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Theorem axhv0cl-zf 31009

Description: Derive Axiom ax-hv0cl 31027 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 30993	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6835	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2760	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6		df-h0v 30994	. . . 4 ⊢ 0_ℎ = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
7	3	fveq2i 6835	. . . 4 ⊢ (0_vec‘𝑈) = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
8	6, 7	eqtr4i 2760	. . 3 ⊢ 0_ℎ = (0_vec‘𝑈)
9	5, 8	hl0cl 30926	. 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 2113 ⟨cop 4584 ‘cfv 6490 BaseSetcba 30610 0_veccn0v 30612 CHil_OLDchlo 30909 ℋchba 30943 +_ℎ cva 30944 ·_ℎ csm 30945 norm_ℎcno 30947 0_ℎc0v 30948

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pr 5375 ax-un 7678

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-1st 7931 df-2nd 7932 df-grpo 30517 df-gid 30518 df-ablo 30569 df-vc 30583 df-nv 30616 df-va 30619 df-ba 30620 df-sm 30621 df-0v 30622 df-nmcv 30624 df-cbn 30887 df-hlo 30910 df-hba 30993 df-h0v 30994

This theorem is referenced by: (None)

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