, z   .\/ , s, t, x, y, z   K, s, t, x, y, z   .<_ , s, t, x, y, z   ./\ , s, t, x, y, z   P, s, t, x, y, z   Q, s, t, x, y, z   R, s, t, x, y, z   U, s, t, x, y, z   W, s, t, x, y, z   D, s   S, s, t, y

Allowed substitution hints:   D( t)   S( x, z)   E( x, z, t, s)   F( x, y, z, t, s)   I( y, t, s)   O( x, y, z, t, s)

Proof of Theorem cdlemefs44

Step	Hyp	Ref	Expression
1		cdlemef44.b	. . 3 |- B = ( Base ` K )
2		cdlemef44.l	. . 3 |- .<_ = ( le ` K )
3		cdlemef44.j	. . 3 |- .\/ = ( join ` K )
4		cdlemef44.m	. . 3 |- ./\ = ( meet ` K )
5		cdlemef44.a	. . 3 |- A = ( Atoms ` K )
6		cdlemef44.h	. . 3 |- H = ( LHyp ` K )
7		cdlemef44.u	. . 3 |- U = ( ( P .\/ Q ) ./\ W )
8		cdlemef44.d	. . 3 |- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9		cdlemefs44.e	. . 3 |- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10		cdlemefs44.i	. . 3 |- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
11		eqid 2622	. . 3 |- if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) = if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D )
12		cdlemef44.o	. . 3 |- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) )
13		cdlemef44.f	. . 3 |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
14		eqid 2622	. . 3 |- ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
15		eqid 2622	. . 3 |- ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) )
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	cdlemefs31fv1 35712	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )
17		simp22l 1180	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A )
18		simp23l 1182	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
19	8, 9, 14, 15	cdleme31sde 35673	. . 3 |- ( ( R e. A /\ S e. A ) -> [_ R / s ]_ [_ S / t ]_ E = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )
20	17, 18, 19	syl2anc 693	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ [_ S / t ]_ E = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )
21	16, 20	eqtr4d 2659	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = [_ R / s ]_ [_ S / t ]_ E )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   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  ax-riotaBAD 34239

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-undef 7399  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:  cdlemefs45  35717