Theorem 4atlem4d 30088

Description: Lemma for 4at 30099. 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 960	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> K e. HL )
2		hllat 29850	. . . 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 2408	. . . . 5 |- ( Base ` K ) = ( Base ` K )
5		4at.j	. . . . 5 |- .\/ = ( join ` K )
6		4at.a	. . . . 5 |- A = ( Atoms ` K )
7	4, 5, 6	hlatjcl 29853	. . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
8	7	adantr 452	. . 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 29776	. . . 4 |- ( R e. A -> R e. ( Base ` K ) )
10	9	ad2antrl 709	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> R e. ( Base ` K ) )
11	4, 6	atbase 29776	. . . 4 |- ( S e. A -> S e. ( Base ` K ) )
12	11	ad2antll 710	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> S e. ( Base ` K ) )
13	4, 5	latjass 14483	. . 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 1186	. 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 14438	. . . 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 1184	. . 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 14447	. . 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 1184	. 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 2442	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 359   /\ w3a 936   = wceq 1649   e. wcel 1721   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   Latclat 14433   Atomscatm 29750   HLchlt 29837

This theorem is referenced by:  4atlem9  30089

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-lub 14390  df-join 14392  df-lat 14434  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838