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Theorem 0ofval 28491
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 6663 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 0oval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2871 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54xpeq1d 5577 . . 3 (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec𝑤)}) = (𝑋 × {(0vec𝑤)}))
6 fveq2 6663 . . . . . 6 (𝑤 = 𝑊 → (0vec𝑤) = (0vec𝑊))
7 0oval.6 . . . . . 6 𝑍 = (0vec𝑊)
86, 7syl6eqr 2871 . . . . 5 (𝑤 = 𝑊 → (0vec𝑤) = 𝑍)
98sneqd 4569 . . . 4 (𝑤 = 𝑊 → {(0vec𝑤)} = {𝑍})
109xpeq2d 5578 . . 3 (𝑤 = 𝑊 → (𝑋 × {(0vec𝑤)}) = (𝑋 × {𝑍}))
11 df-0o 28451 . . 3 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec𝑤)}))
123fvexi 6677 . . . 4 𝑋 ∈ V
13 snex 5322 . . . 4 {𝑍} ∈ V
1412, 13xpex 7465 . . 3 (𝑋 × {𝑍}) ∈ V
155, 10, 11, 14ovmpo 7299 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍}))
161, 15syl5eq 2865 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {csn 4557   × cxp 5546  cfv 6348  (class class class)co 7145  NrmCVeccnv 28288  BaseSetcba 28290  0veccn0v 28292   0op c0o 28447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-0o 28451
This theorem is referenced by:  0oval  28492  0oo  28493  lnon0  28502  blocni  28509  hh0oi  29607
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