Theorem cdlemg2jlemOLDN 35881

Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. f preserves join: f(r \/ s) = f(r) \/ s, p. 115 10th line from bottom. TODO: Combine with cdlemg2jOLDN 35886? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemg2.b	|- B = ( Base ` K )
cdlemg2.l	|- .<_ = ( le ` K )
cdlemg2.j	|- .\/ = ( join ` K )
cdlemg2.m	|- ./\ = ( meet ` K )
cdlemg2.a	|- A = ( Atoms ` K )
cdlemg2.h	|- H = ( LHyp ` K )
cdlemg2.t	|- T = ( ( LTrn ` K ) ` W )
cdlemg2ex.u	|- U = ( ( p .\/ q ) ./\ W )
cdlemg2ex.d	|- D = ( ( t .\/ U ) ./\ ( q .\/ ( ( p .\/ t ) ./\ W ) ) )
cdlemg2ex.e	|- E = ( ( p .\/ q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemg2ex.g	|- G = ( x e. B |-> if ( ( p =/= q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )

Assertion

Ref	Expression
cdlemg2jlemOLDN	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( F ` ( P .\/ Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) )

Distinct variable groups:   t, s, x, y, z, A   B, s, t, x, y, z   D, s, x, y, z   x, E, y, z   H, s, t, x, y, 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   U, s, t, x, y, z   W, s, t, x, y, z   q, p, A   F, p, q   H, p, q   K, p, q   .<_ , p, q   T, p, q   W, p, q, s, t, x, y, z   .\/ , p, q   P, p, q   Q, p, q

Allowed substitution hints:   B( q, p)   D( t, q, p)   T( x, y, z, t, s)   U( q, p)   E( t, s, q, p)   F( x, y, z, t, s)   G( x, y, z, t, s, q, p)   ./\ ( q, p)

Proof of Theorem cdlemg2jlemOLDN

Step	Hyp	Ref	Expression
1		cdlemg2.b	. . 3 |- B = ( Base ` K )
2		cdlemg2.l	. . 3 |- .<_ = ( le ` K )
3		cdlemg2.j	. . 3 |- .\/ = ( join ` K )
4		cdlemg2.m	. . 3 |- ./\ = ( meet ` K )
5		cdlemg2.a	. . 3 |- A = ( Atoms ` K )
6		cdlemg2.h	. . 3 |- H = ( LHyp ` K )
7		cdlemg2.t	. . 3 |- T = ( ( LTrn ` K ) ` W )
8		cdlemg2ex.u	. . 3 |- U = ( ( p .\/ q ) ./\ W )
9		cdlemg2ex.d	. . 3 |- D = ( ( t .\/ U ) ./\ ( q .\/ ( ( p .\/ t ) ./\ W ) ) )
10		cdlemg2ex.e	. . 3 |- E = ( ( p .\/ q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11		cdlemg2ex.g	. . 3 |- G = ( x e. B |-> if ( ( p =/= q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
12		fveq1 6190	. . . 4 |- ( F = G -> ( F ` ( P .\/ Q ) ) = ( G ` ( P .\/ Q ) ) )
13		fveq1 6190	. . . . 5 |- ( F = G -> ( F ` P ) = ( G ` P ) )
14		fveq1 6190	. . . . 5 |- ( F = G -> ( F ` Q ) = ( G ` Q ) )
15	13, 14	oveq12d 6668	. . . 4 |- ( F = G -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( G ` P ) .\/ ( G ` Q ) ) )
16	12, 15	eqeq12d 2637	. . 3 |- ( F = G -> ( ( F ` ( P .\/ Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) <-> ( G ` ( P .\/ Q ) ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) )
17		vex 3203	. . . . 5 |- s e. _V
18		eqid 2622	. . . . . 6 |- ( ( s .\/ U ) ./\ ( q .\/ ( ( p .\/ s ) ./\ W ) ) ) = ( ( s .\/ U ) ./\ ( q .\/ ( ( p .\/ s ) ./\ W ) ) )
19	9, 18	cdleme31sc 35672	. . . . 5 |- ( s e. _V -> [_ s / t ]_ D = ( ( s .\/ U ) ./\ ( q .\/ ( ( p .\/ s ) ./\ W ) ) ) )
20	17, 19	ax-mp 5	. . . 4 |- [_ s / t ]_ D = ( ( s .\/ U ) ./\ ( q .\/ ( ( p .\/ s ) ./\ W ) ) )
21		eqid 2622	. . . 4 |- ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) )
22		eqid 2622	. . . 4 |- if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D ) = if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D )
23		eqid 2622	. . . 4 |- ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( p .\/ q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( p .\/ q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) )
24	1, 2, 3, 4, 5, 6, 8, 20, 9, 10, 21, 22, 23, 11	cdleme42mgN 35776	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ -. p .<_ W ) /\ ( q e. A /\ -. q .<_ W ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( G ` ( P .\/ Q ) ) = ( ( G ` P ) .\/ ( G ` Q ) ) )
25	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 24	cdlemg2ce 35880	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( F ` ( P .\/ Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) )
26	25	3com23 1271	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( F ` ( P .\/ Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   [_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   LTrncltrn 35387

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-map 7859  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  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemg2jOLDN  35886