Theorem hvadd12i 31077

Description: Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
hvadd12i	⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))

Proof of Theorem hvadd12i

Step	Hyp	Ref	Expression
1		hvass.1	. . . 4 ⊢ 𝐴 ∈ ℋ
2		hvass.2	. . . 4 ⊢ 𝐵 ∈ ℋ
3	1, 2	hvcomi 31039	. . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)
4	3	oveq1i 7442	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶)
5		hvass.3	. . 3 ⊢ 𝐶 ∈ ℋ
6	1, 2, 5	hvassi 31073	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))
7	2, 1, 5	hvassi 31073	. 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))
8	4, 6, 7	3eqtr3i 2772	1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))

Syntax hints:   = wceq 1539   ∈ wcel 2107  (class class class)co 7432   ℋchba 30939   +_ℎ cva 30940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-hvcom 31021  ax-hvass 31022

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435

This theorem is referenced by:  hvsubaddi  31086