### Nuprl Lemma : module_act_minus_r

`∀A:Rng. ∀m:A-Module. ∀a:|A|. ∀u:m.car.  ((a m.act (m.minus u)) = (m.minus (a m.act u)) ∈ m.car)`

Proof

Definitions occuring in Statement :  module: `A-Module` alg_act: `a.act` alg_minus: `a.minus` alg_car: `a.car` infix_ap: `x f y` all: `∀x:A. B[x]` apply: `f a` equal: `s = t ∈ T` rng: `Rng` rng_car: `|r|`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` rng: `Rng` module: `A-Module` grp_of_module: `m↓grp` add_grp_of_rng: `r↓+gp` grp_car: `|g|` pi1: `fst(t)` rng_of_alg: `a↓rg` rng_car: `|r|` grp_inv: `~` pi2: `snd(t)` rng_minus: `-r` subtype_rel: `A ⊆r B` guard: `{T}` uimplies: `b supposing a` infix_ap: `x f y` uiff: `uiff(P;Q)` and: `P ∧ Q` rev_uimplies: `rev_uimplies(P;Q)` squash: `↓T` prop: `ℙ` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` grp_op: `*` rng_plus: `+r` grp_id: `e` rng_zero: `0`
Lemmas referenced :  alg_car_wf rng_car_wf module_wf rng_wf grp_eq_op_l grp_of_module_wf2 grp_subtype_igrp abgrp_subtype_grp subtype_rel_transitivity abgrp_wf grp_wf igrp_wf alg_act_wf grp_inv_wf grp_of_module_wf grp_car_wf equal_wf squash_wf true_wf infix_ap_wf grp_op_wf grp_inverse iff_weakening_equal module_act_plus alg_minus_wf alg_zero_wf grp_id_wf module_act_zero_r
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isectElimination setElimination rename hypothesisEquality sqequalRule applyEquality instantiate independent_isectElimination because_Cache lambdaEquality productElimination imageElimination equalityTransitivity equalitySymmetry universeEquality functionEquality natural_numberEquality imageMemberEquality baseClosed independent_functionElimination

Latex:
\mforall{}A:Rng.  \mforall{}m:A-Module.  \mforall{}a:|A|.  \mforall{}u:m.car.    ((a  m.act  (m.minus  u))  =  (m.minus  (a  m.act  u)))

Date html generated: 2017_10_01-AM-09_51_51
Last ObjectModification: 2017_03_03-PM-00_46_44

Theory : algebras_1

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