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Theorem axhvdistr1-zf 31607

Description: Derive Axiom ax-hvdistr1 31625 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 31586	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6899	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2789	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 31509	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hva 31591	. . 3 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)
8	3, 6	h2hsm 31592	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
9	5, 7, 8	hldi 31524	. 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 1569 ∈ wcel 2146 ⟨cop 4838 ‘cfv 6547 (class class class)co 7430 ℂcc 11211 BaseSetcba 31203 CHil_OLDchlo 31502 ℋchba 31536 +_ℎ cva 31537 ·_ℎ csm 31538 norm_ℎcno 31540

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pr 5395 ax-un 7761

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 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 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-ov 7433 df-oprab 7434 df-1st 8011 df-2nd 8012 df-vc 31176 df-nv 31209 df-va 31212 df-ba 31213 df-sm 31214 df-0v 31215 df-nmcv 31217 df-cbn 31480 df-hlo 31503 df-hba 31586

This theorem is used by: (None)

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