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Theorem hvadd12i 30804
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 30766 . . 3 (𝐴 + 𝐵) = (𝐵 + 𝐴)
43oveq1i 7412 . 2 ((𝐴 + 𝐵) + 𝐶) = ((𝐵 + 𝐴) + 𝐶)
5 hvass.3 . . 3 𝐶 ∈ ℋ
61, 2, 5hvassi 30800 . 2 ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
72, 1, 5hvassi 30800 . 2 ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶))
84, 6, 73eqtr3i 2760 1 (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  (class class class)co 7402  chba 30666   + cva 30667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-hvcom 30748  ax-hvass 30749
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405
This theorem is referenced by:  hvsubaddi  30813
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