Theorem clatl 17116

Description: A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.) TODO: use eqrelrdv2 5219 to shorten proof and eliminate joindmss 17007 and meetdmss 17021?

Assertion

Ref	Expression
clatl	|- ( K e. CLat -> K e. Lat )

Proof of Theorem clatl

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
2		eqid 2622	. . . . . . 7 |- ( join ` K ) = ( join ` K )
3		simpl 473	. . . . . . 7 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> K e. Poset )
4	1, 2, 3	joindmss 17007	. . . . . 6 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> dom ( join ` K ) C_ ( ( Base ` K ) X. ( Base ` K ) ) )
5		relxp 5227	. . . . . . . 8 |- Rel ( ( Base ` K ) X. ( Base ` K ) )
6	5	a1i 11	. . . . . . 7 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> Rel ( ( Base ` K ) X. ( Base ` K ) ) )
7		opelxp 5146	. . . . . . . . . . . 12 |- ( <. x , y >. e. ( ( Base ` K ) X. ( Base ` K ) ) <-> ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) )
8		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
9		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
10	8, 9	prss 4351	. . . . . . . . . . . 12 |- ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) <-> { x , y } C_ ( Base ` K ) )
11	7, 10	sylbb 209	. . . . . . . . . . 11 |- ( <. x , y >. e. ( ( Base ` K ) X. ( Base ` K ) ) -> { x , y } C_ ( Base ` K ) )
12		prex 4909	. . . . . . . . . . . 12 |- { x , y } e. _V
13	12	elpw 4164	. . . . . . . . . . 11 |- ( { x , y } e. ~P ( Base ` K ) <-> { x , y } C_ ( Base ` K ) )
14	11, 13	sylibr 224	. . . . . . . . . 10 |- ( <. x , y >. e. ( ( Base ` K ) X. ( Base ` K ) ) -> { x , y } e. ~P ( Base ` K ) )
15		eleq2 2690	. . . . . . . . . 10 |- ( dom ( lub ` K ) = ~P ( Base ` K ) -> ( { x , y } e. dom ( lub ` K ) <-> { x , y } e. ~P ( Base ` K ) ) )
16	14, 15	syl5ibr 236	. . . . . . . . 9 |- ( dom ( lub ` K ) = ~P ( Base ` K ) -> ( <. x , y >. e. ( ( Base ` K ) X. ( Base ` K ) ) -> { x , y } e. dom ( lub ` K ) ) )
17	16	adantl 482	. . . . . . . 8 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> ( <. x , y >. e. ( ( Base ` K ) X. ( Base ` K ) ) -> { x , y } e. dom ( lub ` K ) ) )
18		eqid 2622	. . . . . . . . 9 |- ( lub ` K ) = ( lub ` K )
19	8	a1i 11	. . . . . . . . 9 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> x e. _V )
20	9	a1i 11	. . . . . . . . 9 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> y e. _V )
21	18, 2, 3, 19, 20	joindef 17004	. . . . . . . 8 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> ( <. x , y >. e. dom ( join ` K ) <-> { x , y } e. dom ( lub ` K ) ) )
22	17, 21	sylibrd 249	. . . . . . 7 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> ( <. x , y >. e. ( ( Base ` K ) X. ( Base ` K ) ) -> <. x , y >. e. dom ( join ` K ) ) )
23	6, 22	relssdv 5212	. . . . . 6 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> ( ( Base ` K ) X. ( Base ` K ) ) C_ dom ( join ` K ) )
24	4, 23	eqssd 3620	. . . . 5 |- ( ( K e. Poset /\ dom ( lub ` K ) = ~P ( Base ` K ) ) -> dom ( join ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) )
25	24	ex 450	. . . 4 |- ( K e. Poset -> ( dom ( lub ` K ) = ~P ( Base ` K ) -> dom ( join ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) ) )
26		eqid 2622	. . . . . . 7 |- ( meet ` K ) = ( meet ` K )
27		simpl 473	. . . . . . 7 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> K e. Poset )
