Theorem meetle 16827

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

Hypotheses

Ref	Expression
meetval2.b	|- B = (base` K)
meetval2.s	|- S = (geNEW` K)
meetval2.m	|- M = (meet` K)

Assertion

Ref	Expression
meetle	|- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> ((XMY)SZ <-> (XSZ /\ YSZ)))

Proof of Theorem meetle

Step	Hyp	Ref	Expression
1		meetval2.b	. . . . . . . 8 |- B = (base` K)
2		meetval2.s	. . . . . . . 8 |- S = (geNEW` K)
3		meetval2.m	. . . . . . . 8 |- M = (meet` K)
4	1, 2, 3	lemeet1 16825	. . . . . . 7 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XMY) e. B) -> XS(XMY))
5	4	ex 402	. . . . . 6 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> ((XMY) e. B -> XS(XMY)))
6	5	3adant3r3 1079	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XMY) e. B -> XS(XMY)))
7	6	3impia 1064	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> XS(XMY))
8	1, 2	posgetrNEW 16798	. . . . . . . . 9 |- ((K e. PosetNEW /\ (X e. B /\ (XMY) e. B /\ Z e. B)) -> ((XS(XMY) /\ (XMY)SZ) -> XSZ))
9	8	3exp2 1086	. . . . . . . 8 |- (K e. PosetNEW -> (X e. B -> ((XMY) e. B -> (Z e. B -> ((XS(XMY) /\ (XMY)SZ) -> XSZ)))))
10	9	com34 40	. . . . . . 7 |- (K e. PosetNEW -> (X e. B -> (Z e. B -> ((XMY) e. B -> ((XS(XMY) /\ (XMY)SZ) -> XSZ)))))
11	10	imp32 390	. . . . . 6 |- ((K e. PosetNEW /\ (X e. B /\ Z e. B)) -> ((XMY) e. B -> ((XS(XMY) /\ (XMY)SZ) -> XSZ)))
12	11	3adantr2 1036	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XMY) e. B -> ((XS(XMY) /\ (XMY)SZ) -> XSZ)))
13	12	3impia 1064	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> ((XS(XMY) /\ (XMY)SZ) -> XSZ))
14	7, 13	mpand 765	. . 3 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> ((XMY)SZ -> XSZ))
15	1, 2, 3	lemeet2 16826	. . . . . . 7 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XMY) e. B) -> YS(XMY))
16	15	ex 402	. . . . . 6 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> ((XMY) e. B -> YS(XMY)))
17	16	3adant3r3 1079	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XMY) e. B -> YS(XMY)))
18	17	3impia 1064	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> YS(XMY))
19	1, 2	posgetrNEW 16798	. . . . . . . . 9 |- ((K e. PosetNEW /\ (Y e. B /\ (XMY) e. B /\ Z e. B)) -> ((YS(XMY) /\ (XMY)SZ) -> YSZ))
20	19	3exp2 1086	. . . . . . . 8 |- (K e. PosetNEW -> (Y e. B -> ((XMY) e. B -> (Z e. B -> ((YS(XMY) /\ (XMY)SZ) -> YSZ)))))
21	20	com34 40	. . . . . . 7 |- (K e. PosetNEW -> (Y e. B -> (Z e. B -> ((XMY) e. B -> ((YS(XMY) /\ (XMY)SZ) -> YSZ)))))
22	21	3imp 1061	. . . . . 6 |- ((K e. PosetNEW /\ Y e. B /\ Z e. B) -> ((XMY) e. B -> ((YS(XMY) /\ (XMY)SZ) -> YSZ)))
23	22	3adant3r1 1077	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XMY) e. B -> ((YS(XMY) /\ (XMY)SZ) -> YSZ)))
24	23	3impia 1064	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> ((YS(XMY) /\ (XMY)SZ) -> YSZ))
25	18, 24	mpand 765	. . 3 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> ((XMY)SZ -> YSZ))
26	14, 25	jcad 661	. 2 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> ((XMY)SZ -> (XSZ /\ YSZ)))
27	1, 2, 3	meetlem 16824	. . . . . . . . 9 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XMY) e. B) -> ((XS(XMY) /\ YS(XMY)) /\ A.z e. B ((XSz /\ YSz) -> (XMY)Sz)))
28	27	simprd 352	. . . . . . . 8 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XMY) e. B) -> A.z e. B ((XSz /\ YSz) -> (XMY)Sz))
29		breq2 3342	. . . . . . . . . . 11 |- (z = Z -> (XSz <-> XSZ))
30		breq2 3342	. . . . . . . . . . 11 |- (z = Z -> (YSz <-> YSZ))
31	29, 30	anbi12d 690	. . . . . . . . . 10 |- (z = Z -> ((XSz /\ YSz) <-> (XSZ /\ YSZ)))
32		breq2 3342	. . . . . . . . . 10 |- (z = Z -> ((XMY)Sz <-> (XMY)SZ))
33	31, 32	imbi12d 688	. . . . . . . . 9 |- (z = Z -> (((XSz /\ YSz) -> (XMY)Sz) <-> ((XSZ /\ YSZ) -> (XMY)SZ)))
34	33	rcla4cv 2377	. . . . . . . 8 |- (A.z e. B ((XSz /\ YSz) -> (XMY)Sz) -> (Z e. B -> ((XSZ /\ YSZ) -> (XMY)SZ)))
35	28, 34	syl 12	. . . . . . 7 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ (XMY) e. B) -> (Z e. B -> ((XSZ /\ YSZ) -> (XMY)SZ)))
36	35	ex 402	. . . . . 6 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> ((XMY) e. B -> (Z e. B -> ((XSZ /\ YSZ) -> (XMY)SZ))))
37	36	com23 36	. . . . 5 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (Z e. B -> ((XMY) e. B -> ((XSZ /\ YSZ) -> (XMY)SZ))))
38	37	3exp 1066	. . . 4 |- (K e. PosetNEW -> (X e. B -> (Y e. B -> (Z e. B -> ((XMY) e. B -> ((XSZ /\ YSZ) -> (XMY)SZ))))))
39	38	3impd 1082	. . 3 |- (K e. PosetNEW -> ((X e. B /\ Y e. B /\ Z e. B) -> ((XMY) e. B -> ((XSZ /\ YSZ) -> (XMY)SZ))))
40	39	3imp 1061	. 2 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> ((XSZ /\ YSZ) -> (XMY)SZ))
41	26, 40	impbid 574	1 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ (XMY) e. B) -> ((XMY)SZ <-> (XSZ /\ YSZ)))

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  PosetNEWcpo 16760  geNEWcpge 16762  meetcmee 16767

This theorem is referenced by:  latlem12 16873

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-pge 16792  df-glb 16800  df-meet 16802

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