Theorem paddasslem16 35121

Description: Lemma for paddass 35124. Use elpaddn0 35086 to eliminate x and r from paddasslem15 35120. (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 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
2	1	3ad2ant1 1082	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. Lat )
3		simp21 1094	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> X C_ A )
4		simp1 1061	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. HL )
5		simp22 1095	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> Y C_ A )
6		simp23 1096	. . . . 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 35100	. . . . 5 |- ( ( K e. HL /\ Y C_ A /\ Z C_ A ) -> ( Y .+ Z ) C_ A )
10	4, 5, 6, 9	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( Y .+ Z ) C_ A )
11		simp3l 1089	. . . 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 35086	. . . 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 1329	. . 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 477	. . . . . . . 8 |- ( ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) -> ( Y =/= (/) /\ Z =/= (/) ) )
17	12, 13, 7, 8	paddasslem15 35120	. . . . . . . 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 1376	. . . . . . 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 1285	. . . . . 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 445	. . . . 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 3037	. . . 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 629	. . 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 230	. 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 3609	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   HLchlt 34637   +Pcpadd 35081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-padd 35082

This theorem is referenced by:  paddasslem18  35123