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Step * 1 1 2 1 1 of Lemma dma-lift-compose-assoc


1. I : Type
2. J : Type
3. K : Type
4. H : Type
5. eqi : EqDecider(I)
6. eqj : EqDecider(J)
7. eqk : EqDecider(K)
8. f : J ⟶ Point(free-DeMorgan-algebra(I;eqi))
9. g : K ⟶ Point(free-DeMorgan-algebra(J;eqj))
10. h : H ⟶ Point(free-DeMorgan-algebra(K;eqk))
11. x : H
12. ∀X:Type. ∀eq:EqDecider(X).  (free-dml-deq(X;eq) ∈ EqDecider(Point(free-DeMorgan-algebra(X;eq))))
13. ∀i:J. (<i> ∈ Point(free-DeMorgan-algebra(J;eqj)))
14. ∀[f:K ⟶ Point(free-DeMorgan-algebra(I;eqi))].
    ∀[g:dma-hom(free-DeMorgan-algebra(K;eqk);free-DeMorgan-algebra(I;eqi))].
      ∀x:Point(free-DeMorgan-algebra(K;eqk))
        ((free-dma-lift(K;eqk;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);f) x)
        = (g x)
        ∈ Point(free-DeMorgan-algebra(I;eqi))) 
      supposing ∀i:K. ((g <i>) = (f i) ∈ Point(free-DeMorgan-algebra(I;eqi)))
15. ∀k:K. (<k> ∈ Point(free-DeMorgan-algebra(K;eqk)))
16. v : {g:dma-hom(free-DeMorgan-algebra(J;eqj);free-DeMorgan-algebra(I;eqi))| 
         ∀i:J. ((g <i>) = (f i) ∈ Point(free-DeMorgan-algebra(I;eqi)))} 
17. free-dma-lift(J;eqj;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);f)
= v
∈ {g:dma-hom(free-DeMorgan-algebra(J;eqj);free-DeMorgan-algebra(I;eqi))| 
   ∀i:J. ((g <i>) = (f i) ∈ Point(free-DeMorgan-algebra(I;eqi)))} 
18. v1 : {g1:dma-hom(free-DeMorgan-algebra(K;eqk);free-DeMorgan-algebra(J;eqj))| 
          ∀i:K. ((g1 <i>) = (g i) ∈ Point(free-DeMorgan-algebra(J;eqj)))} 
19. free-dma-lift(K;eqk;free-DeMorgan-algebra(J;eqj);free-dml-deq(J;eqj);g)
= v1
∈ {g1:dma-hom(free-DeMorgan-algebra(K;eqk);free-DeMorgan-algebra(J;eqj))| 
   ∀i:K. ((g1 <i>) = (g i) ∈ Point(free-DeMorgan-algebra(J;eqj)))} 
⊢ v o v1 ∈ dma-hom(free-DeMorgan-algebra(K;eqk);free-DeMorgan-algebra(I;eqi))
BY
{ (BLemma `compose-dma-hom` THEN Auto) }

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Latex:

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Latex:

1.  I  :  Type
2.  J  :  Type
3.  K  :  Type
4.  H  :  Type
5.  eqi  :  EqDecider(I)
6.  eqj  :  EqDecider(J)
7.  eqk  :  EqDecider(K)
8.  f  :  J  {}\mrightarrow{}  Point(free-DeMorgan-algebra(I;eqi))
9.  g  :  K  {}\mrightarrow{}  Point(free-DeMorgan-algebra(J;eqj))
10.  h  :  H  {}\mrightarrow{}  Point(free-DeMorgan-algebra(K;eqk))
11.  x  :  H
12.  \mforall{}X:Type.  \mforall{}eq:EqDecider(X).    (free-dml-deq(X;eq)  \mmember{}  EqDecider(Point(free-DeMorgan-algebra(X;eq))))
13.  \mforall{}i:J.  (<i>  \mmember{}  Point(free-DeMorgan-algebra(J;eqj)))
14.  \mforall{}[f:K  {}\mrightarrow{}  Point(free-DeMorgan-algebra(I;eqi))].
        \mforall{}[g:dma-hom(free-DeMorgan-algebra(K;eqk);free-DeMorgan-algebra(I;eqi))].
            \mforall{}x:Point(free-DeMorgan-algebra(K;eqk))
                ((free-dma-lift(K;eqk;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);f)  x)  =  (g  x)) 
            supposing  \mforall{}i:K.  ((g  <i>)  =  (f  i))
15.  \mforall{}k:K.  (<k>  \mmember{}  Point(free-DeMorgan-algebra(K;eqk)))
16.  v  :  \{g:dma-hom(free-DeMorgan-algebra(J;eqj);free-DeMorgan-algebra(I;eqi))| 
                  \mforall{}i:J.  ((g  <i>)  =  (f  i))\} 
17.  free-dma-lift(J;eqj;free-DeMorgan-algebra(I;eqi);free-dml-deq(I;eqi);f)  =  v
18.  v1  :  \{g1:dma-hom(free-DeMorgan-algebra(K;eqk);free-DeMorgan-algebra(J;eqj))| 
                    \mforall{}i:K.  ((g1  <i>)  =  (g  i))\} 
19.  free-dma-lift(K;eqk;free-DeMorgan-algebra(J;eqj);free-dml-deq(J;eqj);g)  =  v1
\mvdash{}  v  o  v1  \mmember{}  dma-hom(free-DeMorgan-algebra(K;eqk);free-DeMorgan-algebra(I;eqi))


By

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Latex:
(BLemma  `compose-dma-hom`  THEN  Auto)

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