Theorem op01dm 34470

Description: Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018.)

Hypotheses

Ref	Expression
op01dm.b	|- B = ( Base ` K )
op01dm.u	|- U = ( lub ` K )
op01dm.g	|- G = ( glb ` K )

Assertion

Ref	Expression
op01dm	|- ( K e. OP -> ( B e. dom U /\ B e. dom G ) )

Proof of Theorem op01dm

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

Step	Hyp	Ref	Expression
1		op01dm.b	. . 3 |- B = ( Base ` K )
2		op01dm.u	. . 3 |- U = ( lub ` K )
3		op01dm.g	. . 3 |- G = ( glb ` K )
4		eqid 2622	. . 3 |- ( le ` K ) = ( le ` K )
5		eqid 2622	. . 3 |- ( oc ` K ) = ( oc ` K )
6		eqid 2622	. . 3 |- ( join ` K ) = ( join ` K )
7		eqid 2622	. . 3 |- ( meet ` K ) = ( meet ` K )
8		eqid 2622	. . 3 |- ( 0. ` K ) = ( 0. ` K )
9		eqid 2622	. . 3 |- ( 1. ` K ) = ( 1. ` K )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 34467	. 2 |- ( K e. OP <-> ( ( K e. Poset /\ B e. dom U /\ B e. dom G ) /\ A. x e. B A. y e. B ( ( ( ( oc ` K ) ` x ) e. B /\ ( ( oc ` K ) ` ( ( oc ` K ) ` x ) ) = x /\ ( x ( le ` K ) y -> ( ( oc ` K ) ` y ) ( le ` K ) ( ( oc ` K ) ` x ) ) ) /\ ( x ( join ` K ) ( ( oc ` K ) ` x ) ) = ( 1. ` K ) /\ ( x ( meet ` K ) ( ( oc ` K ) ` x ) ) = ( 0. ` K ) ) ) )
11		simpl 473	. . 3 |- ( ( ( B e. dom U /\ B e. dom G ) /\ A. x e. B A. y e. B ( ( ( ( oc ` K ) ` x ) e. B /\ ( ( oc ` K ) ` ( ( oc ` K ) ` x ) ) = x /\ ( x ( le ` K ) y -> ( ( oc ` K ) ` y ) ( le ` K ) ( ( oc ` K ) ` x ) ) ) /\ ( x ( join ` K ) ( ( oc ` K ) ` x ) ) = ( 1. ` K ) /\ ( x ( meet ` K ) ( ( oc ` K ) ` x ) ) = ( 0. ` K ) ) ) -> ( B e. dom U /\ B e. dom G ) )
12	11	3adantl1 1217	. 2 |- ( ( ( K e. Poset /\ B e. dom U /\ B e. dom G ) /\ A. x e. B A. y e. B ( ( ( ( oc ` K ) ` x ) e. B /\ ( ( oc ` K ) ` ( ( oc ` K ) ` x ) ) = x /\ ( x ( le ` K ) y -> ( ( oc ` K ) ` y ) ( le ` K ) ( ( oc ` K ) ` x ) ) ) /\ ( x ( join ` K ) ( ( oc ` K ) ` x ) ) = ( 1. ` K ) /\ ( x ( meet ` K ) ( ( oc ` K ) ` x ) ) = ( 0. ` K ) ) ) -> ( B e. dom U /\ B e. dom G ) )
13	10, 12	sylbi 207	1 |- ( K e. OP -> ( B e. dom U /\ B e. dom G ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   occoc 15949   Posetcpo 16940   lubclub 16942   glbcglb 16943   joincjn 16944   meetcmee 16945   0.cp0 17037   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-ov 6653  df-oposet 34463

This theorem is referenced by:  op0cl  34471  op1cl  34472  op0le  34473  ople1  34478  lhp2lt  35287