Theorem diaffval 36319

Description: The partial isomorphism A for a lattice K. (Contributed by NM, 15-Oct-2013.)

Hypotheses

Ref	Expression
diaval.b	|- B = ( Base ` K )
diaval.l	|- .<_ = ( le ` K )
diaval.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
diaffval	|- ( K e. V -> ( DIsoA ` K ) = ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) )

Distinct variable groups:   x, w, y, .<_   w, B, x, y   w, H   w, f, x, y, K

Allowed substitution hints:   B( f)   H( x, y, f)   .<_ ( f)   V( x, y, w, f)

Proof of Theorem diaffval

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. V -> K e. _V )
2		fveq2 6191	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		diaval.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2674	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 6191	. . . . . . 7 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
6		diaval.b	. . . . . . 7 |- B = ( Base ` K )
7	5, 6	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( Base ` k ) = B )
8		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( le ` k ) = ( le ` K ) )
9		diaval.l	. . . . . . . 8 |- .<_ = ( le ` K )
10	8, 9	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( le ` k ) = .<_ )
11	10	breqd 4664	. . . . . 6 |- ( k = K -> ( y ( le ` k ) w <-> y .<_ w ) )
12	7, 11	rabeqbidv 3195	. . . . 5 |- ( k = K -> { y e. ( Base ` k ) | y ( le ` k ) w } = { y e. B | y .<_ w } )
13		fveq2 6191	. . . . . . 7 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
14	13	fveq1d 6193	. . . . . 6 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
15		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( trL ` k ) = ( trL ` K ) )
16	15	fveq1d 6193	. . . . . . . 8 |- ( k = K -> ( ( trL ` k ) ` w ) = ( ( trL ` K ) ` w ) )
17	16	fveq1d 6193	. . . . . . 7 |- ( k = K -> ( ( ( trL ` k ) ` w ) ` f ) = ( ( ( trL ` K ) ` w ) ` f ) )
18		eqidd 2623	. . . . . . 7 |- ( k = K -> x = x )
19	17, 10, 18	breq123d 4667	. . . . . 6 |- ( k = K -> ( ( ( ( trL ` k ) ` w ) ` f ) ( le ` k ) x <-> ( ( ( trL ` K ) ` w ) ` f ) .<_ x ) )
20	14, 19	rabeqbidv 3195	. . . . 5 |- ( k = K -> { f e. ( ( LTrn ` k ) ` w ) | ( ( ( trL ` k ) ` w ) ` f ) ( le ` k ) x } = { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } )
21	12, 20	mpteq12dv 4733	. . . 4 |- ( k = K -> ( x e. { y e. ( Base ` k ) | y ( le ` k ) w } |-> { f e. ( ( LTrn ` k ) ` w ) | ( ( ( trL ` k ) ` w ) ` f ) ( le ` k ) x } ) = ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) )
22	4, 21	mpteq12dv 4733	. . 3 |- ( k = K -> ( w e. ( LHyp ` k ) |-> ( x e. { y e. ( Base ` k ) | y ( le ` k ) w } |-> { f e. ( ( LTrn ` k ) ` w ) | ( ( ( trL ` k ) ` w ) ` f ) ( le ` k ) x } ) ) = ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) )
23		df-disoa 36318	. . 3 |- DIsoA = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. { y e. ( Base ` k ) | y ( le ` k ) w } |-> { f e. ( ( LTrn ` k ) ` w ) | ( ( ( trL ` k ) ` w ) ` f ) ( le ` k ) x } ) ) )
24		fvex 6201	. . . . 5 |- ( LHyp ` K ) e. _V
25	3, 24	eqeltri 2697	. . . 4 |- H e. _V
26	25	mptex 6486	. . 3 |- ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) e. _V
27	22, 23, 26	fvmpt 6282	. 2 |- ( K e. _V -> ( DIsoA ` K ) = ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) )
28	1, 27	syl 17	1 |- ( K e. V -> ( DIsoA ` K ) = ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   Basecbs 15857   lecple 15948   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   DIsoAcdia 36317

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  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-disoa 36318

This theorem is referenced by:  diafval  36320