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Theorem axhfvadd-zf 28674
Description: Derive axiom ax-hfvadd 28692 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
axhil.1 𝑈 = ⟨⟨ + , · ⟩, norm
axhil.2 𝑈 ∈ CHilOLD
Assertion
Ref Expression
axhfvadd-zf + :( ℋ × ℋ)⟶ ℋ

Proof of Theorem axhfvadd-zf
StepHypRef Expression
1 axhil.2 . 2 𝑈 ∈ CHilOLD
2 df-hba 28661 . . . 4 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
3 axhil.1 . . . . 5 𝑈 = ⟨⟨ + , · ⟩, norm
43fveq2i 6669 . . . 4 (BaseSet‘𝑈) = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
52, 4eqtr4i 2851 . . 3 ℋ = (BaseSet‘𝑈)
61hlnvi 28584 . . . 4 𝑈 ∈ NrmCVec
73, 6h2hva 28666 . . 3 + = ( +𝑣𝑈)
85, 7hladdf 28591 . 2 (𝑈 ∈ CHilOLD → + :( ℋ × ℋ)⟶ ℋ)
91, 8ax-mp 5 1 + :( ℋ × ℋ)⟶ ℋ
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2106  cop 4569   × cxp 5551  wf 6347  cfv 6351  BaseSetcba 28278  CHilOLDchlo 28577  chba 28611   + cva 28612   · csm 28613  normcno 28615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-1st 7683  df-2nd 7684  df-grpo 28185  df-ablo 28237  df-vc 28251  df-nv 28284  df-va 28287  df-ba 28288  df-sm 28289  df-0v 28290  df-nmcv 28292  df-cbn 28555  df-hlo 28578  df-hba 28661
This theorem is referenced by: (None)
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