Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  imsval Unicode version

Theorem imsval 21362
 Description: Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
imsval.3
imsval.6 CV
imsval.8
Assertion
Ref Expression
imsval

Proof of Theorem imsval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5605 . . . 4 CV CV
2 fveq2 5605 . . . 4
31, 2coeq12d 4927 . . 3 CV CV
4 df-ims 21265 . . 3 CV
5 fvex 5619 . . . 4 CV
6 fvex 5619 . . . 4
75, 6coex 5295 . . 3 CV
83, 4, 7fvmpt 5682 . 2 CV
9 imsval.8 . 2
10 imsval.6 . . 3 CV
11 imsval.3 . . 3
1210, 11coeq12i 4926 . 2 CV
138, 9, 123eqtr4g 2415 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1642   wcel 1710   ccom 4772  cfv 5334  cnv 21248  cnsb 21253  CVcnmcv 21254  cims 21255 This theorem is referenced by:  imsdval  21363  imsdf  21366  cnims  21374  hhims  21859  hhssims  21960 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fv 5342  df-ims 21265
 Copyright terms: Public domain W3C validator