### Nuprl Lemma : mk_fabmon

`∀s:DSet. ∀g:AbMon. ∀i:|s| ⟶ |g|. ∀U:g':AbMon ⟶ (|s| ⟶ |g'|) ⟶ |g| ⟶ |g'|.`
`  ((∀g':AbMon. ∀f:|s| ⟶ |g'|.`
`      (IsMonHom{g,g'}(U g' f)`
`      ∧ (((U g' f) o i) = f ∈ (|s| ⟶ |g'|))`
`      ∧ (∀u:|g| ⟶ |g'|. (IsMonHom{g,g'}(u) `` ((u o i) = f ∈ (|s| ⟶ |g'|)) `` (u = (U g' f) ∈ (|g| ⟶ |g'|))))))`
`  `` (<g, i, U> ∈ FAbMon(s)))`

Proof

Definitions occuring in Statement :  free_abmonoid: `FAbMon(S)` compose: `f o g` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` member: `t ∈ T` apply: `f a` function: `x:A ⟶ B[x]` pair: `<a, b>` equal: `s = t ∈ T` monoid_hom_p: `IsMonHom{M1,M2}(f)` abmonoid: `AbMon` grp_car: `|g|` dset: `DSet` set_car: `|p|`
Definitions unfolded in proof :  free_abmonoid: `FAbMon(S)` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` dset: `DSet` subtype_rel: `A ⊆r B` abmonoid: `AbMon` mon: `Mon` so_lambda: `λ2x.t[x]` monoid_hom: `MonHom(M1,M2)` so_apply: `x[s]` prop: `ℙ` and: `P ∧ Q` unique_set: `{!x:T | P[x]}` cand: `A c∧ B` guard: `{T}`
Lemmas referenced :  set_car_wf abmonoid_wf grp_car_wf unique_set_wf monoid_hom_wf equal_wf compose_wf all_wf monoid_hom_p_wf dset_wf monoid_hom_properties
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation cut dependent_pairEquality hypothesisEquality functionExtensionality applyEquality introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename because_Cache hypothesis functionEquality lambdaEquality cumulativity universeEquality productEquality instantiate dependent_set_memberEquality independent_pairFormation dependent_functionElimination productElimination independent_functionElimination

Latex:
\mforall{}s:DSet.  \mforall{}g:AbMon.  \mforall{}i:|s|  {}\mrightarrow{}  |g|.  \mforall{}U:g':AbMon  {}\mrightarrow{}  (|s|  {}\mrightarrow{}  |g'|)  {}\mrightarrow{}  |g|  {}\mrightarrow{}  |g'|.
((\mforall{}g':AbMon.  \mforall{}f:|s|  {}\mrightarrow{}  |g'|.
(IsMonHom\{g,g'\}(U  g'  f)
\mwedge{}  (((U  g'  f)  o  i)  =  f)
\mwedge{}  (\mforall{}u:|g|  {}\mrightarrow{}  |g'|.  (IsMonHom\{g,g'\}(u)  {}\mRightarrow{}  ((u  o  i)  =  f)  {}\mRightarrow{}  (u  =  (U  g'  f))))))
{}\mRightarrow{}  (<g,  i,  U>  \mmember{}  FAbMon(s)))

Date html generated: 2017_10_01-AM-10_01_14
Last ObjectModification: 2017_03_03-PM-01_03_32

Theory : polynom_1

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