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Theorem isdlat 17193

Description: Property of being a distributive lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)

**Hypotheses**
Ref	Expression
isdlat.b
isdlat.j	$.\/$
isdlat.m	$./\$

**Assertion**
Ref	Expression
isdlat	DLat $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Distinct variable groups:

, $.\/$ ,

, $./\$ ,

Proof of Theorem isdlat Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5
2		isdlat.b	. . . . 5
3	1, 2	syl6eqr 2674	. . . 4
4		fveq2 6191	. . . . . 6
5		isdlat.j	. . . . . 6 $.\/$
6	4, 5	syl6eqr 2674	. . . . 5 $.\/$
7		fveq2 6191	. . . . . . 7
8		isdlat.m	. . . . . . 7 $./\$
9	7, 8	syl6eqr 2674	. . . . . 6 $./\$
10	9	sbceq1d 3440	. . . . 5 $./\$
11	6, 10	sbceqbid 3442	. . . 4 $.\/$ $./\$
12	3, 11	sbceqbid 3442	. . 3 $.\/$ $./\$
13		fvex 6201	. . . . 5
14	2, 13	eqeltri 2697	. . . 4
15		fvex 6201	. . . . 5
16	5, 15	eqeltri 2697	. . . 4 $.\/$
17		fvex 6201	. . . . 5
18	8, 17	eqeltri 2697	. . . 4 $./\$
19		raleq 3138	. . . . . . . 8
20	19	raleqbi1dv 3146	. . . . . . 7
21	20	raleqbi1dv 3146	. . . . . 6
22		simpr 477	. . . . . . . . . 10 $.\/$ $/\$ $./\$ $./\$
23		eqidd 2623	. . . . . . . . . 10 $.\/$ $/\$ $./\$
24		simpl 473	. . . . . . . . . . 11 $.\/$ $/\$ $./\$ $.\/$
25	24	oveqd 6667	. . . . . . . . . 10 $.\/$ $/\$ $./\$ $.\/$
26	22, 23, 25	oveq123d 6671	. . . . . . . . 9 $.\/$ $/\$ $./\$ $./\$ $.\/$
27	22	oveqd 6667	. . . . . . . . . 10 $.\/$ $/\$ $./\$ $./\$
28	22	oveqd 6667	. . . . . . . . . 10 $.\/$ $/\$ $./\$ $./\$
29	24, 27, 28	oveq123d 6671	. . . . . . . . 9 $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$
30	26, 29	eqeq12d 2637	. . . . . . . 8 $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
31	30	ralbidv 2986	. . . . . . 7 $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
32	31	2ralbidv 2989	. . . . . 6 $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
33	21, 32	sylan9bb 736	. . . . 5 $/\$ $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
34	33	3impb 1260	. . . 4 $/\$ $.\/$ $/\$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
35	14, 16, 18, 34	sbc3ie 3507	. . 3 $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
36	12, 35	syl6bb 276	. 2 $./\$ $.\/$ $./\$ $.\/$ $./\$
37		df-dlat 17192	. 2 DLat
38	36, 37	elrab2 3366	1 DLat $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

cvv 3200

wsbc 3435

cfv 5888 (class class class)co 6650

cbs 15857

cjn 16944

cmee 16945

clat 17045 DLatcdlat 17191

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-iota 5851 df-fv 5896 df-ov 6653 df-dlat 17192

This theorem is referenced by: dlatmjdi 17194 dlatl 17195 odudlatb 17196

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