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Theorem axhfvmul-zf 28748
 Description: Derive axiom ax-hfvmul 28766 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
axhfvmul-zf · :(ℂ × ℋ)⟶ ℋ

Proof of Theorem axhfvmul-zf
StepHypRef Expression
1 axhil.2 . 2 𝑈 ∈ CHilOLD
2 df-hba 28730 . . . 4 ℋ = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
3 axhil.1 . . . . 5 𝑈 = ⟨⟨ + , · ⟩, norm
43fveq2i 6646 . . . 4 (BaseSet‘𝑈) = (BaseSet‘⟨⟨ + , · ⟩, norm⟩)
52, 4eqtr4i 2847 . . 3 ℋ = (BaseSet‘𝑈)
61hlnvi 28653 . . . 4 𝑈 ∈ NrmCVec
73, 6h2hsm 28736 . . 3 · = ( ·𝑠OLD𝑈)
85, 7hlmulf 28665 . 2 (𝑈 ∈ CHilOLD· :(ℂ × ℋ)⟶ ℋ)
91, 8ax-mp 5 1 · :(ℂ × ℋ)⟶ ℋ
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2115  ⟨cop 4546   × cxp 5526  ⟶wf 6324  ‘cfv 6328  ℂcc 10512  BaseSetcba 28347  CHilOLDchlo 28646   ℋchba 28680   +ℎ cva 28681   ·ℎ csm 28682  normℎcno 28684 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-1st 7664  df-2nd 7665  df-vc 28320  df-nv 28353  df-va 28356  df-ba 28357  df-sm 28358  df-0v 28359  df-nmcv 28361  df-cbn 28624  df-hlo 28647  df-hba 28730 This theorem is referenced by: (None)
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