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Theorem axhv0cl-zf 31017

Description: Derive Axiom ax-hv0cl 31035 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 31001	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6923	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2771	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6		df-h0v 31002	. . . 4 ⊢ 0_ℎ = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
7	3	fveq2i 6923	. . . 4 ⊢ (0_vec‘𝑈) = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
8	6, 7	eqtr4i 2771	. . 3 ⊢ 0_ℎ = (0_vec‘𝑈)
9	5, 8	hl0cl 30934	. 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 1537 ∈ wcel 2108 ⟨cop 4654 ‘cfv 6573 BaseSetcba 30618 0_veccn0v 30620 CHil_OLDchlo 30917 ℋchba 30951 +_ℎ cva 30952 ·_ℎ csm 30953 norm_ℎcno 30955 0_ℎc0v 30956

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-1st 8030 df-2nd 8031 df-grpo 30525 df-gid 30526 df-ablo 30577 df-vc 30591 df-nv 30624 df-va 30627 df-ba 30628 df-sm 30629 df-0v 30630 df-nmcv 30632 df-cbn 30895 df-hlo 30918 df-hba 31001 df-h0v 31002

This theorem is referenced by: (None)

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