Theorem ople0 34474

Description: An element less than or equal to zero equals zero. (chle0 28302 analog.) (Contributed by NM, 21-Oct-2011.)

Hypotheses

Ref	Expression
op0le.b	|- B = ( Base ` K )
op0le.l	|- .<_ = ( le ` K )
op0le.z	|- .0. = ( 0. ` K )

Assertion

Ref	Expression
ople0	|- ( ( K e. OP /\ X e. B ) -> ( X .<_ .0. <-> X = .0. ) )

Proof of Theorem ople0

Step	Hyp	Ref	Expression
1		op0le.b	. . . 4 |- B = ( Base ` K )
2		op0le.l	. . . 4 |- .<_ = ( le ` K )
3		op0le.z	. . . 4 |- .0. = ( 0. ` K )
4	1, 2, 3	op0le 34473	. . 3 |- ( ( K e. OP /\ X e. B ) -> .0. .<_ X )
5	4	biantrud 528	. 2 |- ( ( K e. OP /\ X e. B ) -> ( X .<_ .0. <-> ( X .<_ .0. /\ .0. .<_ X ) ) )
6		opposet 34468	. . . 4 |- ( K e. OP -> K e. Poset )
7	6	adantr 481	. . 3 |- ( ( K e. OP /\ X e. B ) -> K e. Poset )
8		simpr 477	. . 3 |- ( ( K e. OP /\ X e. B ) -> X e. B )
9	1, 3	op0cl 34471	. . . 4 |- ( K e. OP -> .0. e. B )
10	9	adantr 481	. . 3 |- ( ( K e. OP /\ X e. B ) -> .0. e. B )
11	1, 2	posasymb 16952	. . 3 |- ( ( K e. Poset /\ X e. B /\ .0. e. B ) -> ( ( X .<_ .0. /\ .0. .<_ X ) <-> X = .0. ) )
12	7, 8, 10, 11	syl3anc 1326	. 2 |- ( ( K e. OP /\ X e. B ) -> ( ( X .<_ .0. /\ .0. .<_ X ) <-> X = .0. ) )
13	5, 12	bitrd 268	1 |- ( ( K e. OP /\ X e. B ) -> ( X .<_ .0. <-> X = .0. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940   0.cp0 17037   OPcops 34459

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

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-preset 16928  df-poset 16946  df-glb 16975  df-p0 17039  df-oposet 34463

This theorem is referenced by:  lub0N  34476  opoc1  34489  atlatmstc  34606  cvrat4  34729  lhpocnle  35302  cdleme22b  35629  tendoid  36061  tendoex  36263