Metamath Proof Explorer


Theorem cdlemkyu

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

Ref Expression
Hypotheses cdlemk5.b B = Base K
cdlemk5.l ˙ = 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 F -1
cdlemk5.y Y = P ˙ R g ˙ Z ˙ R g b -1
cdlemk5b.s S = f T ι i T | i P = P ˙ R f ˙ N P ˙ R f F -1
cdlemk5b.u1 V = d T , e T ι j T | j P = P ˙ R e ˙ S d P ˙ R e d -1
cdlemk5.o2 Q = S b
cdlemk5.u2 C = e T ι j T | j P = P ˙ R e ˙ Q P ˙ R e b -1
Assertion cdlemkyu K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G G / g Y = C G P

Proof

Step Hyp Ref Expression
1 cdlemk5.b B = Base K
2 cdlemk5.l ˙ = K
3 cdlemk5.j ˙ = join K
4 cdlemk5.m ˙ = meet K
5 cdlemk5.a A = Atoms K
6 cdlemk5.h H = LHyp K
7 cdlemk5.t T = LTrn K W
8 cdlemk5.r R = trL K W
9 cdlemk5.z Z = P ˙ R b ˙ N P ˙ R b F -1
10 cdlemk5.y Y = P ˙ R g ˙ Z ˙ R g b -1
11 cdlemk5b.s S = f T ι i T | i P = P ˙ R f ˙ N P ˙ R f F -1
12 cdlemk5b.u1 V = d T , e T ι j T | j P = P ˙ R e ˙ S d P ˙ R e d -1
13 cdlemk5.o2 Q = S b
14 cdlemk5.u2 C = e T ι j T | j P = P ˙ R e ˙ Q P ˙ R e b -1
15 1 2 3 4 5 6 7 8 9 10 11 12 cdlemky K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G G / g Y = b V G P
16 simp3l K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G b T
17 simp13l K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G G T
18 1 2 3 4 5 6 7 8 11 12 13 14 cdlemkuu b T G T b V G = C G
19 16 17 18 syl2anc K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G b V G = C G
20 19 fveq1d K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G b V G P = C G P
21 15 20 eqtrd K HL W H F T F I B G T G I B N T P A ¬ P ˙ W R F = R N b T b I B R b R F R b R G G / g Y = C G P