Theorem iscvlat2N 34611

Description: The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
iscvlat2.b	|- B = ( Base ` K )
iscvlat2.l	|- .<_ = ( le ` K )
iscvlat2.j	|- .\/ = ( join ` K )
iscvlat2.m	|- ./\ = ( meet ` K )
iscvlat2.z	|- .0. = ( 0. ` K )
iscvlat2.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
iscvlat2N	|- ( K e. CvLat <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )

Distinct variable groups:   q, p, x, A   x, B   K, p, q, x

Allowed substitution hints:   B( q, p)   .\/ ( x, q, p)   .<_ ( x, q, p)   ./\ ( x, q, p)   .0. ( x, q, p)

Proof of Theorem iscvlat2N

Step	Hyp	Ref	Expression
1		iscvlat2.b	. . 3 |- B = ( Base ` K )
2		iscvlat2.l	. . 3 |- .<_ = ( le ` K )
3		iscvlat2.j	. . 3 |- .\/ = ( join ` K )
4		iscvlat2.a	. . 3 |- A = ( Atoms ` K )
5	1, 2, 3, 4	iscvlat 34610	. 2 |- ( K e. CvLat <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
6		simpll 790	. . . . . . . 8 |- ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> K e. AtLat )
7		simplrl 800	. . . . . . . 8 |- ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> p e. A )
8		simpr 477	. . . . . . . 8 |- ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> x e. B )
9		iscvlat2.m	. . . . . . . . 9 |- ./\ = ( meet ` K )
10		iscvlat2.z	. . . . . . . . 9 |- .0. = ( 0. ` K )
11	1, 2, 9, 10, 4	atnle 34604	. . . . . . . 8 |- ( ( K e. AtLat /\ p e. A /\ x e. B ) -> ( -. p .<_ x <-> ( p ./\ x ) = .0. ) )
12	6, 7, 8, 11	syl3anc 1326	. . . . . . 7 |- ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> ( -. p .<_ x <-> ( p ./\ x ) = .0. ) )
13	12	anbi1d 741	. . . . . 6 |- ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) <-> ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) ) )
14	13	imbi1d 331	. . . . 5 |- ( ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) /\ x e. B ) -> ( ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) <-> ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
15	14	ralbidva 2985	. . . 4 |- ( ( K e. AtLat /\ ( p e. A /\ q e. A ) ) -> ( A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) <-> A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
16	15	2ralbidva 2988	. . 3 |- ( K e. AtLat -> ( A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) <-> A. p e. A A. q e. A A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
17	16	pm5.32i 669	. 2 |- ( ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
18	5, 17	bitri 264	1 |- ( K e. CvLat <-> ( K e. AtLat /\ A. p e. A A. q e. A A. x e. B ( ( ( p ./\ x ) = .0. /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   0.cp0 17037   Atomscatm 34550   AtLatcal 34551   CvLatclc 34552

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-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-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609

This theorem is referenced by: (None)