Theorem axhis3-zf

Description: Derive Axiom ax-his3 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
	axhfi.1	⊢ ·ih = ( ·𝑖OLD ‘ 𝑈 )
Assertion	axhis3-zf	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 ·ℎ 𝐵 ) ·ih 𝐶 ) = ( 𝐴 · ( 𝐵 ·ih 𝐶 ) ) )

Proof

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