Theorem hvadd32 31327

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 31294	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) = (𝐶 +ℎ 𝐵))
2	1	oveq2d 7427	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (𝐶 +ℎ 𝐵)))
3	2	3adant1 1146	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (𝐶 +ℎ 𝐵)))
4		ax-hvass 31295	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)))
5		ax-hvass 31295	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 +ℎ 𝐶) +ℎ 𝐵) = (𝐴 +ℎ (𝐶 +ℎ 𝐵)))
6	5	3com23 1142	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐶) +ℎ 𝐵) = (𝐴 +ℎ (𝐶 +ℎ 𝐵)))
7	3, 4, 6	3eqtr4d 2814	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = ((𝐴 +ℎ 𝐶) +ℎ 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  (class class class)co 7411   ℋchba 31212   +_ℎ cva 31213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-hvcom 31294  ax-hvass 31295

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414

This theorem is referenced by:  hvadd4  31329  hvadd32i  31347