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Theorem hladdf 28280
Description: Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
hladdf.1 𝑋 = (BaseSet‘𝑈)
hladdf.2 𝐺 = ( +𝑣𝑈)
Assertion
Ref Expression
hladdf (𝑈 ∈ CHilOLD𝐺:(𝑋 × 𝑋)⟶𝑋)

Proof of Theorem hladdf
StepHypRef Expression
1 hlnv 28272 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
2 hladdf.1 . . 3 𝑋 = (BaseSet‘𝑈)
3 hladdf.2 . . 3 𝐺 = ( +𝑣𝑈)
42, 3nvgf 27998 . 2 (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋)
51, 4syl 17 1 (𝑈 ∈ CHilOLD𝐺:(𝑋 × 𝑋)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157   × cxp 5310  wf 6097  cfv 6101  NrmCVeccnv 27964   +𝑣 cpv 27965  BaseSetcba 27966  CHilOLDchlo 28266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-1st 7401  df-2nd 7402  df-grpo 27873  df-ablo 27925  df-vc 27939  df-nv 27972  df-va 27975  df-ba 27976  df-sm 27977  df-0v 27978  df-nmcv 27980  df-cbn 28244  df-hlo 28267
This theorem is referenced by:  axhfvadd-zf  28364
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