Theorem diafval 36320

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 )
diaval.t	|- T = ( ( LTrn ` K ) ` W )
diaval.r	|- R = ( ( trL ` K ) ` W )
diaval.i	|- I = ( ( DIsoA ` K ) ` W )

Assertion

Ref	Expression
diafval	|- ( ( K e. V /\ W e. H ) -> I = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )

Distinct variable groups:   x, y, .<_   x, B, y   x, f, y, K   x, R   T, f, x   f, W, x, y

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

Proof of Theorem diafval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		diaval.i	. . 3 |- I = ( ( DIsoA ` K ) ` W )
2		diaval.b	. . . . 5 |- B = ( Base ` K )
3		diaval.l	. . . . 5 |- .<_ = ( le ` K )
4		diaval.h	. . . . 5 |- H = ( LHyp ` K )
5	2, 3, 4	diaffval 36319	. . . 4 |- ( 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 } ) ) )
6	5	fveq1d 6193	. . 3 |- ( K e. V -> ( ( DIsoA ` K ) ` W ) = ( ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) ` W ) )
7	1, 6	syl5eq 2668	. 2 |- ( K e. V -> I = ( ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) ` W ) )
8		breq2 4657	. . . . 5 |- ( w = W -> ( y .<_ w <-> y .<_ W ) )
9	8	rabbidv 3189	. . . 4 |- ( w = W -> { y e. B | y .<_ w } = { y e. B | y .<_ W } )
10		fveq2 6191	. . . . . 6 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
11		diaval.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
12	10, 11	syl6eqr 2674	. . . . 5 |- ( w = W -> ( ( LTrn ` K ) ` w ) = T )
13		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( ( trL ` K ) ` w ) = ( ( trL ` K ) ` W ) )
14		diaval.r	. . . . . . . 8 |- R = ( ( trL ` K ) ` W )
15	13, 14	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( ( trL ` K ) ` w ) = R )
16	15	fveq1d 6193	. . . . . 6 |- ( w = W -> ( ( ( trL ` K ) ` w ) ` f ) = ( R ` f ) )
17	16	breq1d 4663	. . . . 5 |- ( w = W -> ( ( ( ( trL ` K ) ` w ) ` f ) .<_ x <-> ( R ` f ) .<_ x ) )
18	12, 17	rabeqbidv 3195	. . . 4 |- ( w = W -> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } = { f e. T | ( R ` f ) .<_ x } )
19	9, 18	mpteq12dv 4733	. . 3 |- ( w = W -> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )
20		eqid 2622	. . 3 |- ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) = ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) )
21		fvex 6201	. . . . 5 |- ( Base ` K ) e. _V
22	2, 21	eqeltri 2697	. . . 4 |- B e. _V
23	22	mptrabex 6488	. . 3 |- ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) e. _V
24	19, 20, 23	fvmpt 6282	. 2 |- ( W e. H -> ( ( w e. H |-> ( x e. { y e. B | y .<_ w } |-> { f e. ( ( LTrn ` K ) ` w ) | ( ( ( trL ` K ) ` w ) ` f ) .<_ x } ) ) ` W ) = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )
25	7, 24	sylan9eq 2676	1 |- ( ( K e. V /\ W e. H ) -> I = ( x e. { y e. B | y .<_ W } |-> { f e. T | ( R ` f ) .<_ x } ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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:  diaval  36321  diafn  36323