Theorem oposlem 16913

Description: Lemma for orthoposet properties.

Hypotheses

Ref	Expression
oposlem.b	|- B = (base` K)
oposlem.l	|- L = (le` K)
oposlem.o	|- O = (oc` K)
oposlem.j	|- J = (join` K)
oposlem.m	|- M = (meet` K)
oposlem.f	|- F = (0.` K)
oposlem.t	|- T = (1.` K)

Assertion

Ref	Expression
oposlem	|- ((K e. OP /\ X e. B /\ Y e. B) -> (((O` X) e. B /\ (O` (O` X)) = X /\ (XLY -> (O` Y)L(O` X))) /\ (XJ(O` X)) = T /\ (XM(O` X)) = F))

Proof of Theorem oposlem

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . 5 |- Struct(3, p, T. ) = Struct(3, p, T. )
2		oposlem.b	. . . . 5 |- B = (base` K)
3		oposlem.l	. . . . 5 |- L = (le` K)
4		oposlem.o	. . . . 5 |- O = (oc` K)
5		oposlem.j	. . . . 5 |- J = (join` K)
6		oposlem.m	. . . . 5 |- M = (meet` K)
7		oposlem.f	. . . . 5 |- F = (0.` K)
8		oposlem.t	. . . . 5 |- T = (1.` K)
9	1, 2, 3, 4, 5, 6, 7, 8	isopos 16909	. . . 4 |- (K e. OP <-> (K e. Struct(3, p, T. ) /\ (K e. PosetNEW /\ F e. B /\ T e. B) /\ A.x e. B A.y e. B (((O` x) e. B /\ (O` (O` x)) = x /\ (xLy -> (O` y)L(O` x))) /\ (xJ(O` x)) = T /\ (xM(O` x)) = F)))
10	9	simp3bi 893	. . 3 |- (K e. OP -> A.x e. B A.y e. B (((O` x) e. B /\ (O` (O` x)) = x /\ (xLy -> (O` y)L(O` x))) /\ (xJ(O` x)) = T /\ (xM(O` x)) = F))
11		fveq2 4681	. . . . . . 7 |- (x = X -> (O` x) = (O` X))
12	11	eleq1d 1963	. . . . . 6 |- (x = X -> ((O` x) e. B <-> (O` X) e. B))
13	11	fveq2d 4685	. . . . . . 7 |- (x = X -> (O` (O` x)) = (O` (O` X)))
14		id 73	. . . . . . 7 |- (x = X -> x = X)
15	13, 14	eqeq12d 1899	. . . . . 6 |- (x = X -> ((O` (O` x)) = x <-> (O` (O` X)) = X))
16		breq1 3341	. . . . . . 7 |- (x = X -> (xLy <-> XLy))
17	11	breq2d 3350	. . . . . . 7 |- (x = X -> ((O` y)L(O` x) <-> (O` y)L(O` X)))
18	16, 17	imbi12d 688	. . . . . 6 |- (x = X -> ((xLy -> (O` y)L(O` x)) <-> (XLy -> (O` y)L(O` X))))
19	12, 15, 18	3anbi123d 1168	. . . . 5 |- (x = X -> (((O` x) e. B /\ (O` (O` x)) = x /\ (xLy -> (O` y)L(O` x))) <-> ((O` X) e. B /\ (O` (O` X)) = X /\ (XLy -> (O` y)L(O` X)))))
20	14, 11	opreq12d 4900	. . . . . 6 |- (x = X -> (xJ(O` x)) = (XJ(O` X)))
21	20	eqeq1d 1892	. . . . 5 |- (x = X -> ((xJ(O` x)) = T <-> (XJ(O` X)) = T))
22	14, 11	opreq12d 4900	. . . . . 6 |- (x = X -> (xM(O` x)) = (XM(O` X)))
23	22	eqeq1d 1892	. . . . 5 |- (x = X -> ((xM(O` x)) = F <-> (XM(O` X)) = F))
24	19, 21, 23	3anbi123d 1168	. . . 4 |- (x = X -> ((((O` x) e. B /\ (O` (O` x)) = x /\ (xLy -> (O` y)L(O` x))) /\ (xJ(O` x)) = T /\ (xM(O` x)) = F) <-> (((O` X) e. B /\ (O` (O` X)) = X /\ (XLy -> (O` y)L(O` X))) /\ (XJ(O` X)) = T /\ (XM(O` X)) = F)))
25		breq2 3342	. . . . . . 7 |- (y = Y -> (XLy <-> XLY))
26		fveq2 4681	. . . . . . . 8 |- (y = Y -> (O` y) = (O` Y))
27	26	breq1d 3348	. . . . . . 7 |- (y = Y -> ((O` y)L(O` X) <-> (O` Y)L(O` X)))
28	25, 27	imbi12d 688	. . . . . 6 |- (y = Y -> ((XLy -> (O` y)L(O` X)) <-> (XLY -> (O` Y)L(O` X))))
29	28	3anbi3d 1174	. . . . 5 |- (y = Y -> (((O` X) e. B /\ (O` (O` X)) = X /\ (XLy -> (O` y)L(O` X))) <-> ((O` X) e. B /\ (O` (O` X)) = X /\ (XLY -> (O` Y)L(O` X)))))
30	29	3anbi1d 1172	. . . 4 |- (y = Y -> ((((O` X) e. B /\ (O` (O` X)) = X /\ (XLy -> (O` y)L(O` X))) /\ (XJ(O` X)) = T /\ (XM(O` X)) = F) <-> (((O` X) e. B /\ (O` (O` X)) = X /\ (XLY -> (O` Y)L(O` X))) /\ (XJ(O` X)) = T /\ (XM(O` X)) = F)))
31	24, 30	rcla42v 2384	. . 3 |- ((X e. B /\ Y e. B) -> (A.x e. B A.y e. B (((O` x) e. B /\ (O` (O` x)) = x /\ (xLy -> (O` y)L(O` x))) /\ (xJ(O` x)) = T /\ (xM(O` x)) = F) -> (((O` X) e. B /\ (O` (O` X)) = X /\ (XLY -> (O` Y)L(O` X))) /\ (XJ(O` X)) = T /\ (XM(O` X)) = F)))
32	10, 31	mpan9 521	. 2 |- ((K e. OP /\ (X e. B /\ Y e. B)) -> (((O` X) e. B /\ (O` (O` X)) = X /\ (XLY -> (O` Y)L(O` X))) /\ (XJ(O` X)) = T /\ (XM(O` X)) = F))
33	32	3impb 1063	1 |- ((K e. OP /\ X e. B /\ Y e. B) -> (((O` X) e. B /\ (O` (O` X)) = X /\ (XLY -> (O` Y)L(O` X))) /\ (XJ(O` X)) = T /\ (XM(O` X)) = F))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   T. wtru 1260   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  3c3 7146  Structcstru 16707  basecbs 16758  lecple 16759  PosetNEWcpo 16760  joincjn 16766  meetcmee 16767  0.cp0 16832  1.cp1 16833  occoc 16836  OPcops 16837

This theorem is referenced by:  opoccl 16921  opococ 16922  oplecon3 16926  opexmid 16934  opnoncon 16935

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-v 2294  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-struct 16708  df-oposet 16905

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