Theorem hvadd12i 30866

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 30828	. . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)
4	3	oveq1i 7430	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶)
5		hvass.3	. . 3 ⊢ 𝐶 ∈ ℋ
6	1, 2, 5	hvassi 30862	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))
7	2, 1, 5	hvassi 30862	. 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))
8	4, 6, 7	3eqtr3i 2764	1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))

Syntax hints:   = wceq 1534   ∈ wcel 2099  (class class class)co 7420   ℋchba 30728   +_ℎ cva 30729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-hvcom 30810  ax-hvass 30811

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423

This theorem is referenced by:  hvsubaddi  30875