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Theorem axhfvadd-zf 27026
Description: Derive axiom ax-hfvadd 27044 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 27013 . . . 4 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
3 axhil.1 . . . . 5 𝑈 = ⟨⟨ + , · ⟩, norm
43fveq2i 6088 . . . 4 (BaseSet‘𝑈) = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
52, 4eqtr4i 2631 . . 3 ℋ = (BaseSet‘𝑈)
61hlnvi 26935 . . . 4 𝑈 ∈ NrmCVec
73, 6h2hva 27018 . . 3 + = ( +𝑣𝑈)
85, 7hladdf 26942 . 2 (𝑈 ∈ CHilOLD → + :( ℋ × ℋ)⟶ ℋ)
91, 8ax-mp 5 1 + :( ℋ × ℋ)⟶ ℋ
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1976  cop 4127   × cxp 5023  wf 5783  cfv 5787  BaseSetcba 26606  CHilOLDchlo 26928  chil 26963   + cva 26964   · csm 26965  normcno 26967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-1st 7033  df-2nd 7034  df-grpo 26494  df-ablo 26549  df-vc 26564  df-nv 26612  df-va 26615  df-ba 26616  df-sm 26617  df-0v 26618  df-nmcv 26620  df-cbn 26906  df-hlo 26929  df-hba 27013
This theorem is referenced by: (None)
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