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Theorem axhvdistr1-zf 31474

Description: Derive Axiom ax-hvdistr1 31492 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
axhil.1	⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
axhil.2	⊢ 𝑈 ∈ CHil_OLD

**Assertion**
Ref	Expression
axhvdistr1-zf	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ 𝐶)))

**Proof of Theorem axhvdistr1-zf**
Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHil_OLD
2		df-hba 31453	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6882	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2786	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 31376	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hva 31458	. . 3 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)
8	3, 6	h2hsm 31459	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
9	5, 7, 8	hldi 31391	. 2 ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝐴 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ 𝐶)))
10	1, 9	mpan 703	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 +_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) +_ℎ (𝐴 ·_ℎ 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 ‘cfv 6533 (class class class)co 7414 ℂcc 11125 BaseSetcba 31070 CHil_OLDchlo 31369 ℋchba 31403 +_ℎ cva 31404 ·_ℎ csm 31405 norm_ℎcno 31407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-oprab 7418 df-1st 7987 df-2nd 7988 df-vc 31043 df-nv 31076 df-va 31079 df-ba 31080 df-sm 31081 df-0v 31082 df-nmcv 31084 df-cbn 31347 df-hlo 31370 df-hba 31453

This theorem is used by: (None)

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