Theorem paddasslem16 32852

Description: Lemma for paddass 32855. Use elpaddn0 32817 to eliminate x and r from paddasslem15 32851. (Contributed by NM, 11-Jan-2012.)

Hypotheses

Ref	Expression
paddasslem.l	|- .<_ = ( le ` K )
paddasslem.j	|- .\/ = ( join ` K )
paddasslem.a	|- A = ( Atoms ` K )
paddasslem.p	|- .+ = ( +P ` K )

Assertion

Ref	Expression
paddasslem16	|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) )

Proof of Theorem paddasslem16

Dummy variables p r x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hllat 32381	. . . . 5 |- ( K e. HL -> K e. Lat )
2	1	3ad2ant1 1018	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. Lat )
3		simp21 1030	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> X C_ A )
4		simp1 997	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. HL )
5		simp22 1031	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> Y C_ A )
6		simp23 1032	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> Z C_ A )
7		paddasslem.a	. . . . . 6 |- A = ( Atoms ` K )
8		paddasslem.p	. . . . . 6 |- .+ = ( +P ` K )
9	7, 8	paddssat 32831	. . . . 5 |- ( ( K e. HL /\ Y C_ A /\ Z C_ A ) -> ( Y .+ Z ) C_ A )
10	4, 5, 6, 9	syl3anc 1230	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( Y .+ Z ) C_ A )
11		simp3l 1025	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) )
12		paddasslem.l	. . . . 5 |- .<_ = ( le ` K )
13		paddasslem.j	. . . . 5 |- .\/ = ( join ` K )
14	12, 13, 7, 8	elpaddn0 32817	. . . 4 |- ( ( ( K e. Lat /\ X C_ A /\ ( Y .+ Z ) C_ A ) /\ ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) ) -> ( p e. ( X .+ ( Y .+ Z ) ) <-> ( p e. A /\ E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) ) ) )
15	2, 3, 10, 11, 14	syl31anc 1233	. . 3 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( p e. ( X .+ ( Y .+ Z ) ) <-> ( p e. A /\ E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) ) ) )
16		simpr 459	. . . . . . . 8 |- ( ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) -> ( Y =/= (/) /\ Z =/= (/) ) )
17	12, 13, 7, 8	paddasslem15 32851	. . . . . . . 8 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) /\ ( p e. A /\ ( x e. X /\ r e. ( Y .+ Z ) ) /\ p .<_ ( x .\/ r ) ) ) -> p e. ( ( X .+ Y ) .+ Z ) )
18	16, 17	syl3anl3 1280	. . . . . . 7 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) /\ ( p e. A /\ ( x e. X /\ r e. ( Y .+ Z ) ) /\ p .<_ ( x .\/ r ) ) ) -> p e. ( ( X .+ Y ) .+ Z ) )
19	18	3exp2 1215	. . . . . 6 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( p e. A -> ( ( x e. X /\ r e. ( Y .+ Z ) ) -> ( p .<_ ( x .\/ r ) -> p e. ( ( X .+ Y ) .+ Z ) ) ) ) )
20	19	imp 427	. . . . 5 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) /\ p e. A ) -> ( ( x e. X /\ r e. ( Y .+ Z ) ) -> ( p .<_ ( x .\/ r ) -> p e. ( ( X .+ Y ) .+ Z ) ) ) )
21	20	rexlimdvv 2902	. . . 4 |- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) /\ p e. A ) -> ( E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) -> p e. ( ( X .+ Y ) .+ Z ) ) )
22	21	expimpd 601	. . 3 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( ( p e. A /\ E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) ) -> p e. ( ( X .+ Y ) .+ Z ) ) )
23	15, 22	sylbid 215	. 2 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( p e. ( X .+ ( Y .+ Z ) ) -> p e. ( ( X .+ Y ) .+ Z ) ) )
24	23	ssrdv 3448	1 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   E.wrex 2755   C_ wss 3414   (/)c0 3738   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   lecple 14916   joincjn 15897   Latclat 15999   Atomscatm 32281   HLchlt 32368   +Pcpadd 32812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-padd 32813

This theorem is referenced by:  paddasslem18  32854