Theorem omllaw 16964

Description: The orthomodular law.

Hypotheses

Ref	Expression
omllaw.b	|- B = (base` K)
omllaw.l	|- L = (le` K)
omllaw.j	|- J = (join` K)
omllaw.m	|- M = (meet` K)
omllaw.o	|- O = (oc` K)

Assertion

Ref	Expression
omllaw	|- ((K e. OML /\ X e. B /\ Y e. B) -> (XLY -> Y = (XJ(YM(O` X)))))

Proof of Theorem omllaw

Step	Hyp	Ref	Expression
1		breq1 3341	. . . . 5 |- (x = X -> (xLy <-> XLy))
2		id 73	. . . . . . 7 |- (x = X -> x = X)
3		fveq2 4681	. . . . . . . 8 |- (x = X -> (O` x) = (O` X))
4	3	opreq2d 4898	. . . . . . 7 |- (x = X -> (yM(O` x)) = (yM(O` X)))
5	2, 4	opreq12d 4900	. . . . . 6 |- (x = X -> (xJ(yM(O` x))) = (XJ(yM(O` X))))
6	5	eqeq2d 1895	. . . . 5 |- (x = X -> (y = (xJ(yM(O` x))) <-> y = (XJ(yM(O` X)))))
7	1, 6	imbi12d 688	. . . 4 |- (x = X -> ((xLy -> y = (xJ(yM(O` x)))) <-> (XLy -> y = (XJ(yM(O` X))))))
8		breq2 3342	. . . . 5 |- (y = Y -> (XLy <-> XLY))
9		id 73	. . . . . 6 |- (y = Y -> y = Y)
10		opreq1 4889	. . . . . . 7 |- (y = Y -> (yM(O` X)) = (YM(O` X)))
11	10	opreq2d 4898	. . . . . 6 |- (y = Y -> (XJ(yM(O` X))) = (XJ(YM(O` X))))
12	9, 11	eqeq12d 1899	. . . . 5 |- (y = Y -> (y = (XJ(yM(O` X))) <-> Y = (XJ(YM(O` X)))))
13	8, 12	imbi12d 688	. . . 4 |- (y = Y -> ((XLy -> y = (XJ(yM(O` X)))) <-> (XLY -> Y = (XJ(YM(O` X))))))
14	7, 13	rcla42v 2384	. . 3 |- ((X e. B /\ Y e. B) -> (A.x e. B A.y e. B (xLy -> y = (xJ(yM(O` x)))) -> (XLY -> Y = (XJ(YM(O` X))))))
15		omllaw.b	. . . . 5 |- B = (base` K)
16		omllaw.l	. . . . 5 |- L = (le` K)
17		omllaw.j	. . . . 5 |- J = (join` K)
18		omllaw.m	. . . . 5 |- M = (meet` K)
19		omllaw.o	. . . . 5 |- O = (oc` K)
20	15, 16, 17, 18, 19	isoml 16959	. . . 4 |- (K e. OML <-> (K e. OL /\ A.x e. B A.y e. B (xLy -> y = (xJ(yM(O` x))))))
21	20	simprbi 353	. . 3 |- (K e. OML -> A.x e. B A.y e. B (xLy -> y = (xJ(yM(O` x)))))
22	14, 21	syl5com 63	. 2 |- (K e. OML -> ((X e. B /\ Y e. B) -> (XLY -> Y = (XJ(YM(O` X))))))
23	22	3impib 1065	1 |- ((K e. OML /\ X e. B /\ Y e. B) -> (XLY -> Y = (XJ(YM(O` X)))))

Syntax hints:   -> wi 3   /\ 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  joincjn 16766  meetcmee 16767  occoc 16836  OLcol 16839  OMLcoml 16840

This theorem is referenced by:  omllaw2 16965  omllaw3 16966  omllaw4 16967

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

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-ral 2109  df-rex 2110  df-rab 2112  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-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-oml 16908

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