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Theorem nvgf 28410
Description: Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvgf.1 𝑋 = (BaseSet‘𝑈)
nvgf.2 𝐺 = ( +𝑣𝑈)
Assertion
Ref Expression
nvgf (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋)

Proof of Theorem nvgf
StepHypRef Expression
1 nvgf.2 . . 3 𝐺 = ( +𝑣𝑈)
21nvgrp 28409 . 2 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3 nvgf.1 . . . 4 𝑋 = (BaseSet‘𝑈)
43, 1bafval 28396 . . 3 𝑋 = ran 𝐺
54grpofo 28291 . 2 (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
6 fof 6583 . 2 (𝐺:(𝑋 × 𝑋)–onto𝑋𝐺:(𝑋 × 𝑋)⟶𝑋)
72, 5, 63syl 18 1 (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115   × cxp 5541  wf 6341  ontowfo 6343  cfv 6345  GrpOpcgr 28281  NrmCVeccnv 28376   +𝑣 cpv 28377  BaseSetcba 28378
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 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
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 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-1st 7686  df-2nd 7687  df-grpo 28285  df-ablo 28337  df-vc 28351  df-nv 28384  df-va 28387  df-ba 28388  df-sm 28389  df-0v 28390  df-nmcv 28392
This theorem is referenced by:  vacn  28486  sspg  28520  hladdf  28691
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