MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0ofval Structured version   Visualization version   GIF version

Theorem 0ofval 30720
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 6901 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 0oval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
42, 3eqtr4di 2784 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54xpeq1d 5711 . . 3 (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec𝑤)}) = (𝑋 × {(0vec𝑤)}))
6 fveq2 6901 . . . . . 6 (𝑤 = 𝑊 → (0vec𝑤) = (0vec𝑊))
7 0oval.6 . . . . . 6 𝑍 = (0vec𝑊)
86, 7eqtr4di 2784 . . . . 5 (𝑤 = 𝑊 → (0vec𝑤) = 𝑍)
98sneqd 4645 . . . 4 (𝑤 = 𝑊 → {(0vec𝑤)} = {𝑍})
109xpeq2d 5712 . . 3 (𝑤 = 𝑊 → (𝑋 × {(0vec𝑤)}) = (𝑋 × {𝑍}))
11 df-0o 30680 . . 3 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec𝑤)}))
123fvexi 6915 . . . 4 𝑋 ∈ V
13 snex 5437 . . . 4 {𝑍} ∈ V
1412, 13xpex 7761 . . 3 (𝑋 × {𝑍}) ∈ V
155, 10, 11, 14ovmpo 7586 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍}))
161, 15eqtrid 2778 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  {csn 4633   × cxp 5680  cfv 6554  (class class class)co 7424  NrmCVeccnv 30517  BaseSetcba 30519  0veccn0v 30521   0op c0o 30676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-0o 30680
This theorem is referenced by:  0oval  30721  0oo  30722  lnon0  30731  blocni  30738  hh0oi  31836
  Copyright terms: Public domain W3C validator