Theorem paddasslem16 33787

Description: Lemma for paddass 33790. Use elpaddn0 33752 to eliminate x and r from paddasslem15 33786. (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 33316	. . . . 5 |- ( K e. HL -> K e. Lat )
2	1	3ad2ant1 1009	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. Lat )
3		simp21 1021	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> X C_ A )
4		simp1 988	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. HL )
5		simp22 1022	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> Y C_ A )
6		simp23 1023	. . . . 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 33766	. . . . 5 |- ( ( K e. HL /\ Y C_ A /\ Z C_ A ) -> ( Y .+ Z ) C_ A )
10	4, 5, 6, 9	syl3anc 1219	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( Y .+ Z ) C_ A )
11		simp3l 1016	. . . 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 33752	. . . 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 1222	. . 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 461	. . . . . . . 8 |- ( ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) -> ( Y =/= (/) /\ Z =/= (/) ) )
17	12, 13, 7, 8	paddasslem15 33786	. . . . . . . 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 1269	. . . . . . 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 1206	. . . . . 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 429	. . . . 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 2945	. . . 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 603	. . 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 3462	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 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   E.wrex 2796   C_ wss 3428   (/)c0 3737   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   lecple 14349   joincjn 15218   Latclat 15319   Atomscatm 33216   HLchlt 33303   +Pcpadd 33747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-padd 33748

This theorem is referenced by:  paddasslem18  33789