Theorem cdlemkyu 36215

Description: Convert between function and explicit forms. C represents Z in cdlemkuu 36183. TODO: Clean all this up. (Contributed by NM, 21-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk5.b	|- B = ( Base ` K )
cdlemk5.l	|- .<_ = ( le ` K )
cdlemk5.j	|- .\/ = ( join ` K )
cdlemk5.m	|- ./\ = ( meet ` K )
cdlemk5.a	|- A = ( Atoms ` K )
cdlemk5.h	|- H = ( LHyp ` K )
cdlemk5.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk5.r	|- R = ( ( trL ` K ) ` W )
cdlemk5.z	|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk5.y	|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdlemk5b.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk5b.u1	|- V = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
cdlemk5.o2	|- Q = ( S ` b )
cdlemk5.u2	|- C = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) )

Assertion

Ref	Expression
cdlemkyu	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> [_ G / g ]_ Y = ( ( C ` G ) ` P ) )

Distinct variable groups:   ./\ , g   .\/ , g   B, g   P, g   R, g   T, g   g, Z   g, b   g, G, d, e   f, g, i, j   e, d, f, i, j, ./\   .<_ , i, j   .\/ , d, e, f, i, j   A, i, j   f, F, i, j   G, d, e, j   i, H, j   i, K, j   f, N, i, j   P, d, e, f, i, j   R, d, e, f, i, j   b, d, e, j, S   T, d, e, f, i, j   W, d, e, f, i, j   f, b, i   Q, d, e

Allowed substitution hints:   A( e, f, g, b, d)   B( e, f, i, j, b, d)   C( e, f, g, i, j, b, d)   P( b)   Q( f, g, i, j, b)   R( b)   S( f, g, i)   T( b)   F( e, g, b, d)   G( f, i, b)   H( e, f, g, b, d)   .\/ ( b)   K( e, f, g, b, d)   .<_ ( e, f, g, b, d)   ./\ ( b)   N( e, g, b, d)   V( e, f, g, i, j, b, d)   W( g, b)   Y( e, f, g, i, j, b, d)   Z( e, f, i, j, b, d)

Proof of Theorem cdlemkyu

Step	Hyp	Ref	Expression
1		cdlemk5.b	. . 3 |- B = ( Base ` K )
2		cdlemk5.l	. . 3 |- .<_ = ( le ` K )
3		cdlemk5.j	. . 3 |- .\/ = ( join ` K )
4		cdlemk5.m	. . 3 |- ./\ = ( meet ` K )
5		cdlemk5.a	. . 3 |- A = ( Atoms ` K )
6		cdlemk5.h	. . 3 |- H = ( LHyp ` K )
7		cdlemk5.t	. . 3 |- T = ( ( LTrn ` K ) ` W )
8		cdlemk5.r	. . 3 |- R = ( ( trL ` K ) ` W )
9		cdlemk5.z	. . 3 |- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
10		cdlemk5.y	. . 3 |- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
11		cdlemk5b.s	. . 3 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
12		cdlemk5b.u1	. . 3 |- V = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemky 36214	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> [_ G / g ]_ Y = ( ( b V G ) ` P ) )
14		simp3l 1089	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> b e. T )
15		simp13l 1176	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> G e. T )
16		cdlemk5.o2	. . . . 5 |- Q = ( S ` b )
17		cdlemk5.u2	. . . . 5 |- C = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) )
18	1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 16, 17	cdlemkuu 36183	. . . 4 |- ( ( b e. T /\ G e. T ) -> ( b V G ) = ( C ` G ) )
19	14, 15, 18	syl2anc 693	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( b V G ) = ( C ` G ) )
20	19	fveq1d 6193	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( b V G ) ` P ) = ( ( C ` G ) ` P ) )
21	13, 20	eqtrd 2656	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> [_ G / g ]_ Y = ( ( C ` G ) ` P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   [_csb 3533   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemkyuu  36216