Theorem latjlej12 17067

Description: Add join to both sides of a lattice ordering. (chlej12i 28334 analog.) (Contributed by NM, 8-Nov-2011.)

Hypotheses

Ref	Expression
latlej.b	|- B = ( Base ` K )
latlej.l	|- .<_ = ( le ` K )
latlej.j	|- .\/ = ( join ` K )

Assertion

Ref	Expression
latjlej12	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .<_ Y /\ Z .<_ W ) -> ( X .\/ Z ) .<_ ( Y .\/ W ) ) )

Proof of Theorem latjlej12

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> K e. Lat )
2		simp2l 1087	. . 3 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> X e. B )
3		simp2r 1088	. . 3 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> Y e. B )
4		simp3l 1089	. . 3 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> Z e. B )
5		latlej.b	. . . 4 |- B = ( Base ` K )
6		latlej.l	. . . 4 |- .<_ = ( le ` K )
7		latlej.j	. . . 4 |- .\/ = ( join ` K )
8	5, 6, 7	latjlej1 17065	. . 3 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( X .\/ Z ) .<_ ( Y .\/ Z ) ) )
9	1, 2, 3, 4, 8	syl13anc 1328	. 2 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( X .<_ Y -> ( X .\/ Z ) .<_ ( Y .\/ Z ) ) )
10		simp3r 1090	. . 3 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> W e. B )
11	5, 6, 7	latjlej2 17066	. . 3 |- ( ( K e. Lat /\ ( Z e. B /\ W e. B /\ Y e. B ) ) -> ( Z .<_ W -> ( Y .\/ Z ) .<_ ( Y .\/ W ) ) )
12	1, 4, 10, 3, 11	syl13anc 1328	. 2 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( Z .<_ W -> ( Y .\/ Z ) .<_ ( Y .\/ W ) ) )
13	5, 7	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ X e. B /\ Z e. B ) -> ( X .\/ Z ) e. B )
14	1, 2, 4, 13	syl3anc 1326	. . 3 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( X .\/ Z ) e. B )
15	5, 7	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ Y e. B /\ Z e. B ) -> ( Y .\/ Z ) e. B )
16	1, 3, 4, 15	syl3anc 1326	. . 3 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( Y .\/ Z ) e. B )
17	5, 7	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ Y e. B /\ W e. B ) -> ( Y .\/ W ) e. B )
18	1, 3, 10, 17	syl3anc 1326	. . 3 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( Y .\/ W ) e. B )
19	5, 6	lattr 17056	. . 3 |- ( ( K e. Lat /\ ( ( X .\/ Z ) e. B /\ ( Y .\/ Z ) e. B /\ ( Y .\/ W ) e. B ) ) -> ( ( ( X .\/ Z ) .<_ ( Y .\/ Z ) /\ ( Y .\/ Z ) .<_ ( Y .\/ W ) ) -> ( X .\/ Z ) .<_ ( Y .\/ W ) ) )
20	1, 14, 16, 18, 19	syl13anc 1328	. 2 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( ( X .\/ Z ) .<_ ( Y .\/ Z ) /\ ( Y .\/ Z ) .<_ ( Y .\/ W ) ) -> ( X .\/ Z ) .<_ ( Y .\/ W ) ) )
21	9, 12, 20	syl2and 500	1 |- ( ( K e. Lat /\ ( X e. B /\ Y e. B ) /\ ( Z e. B /\ W e. B ) ) -> ( ( X .<_ Y /\ Z .<_ W ) -> ( X .\/ Z ) .<_ ( Y .\/ W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   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-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-glb 16975  df-join 16976  df-meet 16977  df-lat 17046

This theorem is referenced by:  latledi  17089  dalem-cly  34957  dalem38  34996  dalem44  35002  cdlema1N  35077  pmapjoin  35138  4atexlemc  35355  cdlemg33a  35994