Theorem joincomALT 16828

Description: The join of a poset commutes. (This may not be a theorem under other definitions of meet.)

Hypotheses

Ref	Expression
joincom.b	|- B = (base` K)
joincom.j	|- J = (join` K)

Assertion

Ref	Expression
joincomALT	|- ((K e. A /\ X e. B /\ Y e. B) -> (XJY) = (YJX))

Proof of Theorem joincomALT

Step	Hyp	Ref	Expression
1		prcom 3097	. . . 4 |- {Y, X} = {X, Y}
2	1	fveq2i 4684	. . 3 |- ((lub` K)` {Y, X}) = ((lub` K)` {X, Y})
3	2	a1i 8	. 2 |- ((K e. A /\ X e. B /\ Y e. B) -> ((lub` K)` {Y, X}) = ((lub` K)` {X, Y}))
4		joincom.b	. . . 4 |- B = (base` K)
5		eqid 1884	. . . 4 |- (lub` K) = (lub` K)
6		joincom.j	. . . 4 |- J = (join` K)
7	4, 5, 6	joinval 16815	. . 3 |- ((K e. A /\ Y e. B /\ X e. B) -> (YJX) = ((lub` K)` {Y, X}))
8	7	3com23 1074	. 2 |- ((K e. A /\ X e. B /\ Y e. B) -> (YJX) = ((lub` K)` {Y, X}))
9	4, 5, 6	joinval 16815	. 2 |- ((K e. A /\ X e. B /\ Y e. B) -> (XJY) = ((lub` K)` {X, Y}))
10	3, 8, 9	3eqtr4rd 1939	1 |- ((K e. A /\ X e. B /\ Y e. B) -> (XJY) = (YJX))

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  {cpr 3045  ` cfv 3998  (class class class)co 4884  basecbs 16758  lubclub 16764  joincjn 16766

This theorem is referenced by:  joincom 16829

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-mpt2 5007  df-join 16801

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