Theorem axhv0cl-zf 28756

Description: Derive axiom ax-hv0cl 28774 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 ⟩, normℎ⟩
axhil.2	⊢ 𝑈 ∈ CHilOLD

Assertion

Ref	Expression
axhv0cl-zf	⊢ 0ℎ ∈ ℋ

Proof of Theorem axhv0cl-zf

Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHilOLD
2		df-hba 28740	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
4	3	fveq2i 6667	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5	2, 4	eqtr4i 2847	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6		df-h0v 28741	. . . 4 ⊢ 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
7	3	fveq2i 6667	. . . 4 ⊢ (0vec‘𝑈) = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
8	6, 7	eqtr4i 2847	. . 3 ⊢ 0ℎ = (0vec‘𝑈)
9	5, 8	hl0cl 28673	. 2 ⊢ (𝑈 ∈ CHilOLD → 0ℎ ∈ ℋ)
10	1, 9	ax-mp 5	1 ⊢ 0ℎ ∈ ℋ

Syntax hints:   = wceq 1533   ∈ wcel 2110  ⟨cop 4566  ‘cfv 6349  BaseSetcba 28357  0_veccn0v 28359  CHil_OLDchlo 28656   ℋchba 28690   +_ℎ cva 28691   ·_ℎ csm 28692  norm_ℎcno 28694  0_ℎc0v 28695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-1st 7683  df-2nd 7684  df-grpo 28264  df-gid 28265  df-ablo 28316  df-vc 28330  df-nv 28363  df-va 28366  df-ba 28367  df-sm 28368  df-0v 28369  df-nmcv 28371  df-cbn 28634  df-hlo 28657  df-hba 28740  df-h0v 28741

This theorem is referenced by: (None)