Theorem islat 17047

Description: The predicate "is a lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)

Hypotheses

Ref	Expression
islat.b	|- B = ( Base ` K )
islat.j	|- .\/ = ( join ` K )
islat.m	|- ./\ = ( meet ` K )

Assertion

Ref	Expression
islat	|- ( K e. Lat <-> ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) )

Proof of Theorem islat

Dummy variable l is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( l = K -> ( join ` l ) = ( join ` K ) )
2		islat.j	. . . . . 6 |- .\/ = ( join ` K )
3	1, 2	syl6eqr 2674	. . . . 5 |- ( l = K -> ( join ` l ) = .\/ )
4	3	dmeqd 5326	. . . 4 |- ( l = K -> dom ( join ` l ) = dom .\/ )
5		fveq2 6191	. . . . . 6 |- ( l = K -> ( Base ` l ) = ( Base ` K ) )
6		islat.b	. . . . . 6 |- B = ( Base ` K )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( l = K -> ( Base ` l ) = B )
8	7	sqxpeqd 5141	. . . 4 |- ( l = K -> ( ( Base ` l ) X. ( Base ` l ) ) = ( B X. B ) )
9	4, 8	eqeq12d 2637	. . 3 |- ( l = K -> ( dom ( join ` l ) = ( ( Base ` l ) X. ( Base ` l ) ) <-> dom .\/ = ( B X. B ) ) )
10		fveq2 6191	. . . . . 6 |- ( l = K -> ( meet ` l ) = ( meet ` K ) )
11		islat.m	. . . . . 6 |- ./\ = ( meet ` K )
12	10, 11	syl6eqr 2674	. . . . 5 |- ( l = K -> ( meet ` l ) = ./\ )
13	12	dmeqd 5326	. . . 4 |- ( l = K -> dom ( meet ` l ) = dom ./\ )
14	13, 8	eqeq12d 2637	. . 3 |- ( l = K -> ( dom ( meet ` l ) = ( ( Base ` l ) X. ( Base ` l ) ) <-> dom ./\ = ( B X. B ) ) )
15	9, 14	anbi12d 747	. 2 |- ( l = K -> ( ( dom ( join ` l ) = ( ( Base ` l ) X. ( Base ` l ) ) /\ dom ( meet ` l ) = ( ( Base ` l ) X. ( Base ` l ) ) ) <-> ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) )
16		df-lat 17046	. 2 |- Lat = { l e. Poset | ( dom ( join ` l ) = ( ( Base ` l ) X. ( Base ` l ) ) /\ dom ( meet ` l ) = ( ( Base ` l ) X. ( Base ` l ) ) ) }
17	15, 16	elrab2 3366	1 |- ( K e. Lat <-> ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   X. cxp 5112   dom cdm 5114   ` cfv 5888   Basecbs 15857   Posetcpo 16940   joincjn 16944   meetcmee 16945   Latclat 17045

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-opab 4713  df-xp 5120  df-dm 5124  df-iota 5851  df-fv 5896  df-lat 17046

This theorem is referenced by:  latcl2  17048  latlem  17049  latpos  17050  latjcom  17059  latmcom  17075  clatl  17116  odulatb  17143