Theorem cmtfvalN 34497

Description: Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cmtfval.b	|- B = ( Base ` K )
cmtfval.j	|- .\/ = ( join ` K )
cmtfval.m	|- ./\ = ( meet ` K )
cmtfval.o	|- ._|_ = ( oc ` K )
cmtfval.c	|- C = ( cm ` K )

Assertion

Ref	Expression
cmtfvalN	|- ( K e. A -> C = { <. x , y >. | ( x e. B /\ y e. B /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) } )

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

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

Proof of Theorem cmtfvalN

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. A -> K e. _V )
2		cmtfval.c	. . 3 |- C = ( cm ` K )
3		fveq2 6191	. . . . . . . 8 |- ( p = K -> ( Base ` p ) = ( Base ` K ) )
4		cmtfval.b	. . . . . . . 8 |- B = ( Base ` K )
5	3, 4	syl6eqr 2674	. . . . . . 7 |- ( p = K -> ( Base ` p ) = B )
6	5	eleq2d 2687	. . . . . 6 |- ( p = K -> ( x e. ( Base ` p ) <-> x e. B ) )
7	5	eleq2d 2687	. . . . . 6 |- ( p = K -> ( y e. ( Base ` p ) <-> y e. B ) )
8		fveq2 6191	. . . . . . . . 9 |- ( p = K -> ( join ` p ) = ( join ` K ) )
9		cmtfval.j	. . . . . . . . 9 |- .\/ = ( join ` K )
10	8, 9	syl6eqr 2674	. . . . . . . 8 |- ( p = K -> ( join ` p ) = .\/ )
11		fveq2 6191	. . . . . . . . . 10 |- ( p = K -> ( meet ` p ) = ( meet ` K ) )
12		cmtfval.m	. . . . . . . . . 10 |- ./\ = ( meet ` K )
13	11, 12	syl6eqr 2674	. . . . . . . . 9 |- ( p = K -> ( meet ` p ) = ./\ )
14	13	oveqd 6667	. . . . . . . 8 |- ( p = K -> ( x ( meet ` p ) y ) = ( x ./\ y ) )
15		eqidd 2623	. . . . . . . . 9 |- ( p = K -> x = x )
16		fveq2 6191	. . . . . . . . . . 11 |- ( p = K -> ( oc ` p ) = ( oc ` K ) )
17		cmtfval.o	. . . . . . . . . . 11 |- ._|_ = ( oc ` K )
18	16, 17	syl6eqr 2674	. . . . . . . . . 10 |- ( p = K -> ( oc ` p ) = ._|_ )
19	18	fveq1d 6193	. . . . . . . . 9 |- ( p = K -> ( ( oc ` p ) ` y ) = ( ._|_ ` y ) )
20	13, 15, 19	oveq123d 6671	. . . . . . . 8 |- ( p = K -> ( x ( meet ` p ) ( ( oc ` p ) ` y ) ) = ( x ./\ ( ._|_ ` y ) ) )
21	10, 14, 20	oveq123d 6671	. . . . . . 7 |- ( p = K -> ( ( x ( meet ` p ) y ) ( join ` p ) ( x ( meet ` p ) ( ( oc ` p ) ` y ) ) ) = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) )
22	21	eqeq2d 2632	. . . . . 6 |- ( p = K -> ( x = ( ( x ( meet ` p ) y ) ( join ` p ) ( x ( meet ` p ) ( ( oc ` p ) ` y ) ) ) <-> x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) )
23	6, 7, 22	3anbi123d 1399	. . . . 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 = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) ) )
24	23	opabbidv 4716	. . . 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 = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) } )
25		df-cmtN 34464	. . . 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 ) ) ) ) } )
26		df-3an 1039	. . . . . 6 |- ( ( x e. B /\ y e. B /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) <-> ( ( x e. B /\ y e. B ) /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) )
27	26	opabbii 4717	. . . . 5 |- { <. x , y >. | ( x e. B /\ y e. B /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) } = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) }
28		fvex 6201	. . . . . . . 8 |- ( Base ` K ) e. _V
29	4, 28	eqeltri 2697	. . . . . . 7 |- B e. _V
30	29, 29	xpex 6962	. . . . . 6 |- ( B X. B ) e. _V
31		opabssxp 5193	. . . . . 6 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) } C_ ( B X. B )
32	30, 31	ssexi 4803	. . . . 5 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) } e. _V
33	27, 32	eqeltri 2697	. . . 4 |- { <. x , y >. | ( x e. B /\ y e. B /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) } e. _V
34	24, 25, 33	fvmpt 6282	. . 3 |- ( K e. _V -> ( cm ` K ) = { <. x , y >. | ( x e. B /\ y e. B /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) } )
35	2, 34	syl5eq 2668	. 2 |- ( K e. _V -> C = { <. x , y >. | ( x e. B /\ y e. B /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) } )
36	1, 35	syl 17	1 |- ( K e. A -> C = { <. x , y >. | ( x e. B /\ y e. B /\ x = ( ( x ./\ y ) .\/ ( x ./\ ( ._|_ ` y ) ) ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {copab 4712   X. cxp 5112   ` cfv 5888  (class class class)co 6650   Basecbs 15857   occoc 15949   joincjn 16944   meetcmee 16945   cmccmtN 34460

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  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-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-cmtN 34464

This theorem is referenced by:  cmtvalN  34498