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Theorem isphg 22318
 Description: The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is , the scalar product is , and the norm is . An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
isphg.1
Assertion
Ref Expression
isphg
Distinct variable groups:   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem isphg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ph 22314 . . 3
21elin2 3531 . 2
3 rneq 5095 . . . . . 6
4 isphg.1 . . . . . 6
53, 4syl6eqr 2486 . . . . 5
6 oveq 6087 . . . . . . . . . 10
76fveq2d 5732 . . . . . . . . 9
87oveq1d 6096 . . . . . . . 8
9 oveq 6087 . . . . . . . . . 10
109fveq2d 5732 . . . . . . . . 9
1110oveq1d 6096 . . . . . . . 8
128, 11oveq12d 6099 . . . . . . 7
1312eqeq1d 2444 . . . . . 6
145, 13raleqbidv 2916 . . . . 5
155, 14raleqbidv 2916 . . . 4
16 oveq 6087 . . . . . . . . . 10
1716oveq2d 6097 . . . . . . . . 9
1817fveq2d 5732 . . . . . . . 8
1918oveq1d 6096 . . . . . . 7
2019oveq2d 6097 . . . . . 6
2120eqeq1d 2444 . . . . 5
22212ralbidv 2747 . . . 4
23 fveq1 5727 . . . . . . . 8
2423oveq1d 6096 . . . . . . 7
25 fveq1 5727 . . . . . . . 8
2625oveq1d 6096 . . . . . . 7
2724, 26oveq12d 6099 . . . . . 6
28 fveq1 5727 . . . . . . . . 9
2928oveq1d 6096 . . . . . . . 8
30 fveq1 5727 . . . . . . . . 9
3130oveq1d 6096 . . . . . . . 8
3229, 31oveq12d 6099 . . . . . . 7
3332oveq2d 6097 . . . . . 6
3427, 33eqeq12d 2450 . . . . 5
35342ralbidv 2747 . . . 4
3615, 22, 35eloprabg 6161 . . 3
3736anbi2d 685 . 2
382, 37syl5bb 249 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  cop 3817   crn 4879  cfv 5454  (class class class)co 6081  coprab 6082  c1 8991   caddc 8993   cmul 8995  cneg 9292  c2 10049  cexp 11382  cnv 22063  ccphlo 22313 This theorem is referenced by:  cncph  22320  isph  22323  phpar  22325  hhph  22680 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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889  df-iota 5418  df-fv 5462  df-ov 6084  df-oprab 6085  df-ph 22314
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