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Theorem 0ofval 28479
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 6666 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 0oval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2878 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54xpeq1d 5582 . . 3 (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec𝑤)}) = (𝑋 × {(0vec𝑤)}))
6 fveq2 6666 . . . . . 6 (𝑤 = 𝑊 → (0vec𝑤) = (0vec𝑊))
7 0oval.6 . . . . . 6 𝑍 = (0vec𝑊)
86, 7syl6eqr 2878 . . . . 5 (𝑤 = 𝑊 → (0vec𝑤) = 𝑍)
98sneqd 4575 . . . 4 (𝑤 = 𝑊 → {(0vec𝑤)} = {𝑍})
109xpeq2d 5583 . . 3 (𝑤 = 𝑊 → (𝑋 × {(0vec𝑤)}) = (𝑋 × {𝑍}))
11 df-0o 28439 . . 3 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec𝑤)}))
123fvexi 6680 . . . 4 𝑋 ∈ V
13 snex 5327 . . . 4 {𝑍} ∈ V
1412, 13xpex 7468 . . 3 (𝑋 × {𝑍}) ∈ V
155, 10, 11, 14ovmpo 7303 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍}))
161, 15syl5eq 2872 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2106  {csn 4563   × cxp 5551  cfv 6351  (class class class)co 7151  NrmCVeccnv 28276  BaseSetcba 28278  0veccn0v 28280   0op c0o 28435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-0o 28439
This theorem is referenced by:  0oval  28480  0oo  28481  lnon0  28490  blocni  28497  hh0oi  29595
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