Theorem cmtfval 16937

Description: Value of commutes relation.

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
cmtfval	|- (K e. A -> C = {<.x, y>. | (x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y))))})

Distinct variable groups:   x,y,B   x,K,y

Proof of Theorem cmtfval

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (K e. A -> K e. _V)
2		fveq2 4681	. . . . . . . 8 |- (p = K -> (base` p) = (base` K))
3		cmtfval.b	. . . . . . . 8 |- B = (base` K)
4	2, 3	syl6eqr 1946	. . . . . . 7 |- (p = K -> (base` p) = B)
5	4	eleq2d 1964	. . . . . 6 |- (p = K -> (x e. (base` p) <-> x e. B))
6	4	eleq2d 1964	. . . . . 6 |- (p = K -> (y e. (base` p) <-> y e. B))
7		fveq2 4681	. . . . . . . . 9 |- (p = K -> (join` p) = (join` K))
8		cmtfval.j	. . . . . . . . 9 |- J = (join` K)
9	7, 8	syl6eqr 1946	. . . . . . . 8 |- (p = K -> (join` p) = J)
10		fveq2 4681	. . . . . . . . . 10 |- (p = K -> (meet` p) = (meet` K))
11		cmtfval.m	. . . . . . . . . 10 |- M = (meet` K)
12	10, 11	syl6eqr 1946	. . . . . . . . 9 |- (p = K -> (meet` p) = M)
13	12	opreqd 4899	. . . . . . . 8 |- (p = K -> (x(meet` p)y) = (xMy))
14		eqidd 1885	. . . . . . . . 9 |- (p = K -> x = x)
15		fveq2 4681	. . . . . . . . . . 11 |- (p = K -> (oc` p) = (oc` K))
16		cmtfval.o	. . . . . . . . . . 11 |- O = (oc` K)
17	15, 16	syl6eqr 1946	. . . . . . . . . 10 |- (p = K -> (oc` p) = O)
18	17	fveq1d 4683	. . . . . . . . 9 |- (p = K -> ((oc` p)` y) = (O` y))
19	12, 14, 18	opreq123d 10153	. . . . . . . 8 |- (p = K -> (x(meet` p)((oc` p)` y)) = (xM(O` y)))
20	9, 13, 19	opreq123d 10153	. . . . . . 7 |- (p = K -> ((x(meet` p)y)(join` p)(x(meet` p)((oc` p)` y))) = ((xMy)J(xM(O` y))))
21	20	eqeq2d 1895	. . . . . 6 |- (p = K -> (x = ((x(meet` p)y)(join` p)(x(meet` p)((oc` p)` y))) <-> x = ((xMy)J(xM(O` y)))))
22	5, 6, 21	3anbi123d 1168	. . . . 5 |- (p = K -> ((x e. (base` p) /\ y e. (base` p) /\ x = ((x(meet` p)y)(join` p)(x(meet` p)((oc` p)` y)))) <-> (x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y))))))
23	22	opabbidv 3401	. . . 4 |- (p = K -> {<.x, y>. | (x e. (base` p) /\ y e. (base` p) /\ x = ((x(meet` p)y)(join` p)(x(meet` p)((oc` p)` y))))} = {<.x, y>. | (x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y))))})
24		df-cmt 16906	. . . 4 |- cm = (p e. _V |-> {<.x, y>. | (x e. (base` p) /\ y e. (base` p) /\ x = ((x(meet` p)y)(join` p)(x(meet` p)((oc` p)` y))))})
25		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)))))
26	25	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))))}
27		fvex 4689	. . . . . . . 8 |- (base` K) e. _V
28	3, 27	eqeltri 1967	. . . . . . 7 |- B e. _V
29	28, 28	xpex 4096	. . . . . 6 |- (B X. B) e. _V
30		opabssxp 4060	. . . . . 6 |- {<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))} C_ (B X. B)
31	29, 30	ssexi 3456	. . . . 5 |- {<.x, y>. | ((x e. B /\ y e. B) /\ x = ((xMy)J(xM(O` y))))} e. _V
32	26, 31	eqeltri 1967	. . . 4 |- {<.x, y>. | (x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y))))} e. _V
33	23, 24, 32	fvmpt 5015	. . 3 |- (K e. _V -> (cm` K) = {<.x, y>. | (x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y))))})
34		cmtfval.c	. . 3 |- C = (cm` K)
35	33, 34	syl5eq 1940	. 2 |- (K e. _V -> C = {<.x, y>. | (x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y))))})
36	1, 35	syl 12	1 |- (K e. A -> C = {<.x, y>. | (x e. B /\ y e. B /\ x = ((xMy)J(xM(O` y))))})

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292  {copab 3395   X. cxp 3984  ` cfv 3998  (class class class)co 4884  basecbs 16758  joincjn 16766  meetcmee 16767  occoc 16836  cmccmt 16838

This theorem is referenced by:  cmtval 16938

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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