Theorem hvadd12i 31716

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 31678	. . 3 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)
4	3	oveq1i 7461	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐵 +ℎ 𝐴) +ℎ 𝐶)
5		hvass.3	. . 3 ⊢ 𝐶 ∈ ℋ
6	1, 2, 5	hvassi 31712	. 2 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))
7	2, 1, 5	hvassi 31712	. 2 ⊢ ((𝐵 +ℎ 𝐴) +ℎ 𝐶) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))
8	4, 6, 7	3eqtr3i 2794	1 ⊢ (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ 𝐶))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  (class class class)co 7451   ℋchba 31578   +_ℎ cva 31579

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-hvcom 31660  ax-hvass 31661

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454

This theorem is used by:  hvsubaddi  31725