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Theorem hvadd12 31055
Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvadd12 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))

Proof of Theorem hvadd12
StepHypRef Expression
1 ax-hvcom 31021 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
21oveq1d 7447 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐵 + 𝐴) + 𝐶))
323adant3 1132 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐵 + 𝐴) + 𝐶))
4 ax-hvass 31022 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
5 ax-hvass 31022 . . 3 ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶)))
653com12 1123 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 + 𝐴) + 𝐶) = (𝐵 + (𝐴 + 𝐶)))
73, 4, 63eqtr3d 2784 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  (class class class)co 7432  chba 30939   + cva 30940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-hvcom 31021  ax-hvass 31022
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  hvaddsub12  31058
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