Theorem cdlemksv 34423

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 5870	. . . . . 6 |- ( f = G -> ( R ` f ) = ( R ` G ) )
2	1	oveq2d 6311	. . . . 5 |- ( f = G -> ( P .\/ ( R ` f ) ) = ( P .\/ ( R ` G ) ) )
3		coeq1 4995	. . . . . . 7 |- ( f = G -> ( f o. `' F ) = ( G o. `' F ) )
4	3	fveq2d 5874	. . . . . 6 |- ( f = G -> ( R ` ( f o. `' F ) ) = ( R ` ( G o. `' F ) ) )
5	4	oveq2d 6311	. . . . 5 |- ( f = G -> ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) = ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) )
6	2, 5	oveq12d 6313	. . . 4 |- ( f = G -> ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
7	6	eqeq2d 2463	. . 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 6259	. 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 6261	. 2 |- ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) e. _V
11	8, 9, 10	fvmpt 5953	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 1446   e. wcel 1889   |-> cmpt 4464   `'ccnv 4836   o. ccom 4841   ` cfv 5585   iota_crio 6256  (class class class)co 6295   Basecbs 15133   lecple 15209   joincjn 16201   meetcmee 16202   Atomscatm 32841   LHypclh 33561   LTrncltrn 33678   trLctrl 33736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5549  df-fun 5587  df-fv 5593  df-riota 6257  df-ov 6298

This theorem is referenced by:  cdlemksel  34424  cdlemksv2  34426  cdlemkuvN  34443  cdlemkuel  34444  cdlemkuv2  34446  cdlemkuv-2N  34462  cdlemkuu  34474