Theorem cmtval 16938

Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (Th. cmbr 11160 analog.)

Hypotheses

Ref	Expression
cmtfval.b	|- B = (base` K)
cmtfval.j	|- J = (join` K)
cmtfval.m	|- M = (meet` K)
cmtfval.o	|- O = (oc` K)
cmtfval.c	|- C = (cm` K)

Assertion

Ref	Expression
cmtval	|- ((K e. A /\ X e. B /\ Y e. B) -> (XCY <-> X = ((XMY)J(XM(O` Y)))))

Proof of Theorem cmtval

Step	Hyp	Ref	Expression
1		cmtfval.b	. . . . . 6 |- B = (base` K)
2		cmtfval.j	. . . . . 6 |- J = (join` K)
3		cmtfval.m	. . . . . 6 |- M = (meet` K)
4		cmtfval.o	. . . . . 6 |- O = (oc` K)
5		cmtfval.c	. . . . . 6 |- C = (cm` K)
6	1, 2, 3, 4, 5	cmtfval 16937	. . . . 5 |- (K e. A -> C = {<.x, y>. | (x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y))))})
7		df-3an 860	. . . . . 6 |- ((x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y)))) <-> ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y)))))
8	7	opabbii 3402	. . . . 5 |- {<.x, y>. | (x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y))))} = {<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))}
9	6, 8	syl6eq 1944	. . . 4 |- (K e. A -> C = {<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))})
10	9	breqd 3349	. . 3 |- (K e. A -> (XCY <-> X{<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))}Y))
11	10	3ad2ant1 897	. 2 |- ((K e. A /\ X e. B /\ Y e. B) -> (XCY <-> X{<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))}Y))
12		id 73	. . . . . 6 |- (x = X -> x = X)
13		opreq1 4889	. . . . . . 7 |- (x = X -> (xMy) = (XMy))
14		opreq1 4889	. . . . . . 7 |- (x = X -> (xM(O` y)) = (XM(O` y)))
15	13, 14	opreq12d 4900	. . . . . 6 |- (x = X -> ((xMy)J(xM(O` y))) = ((XMy)J(XM(O` y))))
16	12, 15	eqeq12d 1899	. . . . 5 |- (x = X -> (x = ((xMy)J(xM(O` y))) <-> X = ((XMy)J(XM(O` y)))))
17		opreq2 4890	. . . . . . 7 |- (y = Y -> (XMy) = (XMY))
18		fveq2 4681	. . . . . . . 8 |- (y = Y -> (O` y) = (O` Y))
19	18	opreq2d 4898	. . . . . . 7 |- (y = Y -> (XM(O` y)) = (XM(O` Y)))
20	17, 19	opreq12d 4900	. . . . . 6 |- (y = Y -> ((XMy)J(XM(O` y))) = ((XMY)J(XM(O` Y))))
21	20	eqeq2d 1895	. . . . 5 |- (y = Y -> (X = ((XMy)J(XM(O` y))) <-> X = ((XMY)J(XM(O` Y)))))
22	16, 21	opelopab2 3569	. . . 4 |- ((X e. B /\ Y e. B) -> (<.X, Y>. e. {<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))} <-> X = ((XMY)J(XM(O` Y)))))
23		df-br 3339	. . . 4 |- (X{<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))}Y <-> <.X, Y>. e. {<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))})
24	22, 23	syl5bb 591	. . 3 |- ((X e. B /\ Y e. B) -> (X{<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))}Y <-> X = ((XMY)J(XM(O` Y)))))
25	24	3adant1 894	. 2 |- ((K e. A /\ X e. B /\ Y e. B) -> (X{<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))}Y <-> X = ((XMY)J(XM(O` Y)))))
26	11, 25	bitrd 587	1 |- ((K e. A /\ X e. B /\ Y e. B) -> (XCY <-> X = ((XMY)J(XM(O` Y)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  <.cop 3046   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  basecbs 16758  joincjn 16766  meetcmee 16767  occoc 16836  cmccmt 16838

This theorem is referenced by:  cmtcomlem 16969  cmt2 16971  cmtbr2 16974  cmtbr3 16975

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-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-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-fv 4014  df-opr 4886  df-mpt 5006  df-cmt 16906

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