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Metamath Proof Explorer

Theorem hvadd12

Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion hvadd12 ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( 𝐵 +ℎ ( 𝐴 +ℎ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 ax-hvcom ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 ) )
2 1 oveq1d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) +ℎ 𝐶 ) = ( ( 𝐵 +ℎ 𝐴 ) +ℎ 𝐶 ) )
3 2 3adant3 ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) +ℎ 𝐶 ) = ( ( 𝐵 +ℎ 𝐴 ) +ℎ 𝐶 ) )
4 ax-hvass ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) +ℎ 𝐶 ) = ( 𝐴 +ℎ ( 𝐵 +ℎ 𝐶 ) ) )
5 ax-hvass ⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐵 +ℎ 𝐴 ) +ℎ 𝐶 ) = ( 𝐵 +ℎ ( 𝐴 +ℎ 𝐶 ) ) )
6 5 3com12 ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐵 +ℎ 𝐴 ) +ℎ 𝐶 ) = ( 𝐵 +ℎ ( 𝐴 +ℎ 𝐶 ) ) )
7 3 4 6 3eqtr3d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( 𝐵 +ℎ ( 𝐴 +ℎ 𝐶 ) ) )