Theorem iscvlat 34610

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

Hypotheses

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

Assertion

Ref	Expression
iscvlat	|- ( 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 ) ) ) )

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

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

Proof of Theorem iscvlat

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
2		iscvlat.a	. . . 4 |- A = ( Atoms ` K )
3	1, 2	syl6eqr 2674	. . 3 |- ( k = K -> ( Atoms ` k ) = A )
4		fveq2 6191	. . . . . 6 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
5		iscvlat.b	. . . . . 6 |- B = ( Base ` K )
6	4, 5	syl6eqr 2674	. . . . 5 |- ( k = K -> ( Base ` k ) = B )
7		fveq2 6191	. . . . . . . . . 10 |- ( k = K -> ( le ` k ) = ( le ` K ) )
8		iscvlat.l	. . . . . . . . . 10 |- .<_ = ( le ` K )
9	7, 8	syl6eqr 2674	. . . . . . . . 9 |- ( k = K -> ( le ` k ) = .<_ )
10	9	breqd 4664	. . . . . . . 8 |- ( k = K -> ( p ( le ` k ) x <-> p .<_ x ) )
11	10	notbid 308	. . . . . . 7 |- ( k = K -> ( -. p ( le ` k ) x <-> -. p .<_ x ) )
12		eqidd 2623	. . . . . . . 8 |- ( k = K -> p = p )
13		fveq2 6191	. . . . . . . . . 10 |- ( k = K -> ( join ` k ) = ( join ` K ) )
14		iscvlat.j	. . . . . . . . . 10 |- .\/ = ( join ` K )
15	13, 14	syl6eqr 2674	. . . . . . . . 9 |- ( k = K -> ( join ` k ) = .\/ )
16	15	oveqd 6667	. . . . . . . 8 |- ( k = K -> ( x ( join ` k ) q ) = ( x .\/ q ) )
17	12, 9, 16	breq123d 4667	. . . . . . 7 |- ( k = K -> ( p ( le ` k ) ( x ( join ` k ) q ) <-> p .<_ ( x .\/ q ) ) )
18	11, 17	anbi12d 747	. . . . . 6 |- ( k = K -> ( ( -. p ( le ` k ) x /\ p ( le ` k ) ( x ( join ` k ) q ) ) <-> ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) ) )
19		eqidd 2623	. . . . . . 7 |- ( k = K -> q = q )
20	15	oveqd 6667	. . . . . . 7 |- ( k = K -> ( x ( join ` k ) p ) = ( x .\/ p ) )
21	19, 9, 20	breq123d 4667	. . . . . 6 |- ( k = K -> ( q ( le ` k ) ( x ( join ` k ) p ) <-> q .<_ ( x .\/ p ) ) )
22	18, 21	imbi12d 334	. . . . 5 |- ( k = K -> ( ( ( -. p ( le ` k ) x /\ p ( le ` k ) ( x ( join ` k ) q ) ) -> q ( le ` k ) ( x ( join ` k ) p ) ) <-> ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
23	6, 22	raleqbidv 3152	. . . 4 |- ( k = K -> ( A. x e. ( Base ` k ) ( ( -. p ( le ` k ) x /\ p ( le ` k ) ( x ( join ` k ) q ) ) -> q ( le ` k ) ( x ( join ` k ) p ) ) <-> A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
24	3, 23	raleqbidv 3152	. . 3 |- ( k = K -> ( A. q e. ( Atoms ` k ) A. x e. ( Base ` k ) ( ( -. p ( le ` k ) x /\ p ( le ` k ) ( x ( join ` k ) q ) ) -> q ( le ` k ) ( x ( join ` k ) p ) ) <-> A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
25	3, 24	raleqbidv 3152	. 2 |- ( k = K -> ( A. p e. ( Atoms ` k ) A. q e. ( Atoms ` k ) A. x e. ( Base ` k ) ( ( -. p ( le ` k ) x /\ p ( le ` k ) ( x ( join ` k ) q ) ) -> q ( le ` k ) ( x ( join ` k ) p ) ) <-> A. p e. A A. q e. A A. x e. B ( ( -. p .<_ x /\ p .<_ ( x .\/ q ) ) -> q .<_ ( x .\/ p ) ) ) )
26		df-cvlat 34609	. 2 |- CvLat = { k e. AtLat | A. p e. ( Atoms ` k ) A. q e. ( Atoms ` k ) A. x e. ( Base ` k ) ( ( -. p ( le ` k ) x /\ p ( le ` k ) ( x ( join ` k ) q ) ) -> q ( le ` k ) ( x ( join ` k ) p ) ) }
27	25, 26	elrab2 3366	1 |- ( 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 ) ) ) )

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   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-cvlat 34609

This theorem is referenced by:  iscvlat2N  34611  cvlatl  34612  cvlexch1  34615  ishlat2  34640