28	1, 26, 27	meetdmss 17021	. . . . . 6 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> dom ( meet ` K ) C_ ( ( Base ` K ) X. ( Base ` K ) ) )
29	5	a1i 11	. . . . . . 7 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> Rel ( ( Base ` K ) X. ( Base ` K ) ) )
30		eleq2 2690	. . . . . . . . . 10 |- ( dom ( glb ` K ) = ~P ( Base ` K ) -> ( { x , y } e. dom ( glb ` K ) <-> { x , y } e. ~P ( Base ` K ) ) )
31	14, 30	syl5ibr 236	. . . . . . . . 9 |- ( dom ( glb ` K ) = ~P ( Base ` K ) -> ( <. x , y >. e. ( ( Base ` K ) X. ( Base ` K ) ) -> { x , y } e. dom ( glb ` K ) ) )
32	31	adantl 482	. . . . . . . 8 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> ( <. x , y >. e. ( ( Base ` K ) X. ( Base ` K ) ) -> { x , y } e. dom ( glb ` K ) ) )
33		eqid 2622	. . . . . . . . 9 |- ( glb ` K ) = ( glb ` K )
34	8	a1i 11	. . . . . . . . 9 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> x e. _V )
35	9	a1i 11	. . . . . . . . 9 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> y e. _V )
36	33, 26, 27, 34, 35	meetdef 17018	. . . . . . . 8 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> ( <. x , y >. e. dom ( meet ` K ) <-> { x , y } e. dom ( glb ` K ) ) )
37	32, 36	sylibrd 249	. . . . . . 7 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> ( <. x , y >. e. ( ( Base ` K ) X. ( Base ` K ) ) -> <. x , y >. e. dom ( meet ` K ) ) )
38	29, 37	relssdv 5212	. . . . . 6 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> ( ( Base ` K ) X. ( Base ` K ) ) C_ dom ( meet ` K ) )
39	28, 38	eqssd 3620	. . . . 5 |- ( ( K e. Poset /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> dom ( meet ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) )
40	39	ex 450	. . . 4 |- ( K e. Poset -> ( dom ( glb ` K ) = ~P ( Base ` K ) -> dom ( meet ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) ) )
41	25, 40	anim12d 586	. . 3 |- ( K e. Poset -> ( ( dom ( lub ` K ) = ~P ( Base ` K ) /\ dom ( glb ` K ) = ~P ( Base ` K ) ) -> ( dom ( join ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) /\ dom ( meet ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) ) ) )
42	41	imdistani 726	. 2 |- ( ( K e. Poset /\ ( dom ( lub ` K ) = ~P ( Base ` K ) /\ dom ( glb ` K ) = ~P ( Base ` K ) ) ) -> ( K e. Poset /\ ( dom ( join ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) /\ dom ( meet ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) ) ) )
43	1, 18, 33	isclat 17109	. 2 |- ( K e. CLat <-> ( K e. Poset /\ ( dom ( lub ` K ) = ~P ( Base ` K ) /\ dom ( glb ` K ) = ~P ( Base ` K ) ) ) )
44	1, 2, 26	islat 17047	. 2 |- ( K e. Lat <-> ( K e. Poset /\ ( dom ( join ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) /\ dom ( meet ` K ) = ( ( Base ` K ) X. ( Base ` K ) ) ) ) )
45	42, 43, 44	3imtr4i 281	1 |- ( K e. CLat -> K e. Lat )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {cpr 4179   <.cop 4183   X. cxp 5112   dom cdm 5114   Rel wrel 5119   ` cfv 5888   Basecbs 15857   Posetcpo 16940   lubclub 16942   glbcglb 16943   joincjn 16944   meetcmee 16945   Latclat 17045   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  ax-un 6949

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-oprab 6654  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-lat 17046  df-clat 17108

This theorem is referenced by:  lubel  17122  lubun  17123  clatleglb  17126