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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 6864	. . . 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 6864	. . . 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 4598 ‘cfv 6514 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 5237 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

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 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-1st 7971 df-2nd 7972 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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