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Theorem nvo00 22267
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 5594 . 2  |-  ( T : X --> Y  ->  T  Fn  X )
2 nvo00.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 eqid 2438 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
42, 3nvzcl 22120 . . 3  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  X )
5 ne0i 3636 . . 3  |-  ( (
0vec `  U )  e.  X  ->  X  =/=  (/) )
64, 5syl 16 . 2  |-  ( U  e.  NrmCVec  ->  X  =/=  (/) )
7 fconst5 5952 . 2  |-  ( ( T  Fn  X  /\  X  =/=  (/) )  ->  ( T  =  ( X  X.  { Z } )  <->  ran  T  =  { Z } ) )
81, 6, 7syl2anr 466 1  |-  ( ( U  e.  NrmCVec  /\  T : X --> Y )  -> 
( T  =  ( X  X.  { Z } )  <->  ran  T  =  { Z } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   (/)c0 3630   {csn 3816    X. cxp 4879   ran crn 4882    Fn wfn 5452   -->wf 5453   ` cfv 5457   NrmCVeccnv 22068   BaseSetcba 22070   0veccn0v 22072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-1st 6352  df-2nd 6353  df-riota 6552  df-grpo 21784  df-gid 21785  df-ablo 21875  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-nmcv 22084
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