Theorem 4atlem4d 30473

Description: Lemma for 4at 30484. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Hypotheses

Ref	Expression
4at.l	|- .<_ = ( le ` K )
4at.j	|- .\/ = ( join ` K )
4at.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
4atlem4d	|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )

Proof of Theorem 4atlem4d

Step	Hyp	Ref	Expression
1		simpl1 961	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> K e. HL )
2		hllat 30235	. . . 4 |- ( K e. HL -> K e. Lat )
3	1, 2	syl 16	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> K e. Lat )
4		eqid 2438	. . . . 5 |- ( Base ` K ) = ( Base ` K )
5		4at.j	. . . . 5 |- .\/ = ( join ` K )
6		4at.a	. . . . 5 |- A = ( Atoms ` K )
7	4, 5, 6	hlatjcl 30238	. . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
8	7	adantr 453	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
9	4, 6	atbase 30161	. . . 4 |- ( R e. A -> R e. ( Base ` K ) )
10	9	ad2antrl 710	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> R e. ( Base ` K ) )
11	4, 6	atbase 30161	. . . 4 |- ( S e. A -> S e. ( Base ` K ) )
12	11	ad2antll 711	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> S e. ( Base ` K ) )
13	4, 5	latjass 14529	. . 3 |- ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( ( P .\/ Q ) .\/ ( R .\/ S ) ) )
14	3, 8, 10, 12, 13	syl13anc 1187	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( ( P .\/ Q ) .\/ ( R .\/ S ) ) )
15	4, 5	latjcl 14484	. . . 4 |- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) )
16	3, 8, 10, 15	syl3anc 1185	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) )
17	4, 5	latjcom 14493	. . 3 |- ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )
18	3, 16, 12, 17	syl3anc 1185	. 2 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )
19	14, 18	eqtr3d 2472	1 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( S .\/ ( ( P .\/ Q ) .\/ R ) ) )

Syntax hints:   -> wi 4   /\ wa 360   /\ w3a 937   = wceq 1653   e. wcel 1726   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   Latclat 14479   Atomscatm 30135   HLchlt 30222

This theorem is referenced by:  4atlem9  30474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-lub 14436  df-join 14438  df-lat 14480  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223