Theorem isoml 34525

Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
isoml.b	|- B = ( Base ` K )
isoml.l	|- .<_ = ( le ` K )
isoml.j	|- .\/ = ( join ` K )
isoml.m	|- ./\ = ( meet ` K )
isoml.o	|- ._|_ = ( oc ` K )

Assertion

Ref	Expression
isoml	|- ( K e. OML <-> ( K e. OL /\ A. x e. B A. y e. B ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) )

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

Allowed substitution hints:   .\/ ( x, y)   .<_ ( x, y)   ./\ ( x, y)   ._|_ ( x, y)

Proof of Theorem isoml

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
2		isoml.b	. . . 4 |- B = ( Base ` K )
3	1, 2	syl6eqr 2674	. . 3 |- ( k = K -> ( Base ` k ) = B )
4		fveq2 6191	. . . . . . 7 |- ( k = K -> ( le ` k ) = ( le ` K ) )
5		isoml.l	. . . . . . 7 |- .<_ = ( le ` K )
6	4, 5	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( le ` k ) = .<_ )
7	6	breqd 4664	. . . . 5 |- ( k = K -> ( x ( le ` k ) y <-> x .<_ y ) )
8		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( join ` k ) = ( join ` K ) )
9		isoml.j	. . . . . . . 8 |- .\/ = ( join ` K )
10	8, 9	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( join ` k ) = .\/ )
11		eqidd 2623	. . . . . . 7 |- ( k = K -> x = x )
12		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( meet ` k ) = ( meet ` K ) )
13		isoml.m	. . . . . . . . 9 |- ./\ = ( meet ` K )
14	12, 13	syl6eqr 2674	. . . . . . . 8 |- ( k = K -> ( meet ` k ) = ./\ )
15		eqidd 2623	. . . . . . . 8 |- ( k = K -> y = y )
16		fveq2 6191	. . . . . . . . . 10 |- ( k = K -> ( oc ` k ) = ( oc ` K ) )
17		isoml.o	. . . . . . . . . 10 |- ._|_ = ( oc ` K )
18	16, 17	syl6eqr 2674	. . . . . . . . 9 |- ( k = K -> ( oc ` k ) = ._|_ )
19	18	fveq1d 6193	. . . . . . . 8 |- ( k = K -> ( ( oc ` k ) ` x ) = ( ._|_ ` x ) )
20	14, 15, 19	oveq123d 6671	. . . . . . 7 |- ( k = K -> ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) = ( y ./\ ( ._|_ ` x ) ) )
21	10, 11, 20	oveq123d 6671	. . . . . 6 |- ( k = K -> ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) )
22	21	eqeq2d 2632	. . . . 5 |- ( k = K -> ( y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) <-> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) )
23	7, 22	imbi12d 334	. . . 4 |- ( k = K -> ( ( x ( le ` k ) y -> y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) ) <-> ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) )
24	3, 23	raleqbidv 3152	. . 3 |- ( k = K -> ( A. y e. ( Base ` k ) ( x ( le ` k ) y -> y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) ) <-> A. y e. B ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) )
25	3, 24	raleqbidv 3152	. 2 |- ( k = K -> ( A. x e. ( Base ` k ) A. y e. ( Base ` k ) ( x ( le ` k ) y -> y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) ) <-> A. x e. B A. y e. B ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) )
26		df-oml 34466	. 2 |- OML = { k e. OL | A. x e. ( Base ` k ) A. y e. ( Base ` k ) ( x ( le ` k ) y -> y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) ) }
27	25, 26	elrab2 3366	1 |- ( K e. OML <-> ( K e. OL /\ A. x e. B A. y e. B ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   occoc 15949   joincjn 16944   meetcmee 16945   OLcol 34461   OMLcoml 34462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-oml 34466

This theorem is referenced by:  isomliN  34526  omlol  34527  omllaw  34530