### Nuprl Lemma : rng_hom_minus

`∀[r,s:Rng]. ∀[f:|r| ⟶ |s|].  ∀[x:|r|]. (f[-r x] = (-s f[x]) ∈ |s|) supposing rng_hom_p(r;s;f)`

Proof

Definitions occuring in Statement :  rng_hom_p: `rng_hom_p(r;s;f)` rng: `Rng` rng_minus: `-r` rng_car: `|r|` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` apply: `f a` function: `x:A ⟶ B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` rng_hom_p: `rng_hom_p(r;s;f)` and: `P ∧ Q` rng: `Rng` prop: `ℙ` fun_thru_2op: `FunThru2op(A;B;opa;opb;f)` true: `True` squash: `↓T` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` implies: `P `` Q` so_apply: `x[s]` infix_ap: `x f y` rev_implies: `P `` Q`
Lemmas referenced :  rng_hom_zero rng_car_wf rng_hom_p_wf rng_wf rng_minus_wf infix_ap_wf rng_plus_wf equal_wf squash_wf true_wf rng_plus_comm rng_plus_inv iff_weakening_equal rng_plus_ac_1 rng_plus_zero
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination productElimination sqequalRule isect_memberEquality axiomEquality setElimination rename functionExtensionality applyEquality because_Cache functionEquality equalityTransitivity equalitySymmetry natural_numberEquality lambdaEquality imageElimination universeEquality imageMemberEquality baseClosed independent_functionElimination

Latex:
\mforall{}[r,s:Rng].  \mforall{}[f:|r|  {}\mrightarrow{}  |s|].    \mforall{}[x:|r|].  (f[-r  x]  =  (-s  f[x]))  supposing  rng\_hom\_p(r;s;f)

Date html generated: 2017_10_01-AM-08_18_23
Last ObjectModification: 2017_02_28-PM-02_03_17

Theory : rings_1

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