Theorem cdlemksv 34488

Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.)

Hypotheses

Ref	Expression
cdlemk.b	|- B = ( Base ` K )
cdlemk.l	|- .<_ = ( le ` K )
cdlemk.j	|- .\/ = ( join ` K )
cdlemk.a	|- A = ( Atoms ` K )
cdlemk.h	|- H = ( LHyp ` K )
cdlemk.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk.r	|- R = ( ( trL ` K ) ` W )
cdlemk.m	|- ./\ = ( meet ` K )
cdlemk.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )

Assertion

Ref	Expression
cdlemksv	|- ( G e. T -> ( S ` G ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )

Distinct variable groups:   ./\ , f   .\/ , f   f, F   f, i, G   f, N   P, f   R, f   T, f   f, W

Allowed substitution hints:   A( f, i)   B( f, i)   P( i)   R( i)   S( f, i)   T( i)   F( i)   H( f, i)   .\/ ( i)   K( f, i)   .<_ ( f, i)   ./\ ( i)   N( i)   W( i)

Proof of Theorem cdlemksv

Step	Hyp	Ref	Expression
1		fveq2 5691	. . . . . 6 |- ( f = G -> ( R ` f ) = ( R ` G ) )
2	1	oveq2d 6107	. . . . 5 |- ( f = G -> ( P .\/ ( R ` f ) ) = ( P .\/ ( R ` G ) ) )
3		coeq1 4997	. . . . . . 7 |- ( f = G -> ( f o. `' F ) = ( G o. `' F ) )
4	3	fveq2d 5695	. . . . . 6 |- ( f = G -> ( R ` ( f o. `' F ) ) = ( R ` ( G o. `' F ) ) )
5	4	oveq2d 6107	. . . . 5 |- ( f = G -> ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) = ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) )
6	2, 5	oveq12d 6109	. . . 4 |- ( f = G -> ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
7	6	eqeq2d 2454	. . 3 |- ( f = G -> ( ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) <-> ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )
8	7	riotabidv 6054	. 2 |- ( f = G -> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )
9		cdlemk.s	. 2 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10		riotaex 6056	. 2 |- ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) e. _V
11	8, 9, 10	fvmpt 5774	1 |- ( G e. T -> ( S ` G ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1369   e. wcel 1756   e. cmpt 4350   `'ccnv 4839   o. ccom 4844   ` cfv 5418   iota_crio 6051  (class class class)co 6091   Basecbs 14174   lecple 14245   joincjn 15114   meetcmee 15115   Atomscatm 32908   LHypclh 33628   LTrncltrn 33745   trLctrl 33802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-riota 6052  df-ov 6094

This theorem is referenced by:  cdlemksel  34489  cdlemksv2  34491  cdlemkuvN  34508  cdlemkuel  34509  cdlemkuv2  34511  cdlemkuv-2N  34527  cdlemkuu  34539