Theorem paddasslem16 17296

Description: Lemma for paddass 17299. Use elpaddn0 17261 to eliminate x and r from paddasslem15 17295.

Hypotheses

Ref	Expression
paddasslem.l	|- L = (le` K)
paddasslem.j	|- J = (join` K)
paddasslem.a	|- A = (AtomsNEW` K)
paddasslem.p	|- P = (+P` K)

Assertion

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

Proof of Theorem paddasslem16

Step	Hyp	Ref	Expression
1		hllat 17026	. . . . . . 7 |- (K e. HL -> K e. LatNEW)
2	1	3ad2ant1 897	. . . . . 6 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> K e. LatNEW)
3		simp21 909	. . . . . 6 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> X C_ A)
4		simp1 876	. . . . . . 7 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> K e. HL)
5		simp22 910	. . . . . . 7 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> Y C_ A)
6		simp23 911	. . . . . . 7 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> Z C_ A)
7		paddasslem.a	. . . . . . . 8 |- A = (AtomsNEW` K)
8		paddasslem.p	. . . . . . . 8 |- P = (+P` K)
9	7, 8	paddssat 17275	. . . . . . 7 |- ((K e. HL /\ Y C_ A /\ Z C_ A) -> (YPZ) C_ A)
10	4, 5, 6, 9	syl111anc 1100	. . . . . 6 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> (YPZ) C_ A)
11		simp3l 904	. . . . . 6 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> (X =/= (/) /\ (YPZ) =/= (/)))
12		paddasslem.l	. . . . . . 7 |- L = (le` K)
13		paddasslem.j	. . . . . . 7 |- J = (join` K)
14	12, 13, 7, 8	elpaddn0 17261	. . . . . 6 |- (((K e. LatNEW /\ X C_ A /\ (YPZ) C_ A) /\ (X =/= (/) /\ (YPZ) =/= (/))) -> (p e. (XP(YPZ)) <-> (p e. A /\ E.x e. X E.r e. (YPZ)pL(xJr))))
15	2, 3, 10, 11, 14	syl31anc 1103	. . . . 5 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> (p e. (XP(YPZ)) <-> (p e. A /\ E.x e. X E.r e. (YPZ)pL(xJr))))
16	12, 13, 7, 8	paddasslem15 17295	. . . . . . . . . 10 |- (((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ (Y =/= (/) /\ Z =/= (/))) /\ (p e. A /\ (x e. X /\ r e. (YPZ)) /\ pL(xJr))) -> p e. ((XPY)PZ))
17		simpr 350	. . . . . . . . . 10 |- (((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/))) -> (Y =/= (/) /\ Z =/= (/)))
18	16, 17	syl3anl3 1147	. . . . . . . . 9 |- (((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) /\ (p e. A /\ (x e. X /\ r e. (YPZ)) /\ pL(xJr))) -> p e. ((XPY)PZ))
19	18	3exp2 1086	. . . . . . . 8 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> (p e. A -> ((x e. X /\ r e. (YPZ)) -> (pL(xJr) -> p e. ((XPY)PZ)))))
20	19	imp 377	. . . . . . 7 |- (((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) /\ p e. A) -> ((x e. X /\ r e. (YPZ)) -> (pL(xJr) -> p e. ((XPY)PZ))))
21	20	r19.23advv 2218	. . . . . 6 |- (((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) /\ p e. A) -> (E.x e. X E.r e. (YPZ)pL(xJr) -> p e. ((XPY)PZ)))
22	21	expimpd 404	. . . . 5 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> ((p e. A /\ E.x e. X E.r e. (YPZ)pL(xJr)) -> p e. ((XPY)PZ)))
23	15, 22	sylbid 220	. . . 4 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> (p e. (XP(YPZ)) -> p e. ((XPY)PZ)))
24	23	imp 377	. . 3 |- (((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) /\ p e. (XP(YPZ))) -> p e. ((XPY)PZ))
25	24	ex 402	. 2 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> (p e. (XP(YPZ)) -> p e. ((XPY)PZ)))
26	25	ssrdv 2622	1 |- ((K e. HL /\ (X C_ A /\ Y C_ A /\ Z C_ A) /\ ((X =/= (/) /\ (YPZ) =/= (/)) /\ (Y =/= (/) /\ Z =/= (/)))) -> (XP(YPZ)) C_ ((XPY)PZ))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106   C_ wss 2593  (/)c0 2875   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  lecple 16759  joincjn 16766  LatNEWclat 16834  AtomsNEWcatm 16981  HLchlt 16983  +Pcpadd 17256

This theorem is referenced by:  paddasslem18 17298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-tru 1262  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-mpt2 5007  df-iota 5089  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-struct 16708  df-poset 16772  df-plt 16780  df-pge 16792  df-lub 16799  df-glb 16800  df-join 16801  df-meet 16802  df-p0 16841  df-lat 16847  df-clat 16848  df-oposet 16905  df-ol 16907  df-oml 16908  df-covers 16984  df-atoms 16985  df-atlat 16986  df-hlat 17017  df-padd 17257

Copyright terms: Public domain