Theorem ncvr1 29755

Description: No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)

Hypotheses

Ref	Expression
ncvr1.b	|- B = ( Base ` K )
ncvr1.u	|- .1. = ( 1. ` K )
ncvr1.c	|- C = ( <o ` K )

Assertion

Ref	Expression
ncvr1	|- ( ( K e. OP /\ X e. B ) -> -. .1. C X )

Proof of Theorem ncvr1

Step	Hyp	Ref	Expression
1		ncvr1.b	. . . 4 |- B = ( Base ` K )
2		eqid 2404	. . . 4 |- ( le ` K ) = ( le ` K )
3		ncvr1.u	. . . 4 |- .1. = ( 1. ` K )
4	1, 2, 3	ople1 29674	. . 3 |- ( ( K e. OP /\ X e. B ) -> X ( le ` K ) .1. )
5		opposet 29665	. . . . . 6 |- ( K e. OP -> K e. Poset )
6	5	ad2antrr 707	. . . . 5 |- ( ( ( K e. OP /\ X e. B ) /\ .1. ( lt ` K ) X ) -> K e. Poset )
7	1, 3	op1cl 29668	. . . . . 6 |- ( K e. OP -> .1. e. B )
8	7	ad2antrr 707	. . . . 5 |- ( ( ( K e. OP /\ X e. B ) /\ .1. ( lt ` K ) X ) -> .1. e. B )
9		simplr 732	. . . . 5 |- ( ( ( K e. OP /\ X e. B ) /\ .1. ( lt ` K ) X ) -> X e. B )
10		simpr 448	. . . . 5 |- ( ( ( K e. OP /\ X e. B ) /\ .1. ( lt ` K ) X ) -> .1. ( lt ` K ) X )
11		eqid 2404	. . . . . 6 |- ( lt ` K ) = ( lt ` K )
12	1, 2, 11	pltnle 14378	. . . . 5 |- ( ( ( K e. Poset /\ .1. e. B /\ X e. B ) /\ .1. ( lt ` K ) X ) -> -. X ( le ` K ) .1. )
13	6, 8, 9, 10, 12	syl31anc 1187	. . . 4 |- ( ( ( K e. OP /\ X e. B ) /\ .1. ( lt ` K ) X ) -> -. X ( le ` K ) .1. )
14	13	ex 424	. . 3 |- ( ( K e. OP /\ X e. B ) -> ( .1. ( lt ` K ) X -> -. X ( le ` K ) .1. ) )
15	4, 14	mt2d 111	. 2 |- ( ( K e. OP /\ X e. B ) -> -. .1. ( lt ` K ) X )
16		simpll 731	. . 3 |- ( ( ( K e. OP /\ X e. B ) /\ .1. C X ) -> K e. OP )
17	7	ad2antrr 707	. . 3 |- ( ( ( K e. OP /\ X e. B ) /\ .1. C X ) -> .1. e. B )
18		simplr 732	. . 3 |- ( ( ( K e. OP /\ X e. B ) /\ .1. C X ) -> X e. B )
19		simpr 448	. . 3 |- ( ( ( K e. OP /\ X e. B ) /\ .1. C X ) -> .1. C X )
20		ncvr1.c	. . . 4 |- C = ( <o ` K )
21	1, 11, 20	cvrlt 29753	. . 3 |- ( ( ( K e. OP /\ .1. e. B /\ X e. B ) /\ .1. C X ) -> .1. ( lt ` K ) X )
22	16, 17, 18, 19, 21	syl31anc 1187	. 2 |- ( ( ( K e. OP /\ X e. B ) /\ .1. C X ) -> .1. ( lt ` K ) X )
23	15, 22	mtand 641	1 |- ( ( K e. OP /\ X e. B ) -> -. .1. C X )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   Posetcpo 14352   ltcplt 14353   1.cp1 14422   OPcops 29655   <o ccvr 29745

This theorem is referenced by:  lhp2lt  30483

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-p1 14424  df-oposet 29659  df-covers 29749