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Theorem hladdf 28837
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 28829 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
2 hladdf.1 . . 3 𝑋 = (BaseSet‘𝑈)
3 hladdf.2 . . 3 𝐺 = ( +𝑣𝑈)
42, 3nvgf 28556 . 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 1542  wcel 2114   × cxp 5524  wf 6336  cfv 6340  NrmCVeccnv 28522   +𝑣 cpv 28523  BaseSetcba 28524  CHilOLDchlo 28823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-1st 7717  df-2nd 7718  df-grpo 28431  df-ablo 28483  df-vc 28497  df-nv 28530  df-va 28533  df-ba 28534  df-sm 28535  df-0v 28536  df-nmcv 28538  df-cbn 28801  df-hlo 28824
This theorem is referenced by:  axhfvadd-zf  28920
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