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Metamath Proof Explorer

Theorem nvnnncan1

Description: Cancellation law for vector subtraction. ( nnncan1 analog.) (Contributed by NM, 7-Mar-2008) (New usage is discouraged.)

Ref Expression
Hypotheses nvmf.1 ⊢ X = BaseSet ⁡ U
nvmf.3 ⊢ M = - v ⁡ U
Assertion nvnnncan1 ⊢ U ∈ NrmCVec ∧ A ∈ X ∧ B ∈ X ∧ C ∈ X → A M B M A M C = C M B

Proof

Step Hyp Ref Expression
1 nvmf.1 ⊢ X = BaseSet ⁡ U
2 nvmf.3 ⊢ M = - v ⁡ U
3 eqid ⊢ + v ⁡ U = + v ⁡ U
4 3 nvablo ⊢ U ∈ NrmCVec → + v ⁡ U ∈ AbelOp
5 1 3 bafval ⊢ X = ran ⁡ + v ⁡ U
6 3 2 vsfval ⊢ M = / g ⁡ + v ⁡ U
7 5 6 ablonnncan1 ⊢ + v ⁡ U ∈ AbelOp ∧ A ∈ X ∧ B ∈ X ∧ C ∈ X → A M B M A M C = C M B
8 4 7 sylan ⊢ U ∈ NrmCVec ∧ A ∈ X ∧ B ∈ X ∧ C ∈ X → A M B M A M C = C M B