Theorem cdleme42h 29921

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Mar-2013.)

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 ) )
cdleme34e.v	|- V = ( ( R .\/ S ) ./\ W )

Assertion

Ref	Expression
cdleme42h	|- ( ( ( ( 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 ` S ) .<_ ( ( F ` R ) .\/ V ) )

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   V, s, t, x, z

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)   V( y)

Proof of Theorem cdleme42h

Step	Hyp	Ref	Expression
1		simp11l 1071	. . . 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 ) ) /\ P =/= Q ) -> K e. HL )
2		hllat 28803	. . . 4 |- ( K e. HL -> K e. Lat )
3	1, 2	syl 17	. . 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. Lat )
4		simp1 960	. . . 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 ) ) /\ P =/= Q ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
5		simp2rl 1029	. . . . 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 ) ) /\ P =/= Q ) -> S e. A )
6		cdleme41.b	. . . . . 6 |- B = ( Base ` K )
7		cdleme41.a	. . . . . 6 |- A = ( Atoms ` K )
8	6, 7	atbase 28729	. . . . 5 |- ( S e. A -> S e. B )
9	5, 8	syl 17	. . . 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 ) ) /\ P =/= Q ) -> S e. B )
10		cdleme41.l	. . . . 5 |- .<_ = ( le ` K )
11		cdleme41.j	. . . . 5 |- .\/ = ( join ` K )
12		cdleme41.m	. . . . 5 |- ./\ = ( meet ` K )
13		cdleme41.h	. . . . 5 |- H = ( LHyp ` K )
14		cdleme41.u	. . . . 5 |- U = ( ( P .\/ Q ) ./\ W )
15		cdleme41.d	. . . . 5 |- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
16		cdleme41.e	. . . . 5 |- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
17		cdleme41.g	. . . . 5 |- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
18		cdleme41.i	. . . . 5 |- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
19		cdleme41.n	. . . . 5 |- N = if ( s .<_ ( P .\/ Q ) , I , D )
20		cdleme41.o	. . . . 5 |- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
21		cdleme41.f	. . . . 5 |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
22	6, 10, 11, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21	cdleme32fvcl 29879	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ S e. B ) -> ( F ` S ) e. B )
23	4, 9, 22	syl2anc 645	. . 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 ) -> ( F ` S ) e. B )
24		cdleme34e.v	. . . 4 |- V = ( ( R .\/ S ) ./\ W )
25		simp2ll 1027	. . . . . 6 |- ( ( ( ( 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 )
26	6, 11, 7	hlatjcl 28806	. . . . . 6 |- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. B )
27	1, 25, 5, 26	syl3anc 1187	. . . . 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 ) ) /\ P =/= Q ) -> ( R .\/ S ) e. B )
28		simp11r 1072	. . . . . 6 |- ( ( ( ( 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 ) -> W e. H )
29	6, 13	lhpbase 29437	. . . . . 6 |- ( W e. H -> W e. B )
30	28, 29	syl 17	. . . . 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 ) ) /\ P =/= Q ) -> W e. B )
31	6, 12	latmcl 14120	. . . . 5 |- ( ( K e. Lat /\ ( R .\/ S ) e. B /\ W e. B ) -> ( ( R .\/ S ) ./\ W ) e. B )
32	3, 27, 30, 31	syl3anc 1187	. . . 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 ) ) /\ P =/= Q ) -> ( ( R .\/ S ) ./\ W ) e. B )
33	24, 32	syl5eqel 2342	. . 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 ) -> V e. B )
34	6, 10, 11	latlej1 14129	. . 3 |- ( ( K e. Lat /\ ( F ` S ) e. B /\ V e. B ) -> ( F ` S ) .<_ ( ( F ` S ) .\/ V ) )
35	3, 23, 33, 34	syl3anc 1187	. 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 ` S ) .<_ ( ( F ` S ) .\/ V ) )
36	11, 7	hlatjcom 28807	. . . . . . . 8 |- ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) = ( S .\/ R ) )
37	1, 25, 5, 36	syl3anc 1187	. . . . . . 7 |- ( ( ( ( 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 .\/ S ) = ( S .\/ R ) )
38	37	oveq1d 5807	. . . . . 6 |- ( ( ( ( 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 .\/ S ) ./\ W ) = ( ( S .\/ R ) ./\ W ) )
39	24, 38	syl5eq 2302	. . . . 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 ) ) /\ P =/= Q ) -> V = ( ( S .\/ R ) ./\ W ) )
40	39	oveq2d 5808	. . . 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 ) ) /\ P =/= Q ) -> ( ( F ` S ) .\/ V ) = ( ( F ` S ) .\/ ( ( S .\/ R ) ./\ W ) ) )
41		simp2r 987	. . . . 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 ) ) /\ P =/= Q ) -> ( S e. A /\ -. S .<_ W ) )
42		simp2l 986	. . . . 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 ) ) /\ P =/= Q ) -> ( R e. A /\ -. R .<_ W ) )
43		simp3 962	. . . . 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 ) ) /\ P =/= Q ) -> P =/= Q )
44		eqid 2258	. . . . . 6 |- ( ( S .\/ R ) ./\ W ) = ( ( S .\/ R ) ./\ W )
45	6, 10, 11, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 44	cdleme42g 29920	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ P =/= Q ) -> ( F ` ( S .\/ R ) ) = ( ( F ` S ) .\/ ( ( S .\/ R ) ./\ W ) ) )
46	4, 41, 42, 43, 45	syl121anc 1192	. . . 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 ) ) /\ P =/= Q ) -> ( F ` ( S .\/ R ) ) = ( ( F ` S ) .\/ ( ( S .\/ R ) ./\ W ) ) )
47	40, 46	eqtr4d 2293	. . 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 ) -> ( ( F ` S ) .\/ V ) = ( F ` ( S .\/ R ) ) )
48	37	fveq2d 5462	. . 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 ) -> ( F ` ( R .\/ S ) ) = ( F ` ( S .\/ R ) ) )
49	6, 10, 11, 12, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24	cdleme42g 29920	. . 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 ) -> ( F ` ( R .\/ S ) ) = ( ( F ` R ) .\/ V ) )
50	47, 48, 49	3eqtr2d 2296	. 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 ` S ) .\/ V ) = ( ( F ` R ) .\/ V ) )
51	35, 50	breqtrd 4021	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 ) ) /\ P =/= Q ) -> ( F ` S ) .<_ ( ( F ` R ) .\/ V ) )

Syntax hints:   -. wn 5   -> wi 6   /\ wa 360   /\ w3a 939   = wceq 1619   e. wcel 1621   =/= wne 2421   A.wral 2518   ifcif 3539   class class class wbr 3997   e. cmpt 4051   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   Latclat 14114   Atomscatm 28703   HLchlt 28790   LHypclh 29423

This theorem is referenced by:  cdleme42i  29922

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427