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Theorem axhv0cl-zf 30965

Description: Derive Axiom ax-hv0cl 30983 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
axhv0cl-zf	⊢ 0_ℎ ∈ ℋ

**Proof of Theorem axhv0cl-zf**
Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHil_OLD
2		df-hba 30949	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6825	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2757	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6		df-h0v 30950	. . . 4 ⊢ 0_ℎ = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
7	3	fveq2i 6825	. . . 4 ⊢ (0_vec‘𝑈) = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
8	6, 7	eqtr4i 2757	. . 3 ⊢ 0_ℎ = (0_vec‘𝑈)
9	5, 8	hl0cl 30882	. 2 ⊢ (𝑈 ∈ CHil_OLD → 0_ℎ ∈ ℋ)
10	1, 9	ax-mp 5	1 ⊢ 0_ℎ ∈ ℋ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 ⟨cop 4579 ‘cfv 6481 BaseSetcba 30566 0_veccn0v 30568 CHil_OLDchlo 30865 ℋchba 30899 +_ℎ cva 30900 ·_ℎ csm 30901 norm_ℎcno 30903 0_ℎc0v 30904

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-1st 7921 df-2nd 7922 df-grpo 30473 df-gid 30474 df-ablo 30525 df-vc 30539 df-nv 30572 df-va 30575 df-ba 30576 df-sm 30577 df-0v 30578 df-nmcv 30580 df-cbn 30843 df-hlo 30866 df-hba 30949 df-h0v 30950

This theorem is referenced by: (None)

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