Theorem nvgf 28389

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 = ( +𝑣 ‘𝑈)

Assertion

Ref	Expression
nvgf	⊢ (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋)

Proof of Theorem nvgf

Step	Hyp	Ref	Expression
1		nvgf.2	. . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
2	1	nvgrp 28388	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvgf.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 28375	. . 3 ⊢ 𝑋 = ran 𝐺
5	4	grpofo 28270	. 2 ⊢ (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
6		fof 6585	. 2 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → 𝐺:(𝑋 × 𝑋)⟶𝑋)
7	2, 5, 6	3syl 18	1 ⊢ (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110   × cxp 5548  ⟶wf 6346  –onto→wfo 6348  ‘cfv 6350  GrpOpcgr 28260  NrmCVeccnv 28355   +_𝑣 cpv 28356  BaseSetcba 28357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-1st 7683  df-2nd 7684  df-grpo 28264  df-ablo 28316  df-vc 28330  df-nv 28363  df-va 28366  df-ba 28367  df-sm 28368  df-0v 28369  df-nmcv 28371

This theorem is referenced by:  vacn  28465  sspg  28499  hladdf  28670