Theorem axhvdistr1-zf

Description: Derive Axiom ax-hvdistr1 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	axhil.1	⊢ 𝑈 = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
	axhil.2	⊢ 𝑈 ∈ CHilOLD
Assertion	axhvdistr1-zf	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) +ℎ ( 𝐴 ·ℎ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		axhil.1	⊢ 𝑈 = ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩
2		axhil.2	⊢ 𝑈 ∈ CHilOLD
3		df-hba	⊢ ℋ = ( BaseSet ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
4	1	fveq2i	⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ ⟨ ⟨ +ℎ , ·ℎ ⟩ , normℎ ⟩ )
5	3 4	eqtr4i	⊢ ℋ = ( BaseSet ‘ 𝑈 )
6	2	hlnvi	⊢ 𝑈 ∈ NrmCVec
7	1 6	h2hva	⊢ +ℎ = ( +𝑣 ‘ 𝑈 )
8	1 6	h2hsm	⊢ ·ℎ = ( ·𝑠OLD ‘ 𝑈 )
9	5 7 8	hldi	⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( 𝐴 ·ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) +ℎ ( 𝐴 ·ℎ 𝐶 ) ) )
10	2 9	mpan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝐵 +ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) +ℎ ( 𝐴 ·ℎ 𝐶 ) ) )