Theorem lub0N 34476

Description: The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lub0.u	|- .1. = ( lub ` K )
lub0.z	|- .0. = ( 0. ` K )

Assertion

Ref	Expression
lub0N	|- ( K e. OP -> ( .1. ` (/) ) = .0. )

Proof of Theorem lub0N

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` K ) = ( Base ` K )
2		eqid 2622	. . 3 |- ( le ` K ) = ( le ` K )
3		lub0.u	. . 3 |- .1. = ( lub ` K )
4		biid 251	. . 3 |- ( ( A. y e. (/) y ( le ` K ) x /\ A. z e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) ) <-> ( A. y e. (/) y ( le ` K ) x /\ A. z e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) ) )
5		id 22	. . 3 |- ( K e. OP -> K e. OP )
6		0ss 3972	. . . 4 |- (/) C_ ( Base ` K )
7	6	a1i 11	. . 3 |- ( K e. OP -> (/) C_ ( Base ` K ) )
8	1, 2, 3, 4, 5, 7	lubval 16984	. 2 |- ( K e. OP -> ( .1. ` (/) ) = ( iota_ x e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) x /\ A. z e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) ) ) )
9		lub0.z	. . . 4 |- .0. = ( 0. ` K )
10	1, 9	op0cl 34471	. . 3 |- ( K e. OP -> .0. e. ( Base ` K ) )
11		ral0 4076	. . . . . . 7 |- A. y e. (/) y ( le ` K ) z
12	11	a1bi 352	. . . . . 6 |- ( x ( le ` K ) z <-> ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) )
13	12	ralbii 2980	. . . . 5 |- ( A. z e. ( Base ` K ) x ( le ` K ) z <-> A. z e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) )
14		ral0 4076	. . . . . 6 |- A. y e. (/) y ( le ` K ) x
15	14	biantrur 527	. . . . 5 |- ( A. z e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) <-> ( A. y e. (/) y ( le ` K ) x /\ A. z e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) ) )
16	13, 15	bitri 264	. . . 4 |- ( A. z e. ( Base ` K ) x ( le ` K ) z <-> ( A. y e. (/) y ( le ` K ) x /\ A. z e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) ) )
17	10	adantr 481	. . . . . . 7 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> .0. e. ( Base ` K ) )
18		breq2 4657	. . . . . . . 8 |- ( z = .0. -> ( x ( le ` K ) z <-> x ( le ` K ) .0. ) )
19	18	rspcv 3305	. . . . . . 7 |- ( .0. e. ( Base ` K ) -> ( A. z e. ( Base ` K ) x ( le ` K ) z -> x ( le ` K ) .0. ) )
20	17, 19	syl 17	. . . . . 6 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> ( A. z e. ( Base ` K ) x ( le ` K ) z -> x ( le ` K ) .0. ) )
21	1, 2, 9	ople0 34474	. . . . . 6 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> ( x ( le ` K ) .0. <-> x = .0. ) )
22	20, 21	sylibd 229	. . . . 5 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> ( A. z e. ( Base ` K ) x ( le ` K ) z -> x = .0. ) )
23	1, 2, 9	op0le 34473	. . . . . . . . . 10 |- ( ( K e. OP /\ z e. ( Base ` K ) ) -> .0. ( le ` K ) z )
24	23	adantlr 751	. . . . . . . . 9 |- ( ( ( K e. OP /\ x e. ( Base ` K ) ) /\ z e. ( Base ` K ) ) -> .0. ( le ` K ) z )
25	24	ex 450	. . . . . . . 8 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> ( z e. ( Base ` K ) -> .0. ( le ` K ) z ) )
26		breq1 4656	. . . . . . . . 9 |- ( x = .0. -> ( x ( le ` K ) z <-> .0. ( le ` K ) z ) )
27	26	biimprcd 240	. . . . . . . 8 |- ( .0. ( le ` K ) z -> ( x = .0. -> x ( le ` K ) z ) )
28	25, 27	syl6 35	. . . . . . 7 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> ( z e. ( Base ` K ) -> ( x = .0. -> x ( le ` K ) z ) ) )
29	28	com23 86	. . . . . 6 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> ( x = .0. -> ( z e. ( Base ` K ) -> x ( le ` K ) z ) ) )
30	29	ralrimdv 2968	. . . . 5 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> ( x = .0. -> A. z e. ( Base ` K ) x ( le ` K ) z ) )
31	22, 30	impbid 202	. . . 4 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> ( A. z e. ( Base ` K ) x ( le ` K ) z <-> x = .0. ) )
32	16, 31	syl5bbr 274	. . 3 |- ( ( K e. OP /\ x e. ( Base ` K ) ) -> ( ( A. y e. (/) y ( le ` K ) x /\ A. z e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) ) <-> x = .0. ) )
33	10, 32	riota5 6637	. 2 |- ( K e. OP -> ( iota_ x e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) x /\ A. z e. ( Base ` K ) ( A. y e. (/) y ( le ` K ) z -> x ( le ` K ) z ) ) ) = .0. )
34	8, 33	eqtrd 2656	1 |- ( K e. OP -> ( .1. ` (/) ) = .0. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888   iota_crio 6610   Basecbs 15857   lecple 15948   lubclub 16942   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-lub 16974  df-glb 16975  df-p0 17039  df-oposet 34463

This theorem is referenced by: (None)