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Theorem imsval 24011
 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 5688 . . . 4 CV CV
2 fveq2 5688 . . . 4
31, 2coeq12d 5000 . . 3 CV CV
4 df-ims 23914 . . 3 CV
5 fvex 5698 . . . 4 CV
6 fvex 5698 . . . 4
75, 6coex 6528 . . 3 CV
83, 4, 7fvmpt 5771 . 2 CV
9 imsval.8 . 2
10 imsval.6 . . 3 CV
11 imsval.3 . . 3
1210, 11coeq12i 4999 . 2 CV
138, 9, 123eqtr4g 2498 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1364   wcel 1761   ccom 4840  cfv 5415  cnv 23897  cnsb 23902  CVcnmcv 23903  cims 23904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fv 5423  df-ims 23914 This theorem is referenced by:  imsdval  24012  imsdf  24015  cnims  24023  hhims  24509  hhssims  24611
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