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Theorem hvadd12i 30048
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 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))

Proof of Theorem hvadd12i
StepHypRef Expression
1 hvass.1 . . . 4 𝐴 ∈ ℋ
2 hvass.2 . . . 4 𝐵 ∈ ℋ
31, 2hvcomi 30010 . . 3 (𝐴 + 𝐵) = (𝐵 + 𝐴)
43oveq1i 7371 . 2 ((𝐴 + 𝐵) + 𝐶) = ((𝐵 + 𝐴) + 𝐶)
5 hvass.3 . . 3 𝐶 ∈ ℋ
61, 2, 5hvassi 30044 . 2 ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
72, 1, 5hvassi 30044 . 2 ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶))
84, 6, 73eqtr3i 2769 1 (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  (class class class)co 7361  chba 29910   + cva 29911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-hvcom 29992  ax-hvass 29993
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364
This theorem is referenced by:  hvsubaddi  30057
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