Theorem hvadd32 31066

Description: Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
hvadd32	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵))

Proof of Theorem hvadd32

Step	Hyp	Ref	Expression
1		ax-hvcom 31033	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) = (𝐶 +ℎ 𝐵))
2	1	oveq2d 7464	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (𝐶 +ℎ 𝐵)))
3	2	3adant1 1130	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (𝐶 +ℎ 𝐵)))
4		ax-hvass 31034	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)))
5		ax-hvass 31034	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐶) +ℎ 𝐵) = (𝐴 +ℎ (𝐶 +ℎ 𝐵)))
6	5	3com23 1126	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐶) +ℎ 𝐵) = (𝐴 +ℎ (𝐶 +ℎ 𝐵)))
7	3, 4, 6	3eqtr4d 2790	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  (class class class)co 7448   ℋchba 30951   +_ℎ cva 30952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-hvcom 31033  ax-hvass 31034

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  hvadd4  31068  hvadd32i  31086