Theorem atlex 34603

Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 29219 analog.) (Contributed by NM, 21-Oct-2011.)

Hypotheses

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

Assertion

Ref	Expression
atlex	|- ( ( K e. AtLat /\ X e. B /\ X =/= .0. ) -> E. y e. A y .<_ X )

Distinct variable groups:   y, A   y, K   y, X

Allowed substitution hints:   B( y)   .<_ ( y)   .0. ( y)

Proof of Theorem atlex

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		atlex.b	. . . . 5 |- B = ( Base ` K )
2		eqid 2622	. . . . 5 |- ( glb ` K ) = ( glb ` K )
3		atlex.l	. . . . 5 |- .<_ = ( le ` K )
4		atlex.z	. . . . 5 |- .0. = ( 0. ` K )
5		atlex.a	. . . . 5 |- A = ( Atoms ` K )
6	1, 2, 3, 4, 5	isatl 34586	. . . 4 |- ( K e. AtLat <-> ( K e. Lat /\ B e. dom ( glb ` K ) /\ A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) )
7	6	simp3bi 1078	. . 3 |- ( K e. AtLat -> A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) )
8		neeq1 2856	. . . . 5 |- ( x = X -> ( x =/= .0. <-> X =/= .0. ) )
9		breq2 4657	. . . . . 6 |- ( x = X -> ( y .<_ x <-> y .<_ X ) )
10	9	rexbidv 3052	. . . . 5 |- ( x = X -> ( E. y e. A y .<_ x <-> E. y e. A y .<_ X ) )
11	8, 10	imbi12d 334	. . . 4 |- ( x = X -> ( ( x =/= .0. -> E. y e. A y .<_ x ) <-> ( X =/= .0. -> E. y e. A y .<_ X ) ) )
12	11	rspccv 3306	. . 3 |- ( A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) -> ( X e. B -> ( X =/= .0. -> E. y e. A y .<_ X ) ) )
13	7, 12	syl 17	. 2 |- ( K e. AtLat -> ( X e. B -> ( X =/= .0. -> E. y e. A y .<_ X ) ) )
14	13	3imp 1256	1 |- ( ( K e. AtLat /\ X e. B /\ X =/= .0. ) -> E. y e. A y .<_ X )

Syntax hints:   -> wi 4   /\ 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:  atnle  34604  atlatmstc  34606  cvratlem  34707  cvrat4  34729  2llnmat  34810  2lnat  35070