latj12 - Metamath Proof Explorer

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Theorem latj12 17096

Description: Swap 1st and 2nd members of lattice join. (chj12 28393 analog.) (Contributed by NM, 4-Jun-2012.)

**Hypotheses**
Ref	Expression
latjass.b
latjass.j	$.\/$

**Assertion**
Ref	Expression
latj12	$/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$

**Proof of Theorem latj12**
Step	Hyp	Ref	Expression
1		latjass.b	. . . . 5
2		latjass.j	. . . . 5 $.\/$
3	1, 2	latjcom 17059	. . . 4 $/\$ $/\$ $.\/$ $.\/$
4	3	3adant3r3 1276	. . 3 $/\$ $/\$ $/\$ $.\/$ $.\/$
5	4	oveq1d 6665	. 2 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
6	1, 2	latjass 17095	. 2 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
7		simpl 473	. . 3 $/\$ $/\$ $/\$
8		simpr2 1068	. . 3 $/\$ $/\$ $/\$
9		simpr1 1067	. . 3 $/\$ $/\$ $/\$
10		simpr3 1069	. . 3 $/\$ $/\$ $/\$
11	1, 2	latjass 17095	. . 3 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
12	7, 8, 9, 10, 11	syl13anc 1328	. 2 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
13	5, 6, 12	3eqtr3d 2664	1 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cfv 5888 (class class class)co 6650

cbs 15857

cjn 16944

clat 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-preset 16928 df-poset 16946 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046

This theorem is referenced by: latj31 17099 latj4 17101 4atlem4b 34886 4atlem4c 34887 dalawlem3 35159 cdleme1 35514 cdleme5 35527 cdleme11g 35552