### Nuprl Lemma : uanti_sym_shift

`∀[A,B:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀[S:B ⟶ B ⟶ ℙ]. ∀[f:A ⟶ B].`
`  (UniformlyAntiSym(A;x,y.R[x;y])) supposing `
`     (UniformlyAntiSym(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)` uanti_sym: `UniformlyAntiSym(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` uanti_sym: `UniformlyAntiSym(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)` prop: `ℙ` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` all: `∀x:A. B[x]` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q`
Lemmas referenced :  uanti_sym_wf rels_iso_wf inject_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution hypothesis applyEquality hypothesisEquality sqequalRule isect_memberEquality isectElimination thin axiomEquality because_Cache equalityTransitivity equalitySymmetry lemma_by_obid lambdaEquality functionEquality cumulativity universeEquality dependent_functionElimination independent_functionElimination independent_isectElimination 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].
(UniformlyAntiSym(A;x,y.R[x;y]))  supposing
(UniformlyAntiSym(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_39
Last ObjectModification: 2015_12_26-PM-11_25_03

Theory : gen_algebra_1

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