Theorem labss2 14539

Description: Absorption law. (P \/ (P /\ Q)) = P.

Hypothesis

Ref	Expression
jop1	|- X = dom dom J

Assertion

Ref	Expression
labss2	|- ((J e. A /\ M e. B /\ <.J, M>. e. LatAlg) -> ((P e. X /\ Q e. X) -> (PJ(PMQ)) = P))

Proof of Theorem labss2

Step	Hyp	Ref	Expression
1		id 73	. . . . 5 |- (x = P -> x = P)
2		opreq1 4889	. . . . 5 |- (x = P -> (xMy) = (PMy))
3	1, 2	opreq12d 4900	. . . 4 |- (x = P -> (xJ(xMy)) = (PJ(PMy)))
4	3, 1	eqeq12d 1899	. . 3 |- (x = P -> ((xJ(xMy)) = x <-> (PJ(PMy)) = P))
5		opreq2 4890	. . . . 5 |- (y = Q -> (PMy) = (PMQ))
6	5	opreq2d 4898	. . . 4 |- (y = Q -> (PJ(PMy)) = (PJ(PMQ)))
7	6	eqeq1d 1892	. . 3 |- (y = Q -> ((PJ(PMy)) = P <-> (PJ(PMQ)) = P))
8	4, 7	rcla42v 2384	. 2 |- ((P e. X /\ Q e. X) -> (A.x e. X A.y e. X (xJ(xMy)) = x -> (PJ(PMQ)) = P))
9		jop1	. . 3 |- X = dom dom J
10	9	labs2 14538	. 2 |- ((J e. A /\ M e. B /\ <.J, M>. e. LatAlg) -> A.x e. X A.y e. X (xJ(xMy)) = x)
11	8, 10	syl5com 63	1 |- ((J e. A /\ M e. B /\ <.J, M>. e. LatAlg) -> ((P e. X /\ Q e. X) -> (PJ(PMQ)) = P))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  <.cop 3046  dom cdm 3986  (class class class)co 4884  LatAlgclatalg 14529

This theorem is referenced by:  jidd 14540  midd 14541

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-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-f 4010  df-fv 4014  df-opr 4886  df-latalg 14530

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