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Theorem axhv0cl-zf 30921

Description: Derive Axiom ax-hv0cl 30939 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 30905	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6868	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2756	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6		df-h0v 30906	. . . 4 ⊢ 0_ℎ = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
7	3	fveq2i 6868	. . . 4 ⊢ (0_vec‘𝑈) = (0_vec‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
8	6, 7	eqtr4i 2756	. . 3 ⊢ 0_ℎ = (0_vec‘𝑈)
9	5, 8	hl0cl 30838	. 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 1540 ∈ wcel 2109 ⟨cop 4603 ‘cfv 6519 BaseSetcba 30522 0_veccn0v 30524 CHil_OLDchlo 30821 ℋchba 30855 +_ℎ cva 30856 ·_ℎ csm 30857 norm_ℎcno 30859 0_ℎc0v 30860

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pr 5395 ax-un 7718

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-ral 3047 df-rex 3056 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-nul 4305 df-if 4497 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-id 5541 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-riota 7351 df-ov 7397 df-oprab 7398 df-1st 7977 df-2nd 7978 df-grpo 30429 df-gid 30430 df-ablo 30481 df-vc 30495 df-nv 30528 df-va 30531 df-ba 30532 df-sm 30533 df-0v 30534 df-nmcv 30536 df-cbn 30799 df-hlo 30822 df-hba 30905 df-h0v 30906

This theorem is referenced by: (None)

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