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Metamath Proof Explorer

Theorem hvadd4

Description: Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999) (New usage is discouraged.)

		Ref	Expression
	Assertion	hvadd4	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ ( 𝐶 +_ℎ 𝐷 ) ) = ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ ( 𝐵 +_ℎ 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hvadd32	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ 𝐶 ) = ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ 𝐵 ) )
2	1	oveq1d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ 𝐶 ) +_ℎ 𝐷 ) = ( ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ 𝐵 ) +_ℎ 𝐷 ) )
3	2	3expa	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ 𝐶 ∈ ℋ ) → ( ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ 𝐶 ) +_ℎ 𝐷 ) = ( ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ 𝐵 ) +_ℎ 𝐷 ) )
4	3	adantrr	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ 𝐶 ) +_ℎ 𝐷 ) = ( ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ 𝐵 ) +_ℎ 𝐷 ) )
5		hvaddcl	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +_ℎ 𝐵 ) ∈ ℋ )
6		ax-hvass	⊢ ( ( ( 𝐴 +_ℎ 𝐵 ) ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) → ( ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ 𝐶 ) +_ℎ 𝐷 ) = ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ ( 𝐶 +_ℎ 𝐷 ) ) )
7	6	3expb	⊢ ( ( ( 𝐴 +_ℎ 𝐵 ) ∈ ℋ ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ 𝐶 ) +_ℎ 𝐷 ) = ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ ( 𝐶 +_ℎ 𝐷 ) ) )
8	5 7	sylan	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ 𝐶 ) +_ℎ 𝐷 ) = ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ ( 𝐶 +_ℎ 𝐷 ) ) )
9		hvaddcl	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 +_ℎ 𝐶 ) ∈ ℋ )
10		ax-hvass	⊢ ( ( ( 𝐴 +_ℎ 𝐶 ) ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ ) → ( ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ 𝐵 ) +_ℎ 𝐷 ) = ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ ( 𝐵 +_ℎ 𝐷 ) ) )
11	10	3expb	⊢ ( ( ( 𝐴 +_ℎ 𝐶 ) ∈ ℋ ∧ ( 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ 𝐵 ) +_ℎ 𝐷 ) = ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ ( 𝐵 +_ℎ 𝐷 ) ) )
12	9 11	sylan	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ∧ ( 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ 𝐵 ) +_ℎ 𝐷 ) = ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ ( 𝐵 +_ℎ 𝐷 ) ) )
13	12	an4s	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ 𝐵 ) +_ℎ 𝐷 ) = ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ ( 𝐵 +_ℎ 𝐷 ) ) )
14	4 8 13	3eqtr3d	⊢ ( ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) ∧ ( 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ ) ) → ( ( 𝐴 +_ℎ 𝐵 ) +_ℎ ( 𝐶 +_ℎ 𝐷 ) ) = ( ( 𝐴 +_ℎ 𝐶 ) +_ℎ ( 𝐵 +_ℎ 𝐷 ) ) )