Theorem cdlemksv 34482

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 5879	. . . . . 6 |- ( f = G -> ( R ` f ) = ( R ` G ) )
2	1	oveq2d 6324	. . . . 5 |- ( f = G -> ( P .\/ ( R ` f ) ) = ( P .\/ ( R ` G ) ) )
3		coeq1 4997	. . . . . . 7 |- ( f = G -> ( f o. `' F ) = ( G o. `' F ) )
4	3	fveq2d 5883	. . . . . 6 |- ( f = G -> ( R ` ( f o. `' F ) ) = ( R ` ( G o. `' F ) ) )
5	4	oveq2d 6324	. . . . 5 |- ( f = G -> ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) = ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) )
6	2, 5	oveq12d 6326	. . . 4 |- ( f = G -> ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
7	6	eqeq2d 2481	. . 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 6272	. 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 6274	. 2 |- ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) e. _V
11	8, 9, 10	fvmpt 5963	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 1452   e. wcel 1904   |-> cmpt 4454   `'ccnv 4838   o. ccom 4843   ` cfv 5589   iota_crio 6269  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   Atomscatm 32900   LHypclh 33620   LTrncltrn 33737   trLctrl 33795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-riota 6270  df-ov 6311

This theorem is referenced by:  cdlemksel  34483  cdlemksv2  34485  cdlemkuvN  34502  cdlemkuel  34503  cdlemkuv2  34505  cdlemkuv-2N  34521  cdlemkuu  34533