Theorem atlrelat1 32881

Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 28009, with /\ swapped, analog.) (Contributed by NM, 4-Dec-2011.)

Hypotheses

Ref	Expression
atlrelat1.b	|- B = ( Base ` K )
atlrelat1.l	|- .<_ = ( le ` K )
atlrelat1.s	|- .< = ( lt ` K )
atlrelat1.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
atlrelat1	|- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> ( X .< Y -> E. p e. A ( -. p .<_ X /\ p .<_ Y ) ) )

Distinct variable groups:   A, p   B, p   K, p   .<_ , p   X, p   Y, p

Allowed substitution hint:   .< ( p)

Proof of Theorem atlrelat1

Step	Hyp	Ref	Expression
1		simp13 1039	. . . 4 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> K e. AtLat )
2		atlpos 32861	. . . 4 |- ( K e. AtLat -> K e. Poset )
3	1, 2	syl 17	. . 3 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> K e. Poset )
4		atlrelat1.b	. . . . 5 |- B = ( Base ` K )
5		atlrelat1.l	. . . . 5 |- .<_ = ( le ` K )
6		atlrelat1.s	. . . . 5 |- .< = ( lt ` K )
7	4, 5, 6	pltnle 16205	. . . 4 |- ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .< Y ) -> -. Y .<_ X )
8	7	ex 436	. . 3 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y -> -. Y .<_ X ) )
9	3, 8	syld3an1 1313	. 2 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> ( X .< Y -> -. Y .<_ X ) )
10		iman 426	. . . . . . 7 |- ( ( p .<_ Y -> p .<_ X ) <-> -. ( p .<_ Y /\ -. p .<_ X ) )
11		ancom 452	. . . . . . 7 |- ( ( p .<_ Y /\ -. p .<_ X ) <-> ( -. p .<_ X /\ p .<_ Y ) )
12	10, 11	xchbinx 312	. . . . . 6 |- ( ( p .<_ Y -> p .<_ X ) <-> -. ( -. p .<_ X /\ p .<_ Y ) )
13	12	ralbii 2818	. . . . 5 |- ( A. p e. A ( p .<_ Y -> p .<_ X ) <-> A. p e. A -. ( -. p .<_ X /\ p .<_ Y ) )
14		atlrelat1.a	. . . . . . . 8 |- A = ( Atoms ` K )
15	4, 5, 14	atlatle 32880	. . . . . . 7 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ Y e. B /\ X e. B ) -> ( Y .<_ X <-> A. p e. A ( p .<_ Y -> p .<_ X ) ) )
16	15	3com23 1213	. . . . . 6 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> ( Y .<_ X <-> A. p e. A ( p .<_ Y -> p .<_ X ) ) )
17	16	biimprd 227	. . . . 5 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> ( A. p e. A ( p .<_ Y -> p .<_ X ) -> Y .<_ X ) )
18	13, 17	syl5bir 222	. . . 4 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> ( A. p e. A -. ( -. p .<_ X /\ p .<_ Y ) -> Y .<_ X ) )
19	18	con3d 139	. . 3 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> ( -. Y .<_ X -> -. A. p e. A -. ( -. p .<_ X /\ p .<_ Y ) ) )
20		dfrex2 2837	. . 3 |- ( E. p e. A ( -. p .<_ X /\ p .<_ Y ) <-> -. A. p e. A -. ( -. p .<_ X /\ p .<_ Y ) )
21	19, 20	syl6ibr 231	. 2 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> ( -. Y .<_ X -> E. p e. A ( -. p .<_ X /\ p .<_ Y ) ) )
22	9, 21	syld 45	1 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> ( X .< Y -> E. p e. A ( -. p .<_ X /\ p .<_ Y ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   A.wral 2736   E.wrex 2737   class class class wbr 4401   ` cfv 5581   Basecbs 15114   lecple 15190   Posetcpo 16178   ltcplt 16179   CLatccla 16346   OMLcoml 32735   Atomscatm 32823   AtLatcal 32824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-lat 16285  df-clat 16347  df-oposet 32736  df-ol 32738  df-oml 32739  df-covers 32826  df-ats 32827  df-atl 32858

This theorem is referenced by:  cvlcvr1  32899  hlrelat1  32959