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Theorem nvo00 25793
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 5639 . 2  |-  ( T : X --> Y  ->  T  Fn  X )
2 nvo00.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 eqid 2382 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
42, 3nvzcl 25646 . . 3  |-  ( U  e.  NrmCVec  ->  ( 0vec `  U
)  e.  X )
5 ne0i 3717 . . 3  |-  ( (
0vec `  U )  e.  X  ->  X  =/=  (/) )
64, 5syl 16 . 2  |-  ( U  e.  NrmCVec  ->  X  =/=  (/) )
7 fconst5 6031 . 2  |-  ( ( T  Fn  X  /\  X  =/=  (/) )  ->  ( T  =  ( X  X.  { Z } )  <->  ran  T  =  { Z } ) )
81, 6, 7syl2anr 476 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 367    = wceq 1399    e. wcel 1826    =/= wne 2577   (/)c0 3711   {csn 3944    X. cxp 4911   ran crn 4914    Fn wfn 5491   -->wf 5492   ` cfv 5496   NrmCVeccnv 25594   BaseSetcba 25596   0veccn0v 25598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-1st 6699  df-2nd 6700  df-grpo 25310  df-gid 25311  df-ablo 25401  df-vc 25556  df-nv 25602  df-va 25605  df-ba 25606  df-sm 25607  df-0v 25608  df-nmcv 25610
This theorem is referenced by: (None)
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