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Theorem nvo00 22219
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 5554 . 2  |-  ( T : X --> Y  ->  T  Fn  X )
2 nvo00.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 eqid 2408 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
42, 3nvzcl 22072 . . 3  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  X )
5 ne0i 3598 . . 3  |-  ( (
0vec `  U )  e.  X  ->  X  =/=  (/) )
64, 5syl 16 . 2  |-  ( U  e.  NrmCVec  ->  X  =/=  (/) )
7 fconst5 5912 . 2  |-  ( ( T  Fn  X  /\  X  =/=  (/) )  ->  ( T  =  ( X  X.  { Z } )  <->  ran  T  =  { Z } ) )
81, 6, 7syl2anr 465 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   (/)c0 3592   {csn 3778    X. cxp 4839   ran crn 4842    Fn wfn 5412   -->wf 5413   ` cfv 5417   NrmCVeccnv 22020   BaseSetcba 22022   0veccn0v 22024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-1st 6312  df-2nd 6313  df-riota 6512  df-grpo 21736  df-gid 21737  df-ablo 21827  df-vc 21982  df-nv 22028  df-va 22031  df-ba 22032  df-sm 22033  df-0v 22034  df-nmcv 22036
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