Theorem hvassi 31416

Description: Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hvass.1	⊢ 𝐴 ∈ ℋ
hvass.2	⊢ 𝐵 ∈ ℋ
hvass.3	⊢ 𝐶 ∈ ℋ

Assertion

Ref	Expression
hvassi	⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))

Proof of Theorem hvassi

Step	Hyp	Ref	Expression
1		hvass.1	. 2 ⊢ 𝐴 ∈ ℋ
2		hvass.2	. 2 ⊢ 𝐵 ∈ ℋ
3		hvass.3	. 2 ⊢ 𝐶 ∈ ℋ
4		ax-hvass 31365	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)))
5	1, 2, 3, 4	mp3an 1489	1 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2142  (class class class)co 7412   ℋchba 31282   +_ℎ cva 31283

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvass 31365

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104

This theorem is used by:  hvadd12i  31420  hvsubeq0i  31426  norm3difi  31510