Theorem op1le 34479

Description: If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 28302 analog.) (Contributed by NM, 5-Dec-2011.)

Hypotheses

Ref	Expression
ople1.b	|- B = ( Base ` K )
ople1.l	|- .<_ = ( le ` K )
ople1.u	|- .1. = ( 1. ` K )

Assertion

Ref	Expression
op1le	|- ( ( K e. OP /\ X e. B ) -> ( .1. .<_ X <-> X = .1. ) )

Proof of Theorem op1le

Step	Hyp	Ref	Expression
1		ople1.b	. . . 4 |- B = ( Base ` K )
2		ople1.l	. . . 4 |- .<_ = ( le ` K )
3		ople1.u	. . . 4 |- .1. = ( 1. ` K )
4	1, 2, 3	ople1 34478	. . 3 |- ( ( K e. OP /\ X e. B ) -> X .<_ .1. )
5	4	biantrurd 529	. 2 |- ( ( K e. OP /\ X e. B ) -> ( .1. .<_ X <-> ( X .<_ .1. /\ .1. .<_ 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	op1cl 34472	. . . 4 |- ( K e. OP -> .1. e. B )
10	9	adantr 481	. . 3 |- ( ( K e. OP /\ X e. B ) -> .1. e. B )
11	1, 2	posasymb 16952	. . 3 |- ( ( K e. Poset /\ X e. B /\ .1. e. B ) -> ( ( X .<_ .1. /\ .1. .<_ X ) <-> X = .1. ) )
12	7, 8, 10, 11	syl3anc 1326	. 2 |- ( ( K e. OP /\ X e. B ) -> ( ( X .<_ .1. /\ .1. .<_ X ) <-> X = .1. ) )
13	5, 12	bitrd 268	1 |- ( ( K e. OP /\ X e. B ) -> ( .1. .<_ X <-> X = .1. ) )

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   1.cp1 17038   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-lub 16974  df-p1 17040  df-oposet 34463

This theorem is referenced by:  glb0N  34480  lhpj1  35308