Theorem cdleme42mgN 36627

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: Use instead of cdleme42mN 36626? Combine with cdleme42mN 36626? (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme41.b	|- B = ( Base ` K )
cdleme41.l	|- .<_ = ( le ` K )
cdleme41.j	|- .\/ = ( join ` K )
cdleme41.m	|- ./\ = ( meet ` K )
cdleme41.a	|- A = ( Atoms ` K )
cdleme41.h	|- H = ( LHyp ` K )
cdleme41.u	|- U = ( ( P .\/ Q ) ./\ W )
cdleme41.d	|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme41.e	|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme41.g	|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
cdleme41.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
cdleme41.n	|- N = if ( s .<_ ( P .\/ Q ) , I , D )
cdleme41.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
cdleme41.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )

Assertion

Ref	Expression
cdleme42mgN	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( F ` ( R .\/ S ) ) = ( ( F ` R ) .\/ ( F ` S ) ) )

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

Allowed substitution hints:   D( x, z, t, s)   E( x, t)   F( x, y, z, t, s)   G( x, z, t, s)   H( x)   I( x, y, z, t, s)   K( x)   N( y, t, s)   O( x, y, z, t, s)

Proof of Theorem cdleme42mgN

Step	Hyp	Ref	Expression
1		simpl1l 1045	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> K e. HL )
2		hllat 35501	. . . . 5 |- ( K e. HL -> K e. Lat )
3	1, 2	syl 16	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> K e. Lat )
4		simprll 761	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> R e. A )
5		cdleme41.b	. . . . . 6 |- B = ( Base ` K )
6		cdleme41.a	. . . . . 6 |- A = ( Atoms ` K )
7	5, 6	atbase 35427	. . . . 5 |- ( R e. A -> R e. B )
8	4, 7	syl 16	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> R e. B )
9		simprrl 763	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> S e. A )
10	5, 6	atbase 35427	. . . . 5 |- ( S e. A -> S e. B )
11	9, 10	syl 16	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> S e. B )
12	3, 8, 11	3jca 1174	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( K e. Lat /\ R e. B /\ S e. B ) )
13		cdleme41.j	. . . . . 6 |- .\/ = ( join ` K )
14	5, 13	latjcl 15798	. . . . 5 |- ( ( K e. Lat /\ R e. B /\ S e. B ) -> ( R .\/ S ) e. B )
15		cdleme41.f	. . . . . 6 |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
16	15	cdleme31id 36533	. . . . 5 |- ( ( ( R .\/ S ) e. B /\ P = Q ) -> ( F ` ( R .\/ S ) ) = ( R .\/ S ) )
17	14, 16	sylan 469	. . . 4 |- ( ( ( K e. Lat /\ R e. B /\ S e. B ) /\ P = Q ) -> ( F ` ( R .\/ S ) ) = ( R .\/ S ) )
18	15	cdleme31id 36533	. . . . . 6 |- ( ( R e. B /\ P = Q ) -> ( F ` R ) = R )
19	18	3ad2antl2 1157	. . . . 5 |- ( ( ( K e. Lat /\ R e. B /\ S e. B ) /\ P = Q ) -> ( F ` R ) = R )
20	15	cdleme31id 36533	. . . . . 6 |- ( ( S e. B /\ P = Q ) -> ( F ` S ) = S )
21	20	3ad2antl3 1158	. . . . 5 |- ( ( ( K e. Lat /\ R e. B /\ S e. B ) /\ P = Q ) -> ( F ` S ) = S )
22	19, 21	oveq12d 6214	. . . 4 |- ( ( ( K e. Lat /\ R e. B /\ S e. B ) /\ P = Q ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( R .\/ S ) )
23	17, 22	eqtr4d 2426	. . 3 |- ( ( ( K e. Lat /\ R e. B /\ S e. B ) /\ P = Q ) -> ( F ` ( R .\/ S ) ) = ( ( F ` R ) .\/ ( F ` S ) ) )
24	12, 23	sylan 469	. 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ P = Q ) -> ( F ` ( R .\/ S ) ) = ( ( F ` R ) .\/ ( F ` S ) ) )
25		simpll 751	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ P =/= Q ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
26		simpr 459	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ P =/= Q ) -> P =/= Q )
27		simplrl 759	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ P =/= Q ) -> ( R e. A /\ -. R .<_ W ) )
28		simplrr 760	. . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ P =/= Q ) -> ( S e. A /\ -. S .<_ W ) )
29		cdleme41.l	. . . 4 |- .<_ = ( le ` K )
30		cdleme41.m	. . . 4 |- ./\ = ( meet ` K )
31		cdleme41.h	. . . 4 |- H = ( LHyp ` K )
32		cdleme41.u	. . . 4 |- U = ( ( P .\/ Q ) ./\ W )
33		cdleme41.d	. . . 4 |- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
34		cdleme41.e	. . . 4 |- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
35		cdleme41.g	. . . 4 |- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
36		cdleme41.i	. . . 4 |- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
37		cdleme41.n	. . . 4 |- N = if ( s .<_ ( P .\/ Q ) , I , D )
38		cdleme41.o	. . . 4 |- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
39	5, 29, 13, 30, 6, 31, 32, 33, 34, 35, 36, 37, 38, 15	cdleme42mN 36626	. . 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 ) ) ) -> ( F ` ( R .\/ S ) ) = ( ( F ` R ) .\/ ( F ` S ) ) )
40	25, 26, 27, 28, 39	syl13anc 1228	. 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) /\ P =/= Q ) -> ( F ` ( R .\/ S ) ) = ( ( F ` R ) .\/ ( F ` S ) ) )
41	24, 40	pm2.61dane 2700	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( F ` ( R .\/ S ) ) = ( ( F ` R ) .\/ ( F ` S ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1399   e. wcel 1826   =/= wne 2577   A.wral 2732   ifcif 3857   class class class wbr 4367   |-> cmpt 4425   ` cfv 5496   iota_crio 6157  (class class class)co 6196   Basecbs 14634   lecple 14709   joincjn 15690   meetcmee 15691   Latclat 15792   Atomscatm 35401   HLchlt 35488   LHypclh 36121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-riotaBAD 35097

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-undef 6920  df-preset 15674  df-poset 15692  df-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-p1 15787  df-lat 15793  df-clat 15855  df-oposet 35314  df-ol 35316  df-oml 35317  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489  df-llines 35635  df-lplanes 35636  df-lvols 35637  df-lines 35638  df-psubsp 35640  df-pmap 35641  df-padd 35933  df-lhyp 36125

This theorem is referenced by:  cdlemg2jlemOLDN  36732