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Theorem 0ofval 30816
Description: The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
0oval.1 𝑋 = (BaseSet‘𝑈)
0oval.6 𝑍 = (0vec𝑊)
0oval.0 𝑂 = (𝑈 0op 𝑊)
Assertion
Ref Expression
0ofval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))

Proof of Theorem 0ofval
Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0oval.0 . 2 𝑂 = (𝑈 0op 𝑊)
2 fveq2 6907 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 0oval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
42, 3eqtr4di 2793 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54xpeq1d 5718 . . 3 (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec𝑤)}) = (𝑋 × {(0vec𝑤)}))
6 fveq2 6907 . . . . . 6 (𝑤 = 𝑊 → (0vec𝑤) = (0vec𝑊))
7 0oval.6 . . . . . 6 𝑍 = (0vec𝑊)
86, 7eqtr4di 2793 . . . . 5 (𝑤 = 𝑊 → (0vec𝑤) = 𝑍)
98sneqd 4643 . . . 4 (𝑤 = 𝑊 → {(0vec𝑤)} = {𝑍})
109xpeq2d 5719 . . 3 (𝑤 = 𝑊 → (𝑋 × {(0vec𝑤)}) = (𝑋 × {𝑍}))
11 df-0o 30776 . . 3 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec𝑤)}))
123fvexi 6921 . . . 4 𝑋 ∈ V
13 snex 5442 . . . 4 {𝑍} ∈ V
1412, 13xpex 7772 . . 3 (𝑋 × {𝑍}) ∈ V
155, 10, 11, 14ovmpo 7593 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍}))
161, 15eqtrid 2787 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {csn 4631   × cxp 5687  cfv 6563  (class class class)co 7431  NrmCVeccnv 30613  BaseSetcba 30615  0veccn0v 30617   0op c0o 30772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-0o 30776
This theorem is referenced by:  0oval  30817  0oo  30818  lnon0  30827  blocni  30834  hh0oi  31932
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