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Theorem hvadd12i 27763
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 27725 . . 3 (𝐴 + 𝐵) = (𝐵 + 𝐴)
43oveq1i 6614 . 2 ((𝐴 + 𝐵) + 𝐶) = ((𝐵 + 𝐴) + 𝐶)
5 hvass.3 . . 3 𝐶 ∈ ℋ
61, 2, 5hvassi 27759 . 2 ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
72, 1, 5hvassi 27759 . 2 ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶))
84, 6, 73eqtr3i 2651 1 (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  (class class class)co 6604  chil 27625   + cva 27626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-hvcom 27707  ax-hvass 27708
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607
This theorem is referenced by:  hvsubaddi  27772
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