Theorem 4atlem4d 29791

Description: Lemma for 4at 29802. 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 958	. . . 4 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> K e. HL )
2		hllat 29553	. . . 4 |- ( K e. HL -> K e. Lat )
3	1, 2	syl 15	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> K e. Lat )
4		eqid 2283	. . . . 5 |- ( Base ` K ) = ( Base ` K )
5		4at.j	. . . . 5 |- .\/ = ( join ` K )
6		4at.a	. . . . 5 |- A = ( Atoms ` K )
7	4, 5, 6	hlatjcl 29556	. . . 4 |- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
8	7	adantr 451	. . 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 29479	. . . 4 |- ( R e. A -> R e. ( Base ` K ) )
10	9	ad2antrl 708	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> R e. ( Base ` K ) )
11	4, 6	atbase 29479	. . . 4 |- ( S e. A -> S e. ( Base ` K ) )
12	11	ad2antll 709	. . 3 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) ) -> S e. ( Base ` K ) )
13	4, 5	latjass 14201	. . 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 1184	. 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 14156	. . . 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 1182	. . 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 14165	. . 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 1182	. 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 2317	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 358   /\ w3a 934   = wceq 1623   e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540

This theorem is referenced by:  4atlem9  29792

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-join 14110  df-lat 14152  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541