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Theorem dicfval 36464
Description: The partial isomorphism C for a lattice K. (Contributed by NM, 15-Dec-2013.)
Hypotheses
Ref Expression
dicval.l |- .<_ = ( le ` K )
dicval.a |- A = ( Atoms ` K )
dicval.h |- H = ( LHyp ` K )
dicval.p |- P = ( ( oc ` K ) ` W )
dicval.t |- T = ( ( LTrn ` K ) ` W )
dicval.e |- E = ( ( TEndo ` K ) ` W )
dicval.i |- I = ( ( DIsoC ` K ) ` W )
Assertion
Ref Expression
dicfval |- ( ( K e. V /\ W e. H ) -> I = ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) )
Distinct variable groups:   A, r   f, g, q, r, s, K   .<_ , q   A, q   T, g   f, W, g, q, r, s
Allowed substitution hints:   A( f, g, s)   P( f, g, s, r, q)   T( f, s, r, q)   E( f, g, s, r, q)   H( f, g, s, r, q)   I( f, g, s, r, q)   .<_ ( f, g, s, r)   V( f, g, s, r, q)
Proof of Theorem dicfval
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 dicval.i . . 3 |- I = ( ( DIsoC ` K ) ` W )
2 dicval.l . . . . 5 |- .<_ = ( le ` K )
3 dicval.a . . . . 5 |- A = ( Atoms ` K )
4 dicval.h . . . . 5 |- H = ( LHyp ` K )
52, 3, 4dicffval 36463 . . . 4 |- ( K e. V -> ( DIsoC ` K ) = ( w e. H |-> ( q e. { r e. A | -. r .<_ w } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` w ) ) } ) ) )
65fveq1d 6193 . . 3 |- ( K e. V -> ( ( DIsoC ` K ) ` W ) = ( ( w e. H |-> ( q e. { r e. A | -. r .<_ w } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` w ) ) } ) ) ` W ) )
71, 6syl5eq 2668 . 2 |- ( K e. V -> I = ( ( w e. H |-> ( q e. { r e. A | -. r .<_ w } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` w ) ) } ) ) ` W ) )
8 breq2 4657 . . . . . 6 |- ( w = W -> ( r .<_ w <-> r .<_ W ) )
98notbid 308 . . . . 5 |- ( w = W -> ( -. r .<_ w <-> -. r .<_ W ) )
109rabbidv 3189 . . . 4 |- ( w = W -> { r e. A | -. r .<_ w } = { r e. A | -. r .<_ W } )
11 fveq2 6191 . . . . . . . . . 10 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
12 dicval.t . . . . . . . . . 10 |- T = ( ( LTrn ` K ) ` W )
1311, 12syl6eqr 2674 . . . . . . . . 9 |- ( w = W -> ( ( LTrn ` K ) ` w ) = T )
14 fveq2 6191 . . . . . . . . . . . 12 |- ( w = W -> ( ( oc ` K ) ` w ) = ( ( oc ` K ) ` W ) )
15 dicval.p . . . . . . . . . . . 12 |- P = ( ( oc ` K ) ` W )
1614, 15syl6eqr 2674 . . . . . . . . . . 11 |- ( w = W -> ( ( oc ` K ) ` w ) = P )
1716fveq2d 6195 . . . . . . . . . 10 |- ( w = W -> ( g ` ( ( oc ` K ) ` w ) ) = ( g ` P ) )
1817eqeq1d 2624 . . . . . . . . 9 |- ( w = W -> ( ( g ` ( ( oc ` K ) ` w ) ) = q <-> ( g ` P ) = q ) )
1913, 18riotaeqbidv 6614 . . . . . . . 8 |- ( w = W -> ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) = ( iota_ g e. T ( g ` P ) = q ) )
2019fveq2d 6195 . . . . . . 7 |- ( w = W -> ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) )
2120eqeq2d 2632 . . . . . 6 |- ( w = W -> ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) <-> f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) ) )
22 fveq2 6191 . . . . . . . 8 |- ( w = W -> ( ( TEndo ` K ) ` w ) = ( ( TEndo ` K ) ` W ) )
23 dicval.e . . . . . . . 8 |- E = ( ( TEndo ` K ) ` W )
2422, 23syl6eqr 2674 . . . . . . 7 |- ( w = W -> ( ( TEndo ` K ) ` w ) = E )
2524eleq2d 2687 . . . . . 6 |- ( w = W -> ( s e. ( ( TEndo ` K ) ` w ) <-> s e. E ) )
2621, 25anbi12d 747 . . . . 5 |- ( w = W -> ( ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` w ) ) <-> ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) ) )
2726opabbidv 4716 . . . 4 |- ( w = W -> { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` w ) ) } = { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } )
2810, 27mpteq12dv 4733 . . 3 |- ( w = W -> ( q e. { r e. A | -. r .<_ w } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` w ) ) } ) = ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) )
29 eqid 2622 . . 3 |- ( w e. H |-> ( q e. { r e. A | -. r .<_ w } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` w ) ) } ) ) = ( w e. H |-> ( q e. { r e. A | -. r .<_ w } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` w ) ) } ) )
30 fvex 6201 . . . . 5 |- ( Atoms ` K ) e. _V
313, 30eqeltri 2697 . . . 4 |- A e. _V
3231mptrabex 6488 . . 3 |- ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) e. _V
3328, 29, 32fvmpt 6282 . 2 |- ( W e. H -> ( ( w e. H |-> ( q e. { r e. A | -. r .<_ w } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` w ) ( g ` ( ( oc ` K ) ` w ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` w ) ) } ) ) ` W ) = ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) )
347, 33sylan9eq 2676 1 |- ( ( K e. V /\ W e. H ) -> I = ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   {copab 4712   |-> cmpt 4729   ` cfv 5888   iota_crio 6610   lecple 15948   occoc 15949   Atomscatm 34550   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   DIsoCcdic 36461
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-riota 6611  df-dic 36462
This theorem is referenced by:  dicval  36465  dicfnN  36472
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