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Theorem nvo00 21773
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 5495 . 2  |-  ( T : X --> Y  ->  T  Fn  X )
2 nvo00.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 eqid 2366 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
42, 3nvzcl 21626 . . 3  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  X )
5 ne0i 3549 . . 3  |-  ( (
0vec `  U )  e.  X  ->  X  =/=  (/) )
64, 5syl 15 . 2  |-  ( U  e.  NrmCVec  ->  X  =/=  (/) )
7 fconst5 5849 . 2  |-  ( ( T  Fn  X  /\  X  =/=  (/) )  ->  ( T  =  ( X  X.  { Z } )  <->  ran  T  =  { Z } ) )
81, 6, 7syl2anr 464 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 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   (/)c0 3543   {csn 3729    X. cxp 4790   ran crn 4793    Fn wfn 5353   -->wf 5354   ` cfv 5358   NrmCVeccnv 21574   BaseSetcba 21576   0veccn0v 21578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-1st 6249  df-2nd 6250  df-riota 6446  df-grpo 21290  df-gid 21291  df-ablo 21381  df-vc 21536  df-nv 21582  df-va 21585  df-ba 21586  df-sm 21587  df-0v 21588  df-nmcv 21590
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