### Nuprl Lemma : anti_sym_shift

`∀[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀[S:B ⟶ B ⟶ ℙ]. ∀[f:A ⟶ B].`
`  (AntiSym(A;x,y.R[x;y])) supposing (AntiSym(B;x,y.S[x;y]) and RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f) and Inj(A;B;f))`

Proof

Definitions occuring in Statement :  rels_iso: `RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f)` anti_sym: `AntiSym(T;x,y.R[x; y])` inject: `Inj(A;B;f)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` anti_sym: `AntiSym(T;x,y.R[x; y])` rels_iso: `RelsIso(T;T';x,y.R[x; y];x,y.R'[x; y];f)` inject: `Inj(A;B;f)` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` so_lambda: `λ2x y.t[x; y]` iff: `P `⇐⇒` Q` and: `P ∧ Q`
Lemmas referenced :  anti_sym_wf rels_iso_wf inject_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution lambdaFormation hypothesis applyEquality hypothesisEquality sqequalRule lambdaEquality dependent_functionElimination thin axiomEquality universeEquality because_Cache lemma_by_obid isectElimination isect_memberEquality equalityTransitivity equalitySymmetry functionEquality cumulativity independent_functionElimination productElimination

Latex:
\mforall{}[A,B:Type].  \mforall{}[R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[S:B  {}\mrightarrow{}  B  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[f:A  {}\mrightarrow{}  B].
(AntiSym(A;x,y.R[x;y]))  supposing
(AntiSym(B;x,y.S[x;y])  and
RelsIso(A;B;x,y.R[x;y];x,y.S[x;y];f)  and
Inj(A;B;f))

Date html generated: 2016_05_15-PM-00_03_36
Last ObjectModification: 2015_12_26-PM-11_25_04

Theory : gen_algebra_1

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