Theorem latcl2 17048

Description: The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018.)

Hypotheses

Ref	Expression
latcl2.b	|- B = ( Base ` K )
latcl2.j	|- .\/ = ( join ` K )
latcl2.m	|- ./\ = ( meet ` K )
latcl2.k	|- ( ph -> K e. Lat )
latcl2.x	|- ( ph -> X e. B )
latcl2.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
latcl2	|- ( ph -> ( <. X , Y >. e. dom .\/ /\ <. X , Y >. e. dom ./\ ) )

Proof of Theorem latcl2

Step	Hyp	Ref	Expression
1		latcl2.x	. . . 4 |- ( ph -> X e. B )
2		latcl2.y	. . . 4 |- ( ph -> Y e. B )
3		opelxpi 5148	. . . 4 |- ( ( X e. B /\ Y e. B ) -> <. X , Y >. e. ( B X. B ) )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ph -> <. X , Y >. e. ( B X. B ) )
5		latcl2.k	. . . . 5 |- ( ph -> K e. Lat )
6		latcl2.b	. . . . . 6 |- B = ( Base ` K )
7		latcl2.j	. . . . . 6 |- .\/ = ( join ` K )
8		latcl2.m	. . . . . 6 |- ./\ = ( meet ` K )
9	6, 7, 8	islat 17047	. . . . 5 |- ( K e. Lat <-> ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) )
10	5, 9	sylib 208	. . . 4 |- ( ph -> ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) )
11		simprl 794	. . . 4 |- ( ( K e. Poset /\ ( dom .\/ = ( B X. B ) /\ dom ./\ = ( B X. B ) ) ) -> dom .\/ = ( B X. B ) )
12	10, 11	syl 17	. . 3 |- ( ph -> dom .\/ = ( B X. B ) )
13	4, 12	eleqtrrd 2704	. 2 |- ( ph -> <. X , Y >. e. dom .\/ )
14	10	simprrd 797	. . 3 |- ( ph -> dom ./\ = ( B X. B ) )
15	4, 14	eleqtrrd 2704	. 2 |- ( ph -> <. X , Y >. e. dom ./\ )
16	13, 15	jca 554	1 |- ( ph -> ( <. X , Y >. e. dom .\/ /\ <. X , Y >. e. dom ./\ ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   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  ax-sep 4781  ax-nul 4789  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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  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:  latlej1  17060  latlej2  17061  latjle12  17062  latmle1  17076  latmle2  17077  latlem12  17078