Theorem joinle 17014

Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value. (Contributed by NM, 16-Sep-2011.)

Hypotheses

Ref	Expression
joinle.b	|- B = ( Base ` K )
joinle.l	|- .<_ = ( le ` K )
joinle.j	|- .\/ = ( join ` K )
joinle.k	|- ( ph -> K e. Poset )
joinle.x	|- ( ph -> X e. B )
joinle.y	|- ( ph -> Y e. B )
joinle.z	|- ( ph -> Z e. B )
joinle.e	|- ( ph -> <. X , Y >. e. dom .\/ )

Assertion

Ref	Expression
joinle	|- ( ph -> ( ( X .<_ Z /\ Y .<_ Z ) <-> ( X .\/ Y ) .<_ Z ) )

Proof of Theorem joinle

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		joinle.z	. . 3 |- ( ph -> Z e. B )
2		joinle.b	. . . . 5 |- B = ( Base ` K )
3		joinle.l	. . . . 5 |- .<_ = ( le ` K )
4		joinle.j	. . . . 5 |- .\/ = ( join ` K )
5		joinle.k	. . . . 5 |- ( ph -> K e. Poset )
6		joinle.x	. . . . 5 |- ( ph -> X e. B )
7		joinle.y	. . . . 5 |- ( ph -> Y e. B )
8		joinle.e	. . . . 5 |- ( ph -> <. X , Y >. e. dom .\/ )
9	2, 3, 4, 5, 6, 7, 8	joinlem 17011	. . . 4 |- ( ph -> ( ( X .<_ ( X .\/ Y ) /\ Y .<_ ( X .\/ Y ) ) /\ A. z e. B ( ( X .<_ z /\ Y .<_ z ) -> ( X .\/ Y ) .<_ z ) ) )
10	9	simprd 479	. . 3 |- ( ph -> A. z e. B ( ( X .<_ z /\ Y .<_ z ) -> ( X .\/ Y ) .<_ z ) )
11		breq2 4657	. . . . . 6 |- ( z = Z -> ( X .<_ z <-> X .<_ Z ) )
12		breq2 4657	. . . . . 6 |- ( z = Z -> ( Y .<_ z <-> Y .<_ Z ) )
13	11, 12	anbi12d 747	. . . . 5 |- ( z = Z -> ( ( X .<_ z /\ Y .<_ z ) <-> ( X .<_ Z /\ Y .<_ Z ) ) )
14		breq2 4657	. . . . 5 |- ( z = Z -> ( ( X .\/ Y ) .<_ z <-> ( X .\/ Y ) .<_ Z ) )
15	13, 14	imbi12d 334	. . . 4 |- ( z = Z -> ( ( ( X .<_ z /\ Y .<_ z ) -> ( X .\/ Y ) .<_ z ) <-> ( ( X .<_ Z /\ Y .<_ Z ) -> ( X .\/ Y ) .<_ Z ) ) )
16	15	rspcva 3307	. . 3 |- ( ( Z e. B /\ A. z e. B ( ( X .<_ z /\ Y .<_ z ) -> ( X .\/ Y ) .<_ z ) ) -> ( ( X .<_ Z /\ Y .<_ Z ) -> ( X .\/ Y ) .<_ Z ) )
17	1, 10, 16	syl2anc 693	. 2 |- ( ph -> ( ( X .<_ Z /\ Y .<_ Z ) -> ( X .\/ Y ) .<_ Z ) )
18	2, 3, 4, 5, 6, 7, 8	lejoin1 17012	. . . 4 |- ( ph -> X .<_ ( X .\/ Y ) )
19	2, 4, 5, 6, 7, 8	joincl 17006	. . . . 5 |- ( ph -> ( X .\/ Y ) e. B )
20	2, 3	postr 16953	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ ( X .\/ Y ) e. B /\ Z e. B ) ) -> ( ( X .<_ ( X .\/ Y ) /\ ( X .\/ Y ) .<_ Z ) -> X .<_ Z ) )
21	5, 6, 19, 1, 20	syl13anc 1328	. . . 4 |- ( ph -> ( ( X .<_ ( X .\/ Y ) /\ ( X .\/ Y ) .<_ Z ) -> X .<_ Z ) )
22	18, 21	mpand 711	. . 3 |- ( ph -> ( ( X .\/ Y ) .<_ Z -> X .<_ Z ) )
23	2, 3, 4, 5, 6, 7, 8	lejoin2 17013	. . . 4 |- ( ph -> Y .<_ ( X .\/ Y ) )
24	2, 3	postr 16953	. . . . 5 |- ( ( K e. Poset /\ ( Y e. B /\ ( X .\/ Y ) e. B /\ Z e. B ) ) -> ( ( Y .<_ ( X .\/ Y ) /\ ( X .\/ Y ) .<_ Z ) -> Y .<_ Z ) )
25	5, 7, 19, 1, 24	syl13anc 1328	. . . 4 |- ( ph -> ( ( Y .<_ ( X .\/ Y ) /\ ( X .\/ Y ) .<_ Z ) -> Y .<_ Z ) )
26	23, 25	mpand 711	. . 3 |- ( ph -> ( ( X .\/ Y ) .<_ Z -> Y .<_ Z ) )
27	22, 26	jcad 555	. 2 |- ( ph -> ( ( X .\/ Y ) .<_ Z -> ( X .<_ Z /\ Y .<_ Z ) ) )
28	17, 27	impbid 202	1 |- ( ph -> ( ( X .<_ Z /\ Y .<_ Z ) <-> ( X .\/ Y ) .<_ Z ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   Posetcpo 16940   joincjn 16944

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-ov 6653  df-oprab 6654  df-poset 16946  df-lub 16974  df-join 16976

This theorem is referenced by:  latjle12  17062