Theorem hvadd32i 31145

Description: Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
hvadd32i	⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵)

Proof of Theorem hvadd32i

Step	Hyp	Ref	Expression
1		hvass.1	. 2 ⊢ 𝐴 ∈ ℋ
2		hvass.2	. 2 ⊢ 𝐵 ∈ ℋ
3		hvass.3	. 2 ⊢ 𝐶 ∈ ℋ
4		hvadd32 31125	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵))
5	1, 2, 3, 4	mp3an 1470	1 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵)

Syntax hints:   = wceq 1548   ∈ wcel 2121  (class class class)co 7359   ℋchba 31010   +_ℎ cva 31011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-hvcom 31092  ax-hvass 31093

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6444  df-fv 6496  df-ov 7362

This theorem is referenced by:  hvsubeq0i  31154  hvaddcani  31156  normpar2i  31247