Theorem hvadd12i 31029

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 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))

Proof of Theorem hvadd12i

Step	Hyp	Ref	Expression
1		hvass.1	. . . 4 ⊢ 𝐴 ∈ ℋ
2		hvass.2	. . . 4 ⊢ 𝐵 ∈ ℋ
3	1, 2	hvcomi 30991	. . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)
4	3	oveq1i 7351	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶)
5		hvass.3	. . 3 ⊢ 𝐶 ∈ ℋ
6	1, 2, 5	hvassi 31025	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))
7	2, 1, 5	hvassi 31025	. 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))
8	4, 6, 7	3eqtr3i 2762	1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))

Syntax hints:   = wceq 1541   ∈ wcel 2111  (class class class)co 7341   ℋchba 30891   +_ℎ cva 30892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-hvcom 30973  ax-hvass 30974

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-ov 7344

This theorem is referenced by:  hvsubaddi  31038