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Theorem 0ofval 28580
 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 6646 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 0oval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
42, 3eqtr4di 2851 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54xpeq1d 5549 . . 3 (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec𝑤)}) = (𝑋 × {(0vec𝑤)}))
6 fveq2 6646 . . . . . 6 (𝑤 = 𝑊 → (0vec𝑤) = (0vec𝑊))
7 0oval.6 . . . . . 6 𝑍 = (0vec𝑊)
86, 7eqtr4di 2851 . . . . 5 (𝑤 = 𝑊 → (0vec𝑤) = 𝑍)
98sneqd 4537 . . . 4 (𝑤 = 𝑊 → {(0vec𝑤)} = {𝑍})
109xpeq2d 5550 . . 3 (𝑤 = 𝑊 → (𝑋 × {(0vec𝑤)}) = (𝑋 × {𝑍}))
11 df-0o 28540 . . 3 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec𝑤)}))
123fvexi 6660 . . . 4 𝑋 ∈ V
13 snex 5298 . . . 4 {𝑍} ∈ V
1412, 13xpex 7459 . . 3 (𝑋 × {𝑍}) ∈ V
155, 10, 11, 14ovmpo 7291 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍}))
161, 15syl5eq 2845 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {csn 4525   × cxp 5518  ‘cfv 6325  (class class class)co 7136  NrmCVeccnv 28377  BaseSetcba 28379  0veccn0v 28381   0op c0o 28536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-0o 28540 This theorem is referenced by:  0oval  28581  0oo  28582  lnon0  28591  blocni  28598  hh0oi  29696
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