Theorem cdlemkuvN 36152

Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma₁ (p) function U. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemk1.b	|- B = ( Base ` K )
cdlemk1.l	|- .<_ = ( le ` K )
cdlemk1.j	|- .\/ = ( join ` K )
cdlemk1.m	|- ./\ = ( meet ` K )
cdlemk1.a	|- A = ( Atoms ` K )
cdlemk1.h	|- H = ( LHyp ` K )
cdlemk1.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk1.r	|- R = ( ( trL ` K ) ` W )
cdlemk1.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk1.o	|- O = ( S ` D )
cdlemk1.u	|- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )

Assertion

Ref	Expression
cdlemkuvN	|- ( G e. T -> ( U ` G ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) )

Distinct variable groups:   f, i, ./\   .<_ , i   .\/ , f, i   A, i   D, f, i   f, F, i   i, H   i, K   f, N, i   P, f, i   R, f, i   T, f, i   f, W, i   ./\ , e   .\/ , e   D, e   e, j, G   e, O   P, e   R, e   T, e   e, W

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

Proof of Theorem cdlemkuvN

Step	Hyp	Ref	Expression
1		cdlemk1.b	. 2 |- B = ( Base ` K )
2		cdlemk1.l	. 2 |- .<_ = ( le ` K )
3		cdlemk1.j	. 2 |- .\/ = ( join ` K )
4		cdlemk1.a	. 2 |- A = ( Atoms ` K )
5		cdlemk1.h	. 2 |- H = ( LHyp ` K )
6		cdlemk1.t	. 2 |- T = ( ( LTrn ` K ) ` W )
7		cdlemk1.r	. 2 |- R = ( ( trL ` K ) ` W )
8		cdlemk1.m	. 2 |- ./\ = ( meet ` K )
9		cdlemk1.u	. 2 |- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	cdlemksv 36132	1 |- ( G e. T -> ( U ` G ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |-> cmpt 4729   `'ccnv 5113   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653

This theorem is referenced by: (None)