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Theorem nvo00 30780
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 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑇 = (𝑋 × {𝑍}) ↔ ran 𝑇 = {𝑍}))

Proof of Theorem nvo00
StepHypRef Expression
1 ffn 6736 . 2 (𝑇:𝑋𝑌𝑇 Fn 𝑋)
2 nvo00.1 . . . 4 𝑋 = (BaseSet‘𝑈)
3 eqid 2737 . . . 4 (0vec𝑈) = (0vec𝑈)
42, 3nvzcl 30653 . . 3 (𝑈 ∈ NrmCVec → (0vec𝑈) ∈ 𝑋)
54ne0d 4342 . 2 (𝑈 ∈ NrmCVec → 𝑋 ≠ ∅)
6 fconst5 7226 . 2 ((𝑇 Fn 𝑋𝑋 ≠ ∅) → (𝑇 = (𝑋 × {𝑍}) ↔ ran 𝑇 = {𝑍}))
71, 5, 6syl2anr 597 1 ((𝑈 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑇 = (𝑋 × {𝑍}) ↔ ran 𝑇 = {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  c0 4333  {csn 4626   × cxp 5683  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  NrmCVeccnv 30603  BaseSetcba 30605  0veccn0v 30607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ablo 30564  df-vc 30578  df-nv 30611  df-va 30614  df-ba 30615  df-sm 30616  df-0v 30617  df-nmcv 30619
This theorem is referenced by: (None)
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