Theorem clatlem 17111

Description: Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.)

Hypotheses

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

Assertion

Ref	Expression
clatlem	|- ( ( K e. CLat /\ S C_ B ) -> ( ( U ` S ) e. B /\ ( G ` S ) e. B ) )

Proof of Theorem clatlem

Step	Hyp	Ref	Expression
1		clatlem.b	. . 3 |- B = ( Base ` K )
2		clatlem.u	. . 3 |- U = ( lub ` K )
3		simpl 473	. . 3 |- ( ( K e. CLat /\ S C_ B ) -> K e. CLat )
4		fvex 6201	. . . . . . . 8 |- ( Base ` K ) e. _V
5	1, 4	eqeltri 2697	. . . . . . 7 |- B e. _V
6	5	elpw2 4828	. . . . . 6 |- ( S e. ~P B <-> S C_ B )
7	6	biimpri 218	. . . . 5 |- ( S C_ B -> S e. ~P B )
8	7	adantl 482	. . . 4 |- ( ( K e. CLat /\ S C_ B ) -> S e. ~P B )
9		clatlem.g	. . . . . . . 8 |- G = ( glb ` K )
10	1, 2, 9	isclat 17109	. . . . . . 7 |- ( K e. CLat <-> ( K e. Poset /\ ( dom U = ~P B /\ dom G = ~P B ) ) )
11	10	biimpi 206	. . . . . 6 |- ( K e. CLat -> ( K e. Poset /\ ( dom U = ~P B /\ dom G = ~P B ) ) )
12	11	adantr 481	. . . . 5 |- ( ( K e. CLat /\ S C_ B ) -> ( K e. Poset /\ ( dom U = ~P B /\ dom G = ~P B ) ) )
13		simpl 473	. . . . . 6 |- ( ( dom U = ~P B /\ dom G = ~P B ) -> dom U = ~P B )
14	13	adantl 482	. . . . 5 |- ( ( K e. Poset /\ ( dom U = ~P B /\ dom G = ~P B ) ) -> dom U = ~P B )
15	12, 14	syl 17	. . . 4 |- ( ( K e. CLat /\ S C_ B ) -> dom U = ~P B )
16	8, 15	eleqtrrd 2704	. . 3 |- ( ( K e. CLat /\ S C_ B ) -> S e. dom U )
17	1, 2, 3, 16	lubcl 16985	. 2 |- ( ( K e. CLat /\ S C_ B ) -> ( U ` S ) e. B )
18	12	simprrd 797	. . . 4 |- ( ( K e. CLat /\ S C_ B ) -> dom G = ~P B )
19	8, 18	eleqtrrd 2704	. . 3 |- ( ( K e. CLat /\ S C_ B ) -> S e. dom G )
20	1, 9, 3, 19	glbcl 16998	. 2 |- ( ( K e. CLat /\ S C_ B ) -> ( G ` S ) e. B )
21	17, 20	jca 554	1 |- ( ( K e. CLat /\ S C_ B ) -> ( ( U ` S ) e. B /\ ( G ` S ) e. B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   dom cdm 5114   ` cfv 5888   Basecbs 15857   Posetcpo 16940   lubclub 16942   glbcglb 16943   CLatccla 17107

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-lub 16974  df-glb 16975  df-clat 17108

This theorem is referenced by:  clatlubcl  17112  clatglbcl  17114