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Theorem nvo00 24282
Description: Two ways to express a zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
nvo00.1  |-  X  =  ( BaseSet `  U )
Assertion
Ref Expression
nvo00  |-  ( ( U  e.  NrmCVec  /\  T : X --> Y )  -> 
( T  =  ( X  X.  { Z } )  <->  ran  T  =  { Z } ) )

Proof of Theorem nvo00
StepHypRef Expression
1 ffn 5643 . 2  |-  ( T : X --> Y  ->  T  Fn  X )
2 nvo00.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 eqid 2450 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
42, 3nvzcl 24135 . . 3  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  X )
5 ne0i 3727 . . 3  |-  ( (
0vec `  U )  e.  X  ->  X  =/=  (/) )
64, 5syl 16 . 2  |-  ( U  e.  NrmCVec  ->  X  =/=  (/) )
7 fconst5 6020 . 2  |-  ( ( T  Fn  X  /\  X  =/=  (/) )  ->  ( T  =  ( X  X.  { Z } )  <->  ran  T  =  { Z } ) )
81, 6, 7syl2anr 478 1  |-  ( ( U  e.  NrmCVec  /\  T : X --> Y )  -> 
( T  =  ( X  X.  { Z } )  <->  ran  T  =  { Z } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1757    =/= wne 2641   (/)c0 3721   {csn 3961    X. cxp 4922   ran crn 4925    Fn wfn 5497   -->wf 5498   ` cfv 5502   NrmCVeccnv 24083   BaseSetcba 24085   0veccn0v 24087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-1st 6663  df-2nd 6664  df-grpo 23799  df-gid 23800  df-ablo 23890  df-vc 24045  df-nv 24091  df-va 24094  df-ba 24095  df-sm 24096  df-0v 24097  df-nmcv 24099
This theorem is referenced by: (None)
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