Theorem isatl 34586

Description: The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)

Hypotheses

Ref	Expression
isatlat.b	|- B = ( Base ` K )
isatlat.g	|- G = ( glb ` K )
isatlat.l	|- .<_ = ( le ` K )
isatlat.z	|- .0. = ( 0. ` K )
isatlat.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
isatl	|- ( K e. AtLat <-> ( K e. Lat /\ B e. dom G /\ A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) )

Distinct variable groups:   y, A   x, B   x, y, K

Allowed substitution hints:   A( x)   B( y)   G( x, y)   .<_ ( x, y)   .0. ( x, y)

Proof of Theorem isatl

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
2		isatlat.b	. . . . . 6 |- B = ( Base ` K )
3	1, 2	syl6eqr 2674	. . . . 5 |- ( k = K -> ( Base ` k ) = B )
4		fveq2 6191	. . . . . . 7 |- ( k = K -> ( glb ` k ) = ( glb ` K ) )
5		isatlat.g	. . . . . . 7 |- G = ( glb ` K )
6	4, 5	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( glb ` k ) = G )
7	6	dmeqd 5326	. . . . 5 |- ( k = K -> dom ( glb ` k ) = dom G )
8	3, 7	eleq12d 2695	. . . 4 |- ( k = K -> ( ( Base ` k ) e. dom ( glb ` k ) <-> B e. dom G ) )
9		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( 0. ` k ) = ( 0. ` K ) )
10		isatlat.z	. . . . . . . 8 |- .0. = ( 0. ` K )
11	9, 10	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( 0. ` k ) = .0. )
12	11	neeq2d 2854	. . . . . 6 |- ( k = K -> ( x =/= ( 0. ` k ) <-> x =/= .0. ) )
13		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
14		isatlat.a	. . . . . . . 8 |- A = ( Atoms ` K )
15	13, 14	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( Atoms ` k ) = A )
16		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( le ` k ) = ( le ` K ) )
17		isatlat.l	. . . . . . . . 9 |- .<_ = ( le ` K )
18	16, 17	syl6eqr 2674	. . . . . . . 8 |- ( k = K -> ( le ` k ) = .<_ )
19	18	breqd 4664	. . . . . . 7 |- ( k = K -> ( y ( le ` k ) x <-> y .<_ x ) )
20	15, 19	rexeqbidv 3153	. . . . . 6 |- ( k = K -> ( E. y e. ( Atoms ` k ) y ( le ` k ) x <-> E. y e. A y .<_ x ) )
21	12, 20	imbi12d 334	. . . . 5 |- ( k = K -> ( ( x =/= ( 0. ` k ) -> E. y e. ( Atoms ` k ) y ( le ` k ) x ) <-> ( x =/= .0. -> E. y e. A y .<_ x ) ) )
22	3, 21	raleqbidv 3152	. . . 4 |- ( k = K -> ( A. x e. ( Base ` k ) ( x =/= ( 0. ` k ) -> E. y e. ( Atoms ` k ) y ( le ` k ) x ) <-> A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) )
23	8, 22	anbi12d 747	. . 3 |- ( k = K -> ( ( ( Base ` k ) e. dom ( glb ` k ) /\ A. x e. ( Base ` k ) ( x =/= ( 0. ` k ) -> E. y e. ( Atoms ` k ) y ( le ` k ) x ) ) <-> ( B e. dom G /\ A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) ) )
24		df-atl 34585	. . 3 |- AtLat = { k e. Lat | ( ( Base ` k ) e. dom ( glb ` k ) /\ A. x e. ( Base ` k ) ( x =/= ( 0. ` k ) -> E. y e. ( Atoms ` k ) y ( le ` k ) x ) ) }
25	23, 24	elrab2 3366	. 2 |- ( K e. AtLat <-> ( K e. Lat /\ ( B e. dom G /\ A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) ) )
26		3anass 1042	. 2 |- ( ( K e. Lat /\ B e. dom G /\ A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) <-> ( K e. Lat /\ ( B e. dom G /\ A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) ) )
27	25, 26	bitr4i 267	1 |- ( K e. AtLat <-> ( K e. Lat /\ B e. dom G /\ A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   class class class wbr 4653   dom cdm 5114   ` cfv 5888   Basecbs 15857   lecple 15948   glbcglb 16943   0.cp0 17037   Latclat 17045   Atomscatm 34550   AtLatcal 34551

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-ne 2795  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-dm 5124  df-iota 5851  df-fv 5896  df-atl 34585

This theorem is referenced by:  atllat  34587  atl0dm  34589  atlex  34603