Theorem atlrelat1 32800

Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 27992, 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 1037	. . . 4 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> K e. AtLat )
2		atlpos 32780	. . . 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 16190	. . . 4 |- ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .< Y ) -> -. Y .<_ X )
8	7	ex 435	. . 3 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y -> -. Y .<_ X ) )
9	3, 8	syld3an1 1310	. 2 |- ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B /\ Y e. B ) -> ( X .< Y -> -. Y .<_ X ) )
10		iman 425	. . . . . . 7 |- ( ( p .<_ Y -> p .<_ X ) <-> -. ( p .<_ Y /\ -. p .<_ X ) )
11		ancom 451	. . . . . . 7 |- ( ( p .<_ Y /\ -. p .<_ X ) <-> ( -. p .<_ X /\ p .<_ Y ) )
12	10, 11	xchbinx 311	. . . . . 6 |- ( ( p .<_ Y -> p .<_ X ) <-> -. ( -. p .<_ X /\ p .<_ Y ) )
13	12	ralbii 2854	. . . . 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 32799	. . . . . . 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 1211	. . . . . 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 226	. . . . 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 221	. . . 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 138	. . 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 2874	. . 3 |- ( E. p e. A ( -. p .<_ X /\ p .<_ Y ) <-> -. A. p e. A -. ( -. p .<_ X /\ p .<_ Y ) )
21	19, 20	syl6ibr 230	. 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 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1867   A.wral 2773   E.wrex 2774   class class class wbr 4417   ` cfv 5593   Basecbs 15099   lecple 15175   Posetcpo 16163   ltcplt 16164   CLatccla 16331   OMLcoml 32654   Atomscatm 32742   AtLatcal 32743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6259  df-ov 6300  df-oprab 6301  df-preset 16151  df-poset 16169  df-plt 16182  df-lub 16198  df-glb 16199  df-join 16200  df-meet 16201  df-p0 16263  df-lat 16270  df-clat 16332  df-oposet 32655  df-ol 32657  df-oml 32658  df-covers 32745  df-ats 32746  df-atl 32777

This theorem is referenced by:  cvlcvr1  32818  hlrelat1  32878