Theorem axhfvadd-zf 28865

Description: Derive Axiom ax-hfvadd 28883 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
axhfvadd-zf	⊢ +ℎ :( ℋ × ℋ)⟶ ℋ

Proof of Theorem axhfvadd-zf

Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHilOLD
2		df-hba 28852	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
4	3	fveq2i 6662	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
5	2, 4	eqtr4i 2785	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 28775	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hva 28857	. . 3 ⊢ +ℎ = ( +𝑣 ‘𝑈)
8	5, 7	hladdf 28782	. 2 ⊢ (𝑈 ∈ CHilOLD → +ℎ :( ℋ × ℋ)⟶ ℋ)
9	1, 8	ax-mp 5	1 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ

Syntax hints:   = wceq 1539   ∈ wcel 2112  ⟨cop 4529   × cxp 5523  ⟶wf 6332  ‘cfv 6336  BaseSetcba 28469  CHil_OLDchlo 28768   ℋchba 28802   +_ℎ cva 28803   ·_ℎ csm 28804  norm_ℎcno 28806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-1st 7694  df-2nd 7695  df-grpo 28376  df-ablo 28428  df-vc 28442  df-nv 28475  df-va 28478  df-ba 28479  df-sm 28480  df-0v 28481  df-nmcv 28483  df-cbn 28746  df-hlo 28769  df-hba 28852

This theorem is referenced by: (None)