Theorem joinle 16820

Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value.

Hypotheses

Ref	Expression
joinval2.b	|- B = (base` K)
joinval2.l	|- L = (le` K)
joinval2.j	|- J = (join` K)

Assertion

Ref	Expression
joinle	|- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> ((XLZ /\ YLZ) <-> (XJY)LZ))

Proof of Theorem joinle

Step	Hyp	Ref	Expression
1		joinval2.b	. . . . . . . . . 10 |- B = (base` K)
2		joinval2.l	. . . . . . . . . 10 |- L = (le` K)
3		joinval2.j	. . . . . . . . . 10 |- J = (join` K)
4	1, 2, 3	joinlem 16817	. . . . . . . . 9 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XJY) e. B) -> ((XL(XJY) /\ YL(XJY)) /\ A.z e. B ((XLz /\ YLz) -> (XJY)Lz)))
5	4	simprd 352	. . . . . . . 8 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XJY) e. B) -> A.z e. B ((XLz /\ YLz) -> (XJY)Lz))
6		breq2 3342	. . . . . . . . . . 11 |- (z = Z -> (XLz <-> XLZ))
7		breq2 3342	. . . . . . . . . . 11 |- (z = Z -> (YLz <-> YLZ))
8	6, 7	anbi12d 690	. . . . . . . . . 10 |- (z = Z -> ((XLz /\ YLz) <-> (XLZ /\ YLZ)))
9		breq2 3342	. . . . . . . . . 10 |- (z = Z -> ((XJY)Lz <-> (XJY)LZ))
10	8, 9	imbi12d 688	. . . . . . . . 9 |- (z = Z -> (((XLz /\ YLz) -> (XJY)Lz) <-> ((XLZ /\ YLZ) -> (XJY)LZ)))
11	10	rcla4cv 2377	. . . . . . . 8 |- (A.z e. B ((XLz /\ YLz) -> (XJY)Lz) -> (Z e. B -> ((XLZ /\ YLZ) -> (XJY)LZ)))
12	5, 11	syl 12	. . . . . . 7 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XJY) e. B) -> (Z e. B -> ((XLZ /\ YLZ) -> (XJY)LZ)))
13	12	ex 402	. . . . . 6 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> ((XJY) e. B -> (Z e. B -> ((XLZ /\ YLZ) -> (XJY)LZ))))
14	13	com23 36	. . . . 5 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (Z e. B -> ((XJY) e. B -> ((XLZ /\ YLZ) -> (XJY)LZ))))
15	14	3exp 1066	. . . 4 |- (K e. PosetNEW -> (X e. B -> (Y e. B -> (Z e. B -> ((XJY) e. B -> ((XLZ /\ YLZ) -> (XJY)LZ))))))
16	15	3impd 1082	. . 3 |- (K e. PosetNEW -> ((X e. B /\ Y e. B /\ Z e. B) -> ((XJY) e. B -> ((XLZ /\ YLZ) -> (XJY)LZ))))
17	16	3imp 1061	. 2 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> ((XLZ /\ YLZ) -> (XJY)LZ))
18	1, 2, 3	lejoin1 16818	. . . . . . 7 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XJY) e. B) -> XL(XJY))
19	18	ex 402	. . . . . 6 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> ((XJY) e. B -> XL(XJY)))
20	19	3adant3r3 1079	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XJY) e. B -> XL(XJY)))
21	20	3impia 1064	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> XL(XJY))
22	1, 2	postrNEW 16777	. . . . . . . . 9 |- ((K e. PosetNEW /\ (X e. B /\ (XJY) e. B /\ Z e. B)) -> ((XL(XJY) /\ (XJY)LZ) -> XLZ))
23	22	3exp2 1086	. . . . . . . 8 |- (K e. PosetNEW -> (X e. B -> ((XJY) e. B -> (Z e. B -> ((XL(XJY) /\ (XJY)LZ) -> XLZ)))))
24	23	com34 40	. . . . . . 7 |- (K e. PosetNEW -> (X e. B -> (Z e. B -> ((XJY) e. B -> ((XL(XJY) /\ (XJY)LZ) -> XLZ)))))
25	24	imp32 390	. . . . . 6 |- ((K e. PosetNEW /\ (X e. B /\ Z e. B)) -> ((XJY) e. B -> ((XL(XJY) /\ (XJY)LZ) -> XLZ)))
26	25	3adantr2 1036	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XJY) e. B -> ((XL(XJY) /\ (XJY)LZ) -> XLZ)))
27	26	3impia 1064	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> ((XL(XJY) /\ (XJY)LZ) -> XLZ))
28	21, 27	mpand 765	. . 3 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> ((XJY)LZ -> XLZ))
29	1, 2, 3	lejoin2 16819	. . . . . . 7 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XJY) e. B) -> YL(XJY))
30	29	ex 402	. . . . . 6 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> ((XJY) e. B -> YL(XJY)))
31	30	3adant3r3 1079	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XJY) e. B -> YL(XJY)))
32	31	3impia 1064	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> YL(XJY))
33	1, 2	postrNEW 16777	. . . . . . . . 9 |- ((K e. PosetNEW /\ (Y e. B /\ (XJY) e. B /\ Z e. B)) -> ((YL(XJY) /\ (XJY)LZ) -> YLZ))
34	33	3exp2 1086	. . . . . . . 8 |- (K e. PosetNEW -> (Y e. B -> ((XJY) e. B -> (Z e. B -> ((YL(XJY) /\ (XJY)LZ) -> YLZ)))))
35	34	com34 40	. . . . . . 7 |- (K e. PosetNEW -> (Y e. B -> (Z e. B -> ((XJY) e. B -> ((YL(XJY) /\ (XJY)LZ) -> YLZ)))))
36	35	3imp 1061	. . . . . 6 |- ((K e. PosetNEW /\ Y e. B /\ Z e. B) -> ((XJY) e. B -> ((YL(XJY) /\ (XJY)LZ) -> YLZ)))
37	36	3adant3r1 1077	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XJY) e. B -> ((YL(XJY) /\ (XJY)LZ) -> YLZ)))
38	37	3impia 1064	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> ((YL(XJY) /\ (XJY)LZ) -> YLZ))
39	32, 38	mpand 765	. . 3 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> ((XJY)LZ -> YLZ))
40	28, 39	jcad 661	. 2 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> ((XJY)LZ -> (XLZ /\ YLZ)))
41	17, 40	impbid 574	1 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XJY) e. B) -> ((XLZ /\ YLZ) <-> (XJY)LZ))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  basecbs 16758  lecple 16759  PosetNEWcpo 16760  joincjn 16766

This theorem is referenced by:  latjle12 16863

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-tru 1262  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-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  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-if 2983  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-mpt2 5007  df-iota 5089  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-struct 16708  df-poset 16772  df-lub 16799  df-join 16801

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