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Theorem hvadd32i 29135
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 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)

Proof of Theorem hvadd32i
StepHypRef Expression
1 hvass.1 . 2 𝐴 ∈ ℋ
2 hvass.2 . 2 𝐵 ∈ ℋ
3 hvass.3 . 2 𝐶 ∈ ℋ
4 hvadd32 29115 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))
51, 2, 3, 4mp3an 1463 1 ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2110  (class class class)co 7213  chba 29000   + cva 29001
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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-hvcom 29082  ax-hvass 29083
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216
This theorem is referenced by:  hvsubeq0i  29144  hvaddcani  29146  normpar2i  29237
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