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Theorem cphnm 19148
 Description: The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
nmsq.v
nmsq.h
nmsq.n
Assertion
Ref Expression
cphnm

Proof of Theorem cphnm
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nmsq.v . . . 4
2 nmsq.h . . . 4
3 nmsq.n . . . 4
41, 2, 3cphnmfval 19147 . . 3
54fveq1d 5722 . 2
6 oveq12 6082 . . . . 5
76anidms 627 . . . 4
87fveq2d 5724 . . 3
9 eqid 2435 . . 3
10 fvex 5734 . . 3
118, 9, 10fvmpt 5798 . 2
125, 11sylan9eq 2487 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725   cmpt 4258  cfv 5446  (class class class)co 6073  csqr 12030  cbs 13461  cip 13526  cnm 18616  ccph 19121 This theorem is referenced by:  nmsq  19149 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-cph 19123
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