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Theorem 0ofval 21365
 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
0oval.6
0oval.0
Assertion
Ref Expression
0ofval

Proof of Theorem 0ofval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0oval.0 . 2
2 fveq2 5525 . . . . 5
3 0oval.1 . . . . 5
42, 3syl6eqr 2333 . . . 4
54xpeq1d 4712 . . 3
6 fveq2 5525 . . . . . 6
7 0oval.6 . . . . . 6
86, 7syl6eqr 2333 . . . . 5
98sneqd 3653 . . . 4
109xpeq2d 4713 . . 3
11 df-0o 21325 . . 3
12 fvex 5539 . . . . 5
133, 12eqeltri 2353 . . . 4
14 snex 4216 . . . 4
1513, 14xpex 4801 . . 3
165, 10, 11, 15ovmpt2 5983 . 2
171, 16syl5eq 2327 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684  cvv 2788  csn 3640   cxp 4687  cfv 5255  (class class class)co 5858  cnv 21140  cba 21142  cn0v 21144   c0o 21321 This theorem is referenced by:  0oval  21366  0oo  21367  lnon0  21376  blocni  21383  hh0oi  22483 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-0o 21325
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