Theorem hvadd12i 31260

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 31222	. . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)
4	3	oveq1i 7406	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶)
5		hvass.3	. . 3 ⊢ 𝐶 ∈ ℋ
6	1, 2, 5	hvassi 31256	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))
7	2, 1, 5	hvassi 31256	. 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))
8	4, 6, 7	3eqtr3i 2793	1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))

Syntax hints:   = wceq 1560   ∈ wcel 2142  (class class class)co 7396   ℋchba 31122   +_ℎ cva 31123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-hvcom 31204  ax-hvass 31205

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399

This theorem is referenced by:  hvsubaddi  31269