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Theorem isphg 25555
 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 25551 . . 3
21elin2 3694 . 2
3 rneq 5234 . . . . . 6
4 isphg.1 . . . . . 6
53, 4syl6eqr 2526 . . . . 5
6 oveq 6301 . . . . . . . . . 10
76fveq2d 5876 . . . . . . . . 9
87oveq1d 6310 . . . . . . . 8
9 oveq 6301 . . . . . . . . . 10
109fveq2d 5876 . . . . . . . . 9
1110oveq1d 6310 . . . . . . . 8
128, 11oveq12d 6313 . . . . . . 7
1312eqeq1d 2469 . . . . . 6
145, 13raleqbidv 3077 . . . . 5
155, 14raleqbidv 3077 . . . 4
16 oveq 6301 . . . . . . . . . 10
1716oveq2d 6311 . . . . . . . . 9
1817fveq2d 5876 . . . . . . . 8
1918oveq1d 6310 . . . . . . 7
2019oveq2d 6311 . . . . . 6
2120eqeq1d 2469 . . . . 5
22212ralbidv 2911 . . . 4
23 fveq1 5871 . . . . . . . 8
2423oveq1d 6310 . . . . . . 7
25 fveq1 5871 . . . . . . . 8
2625oveq1d 6310 . . . . . . 7
2724, 26oveq12d 6313 . . . . . 6
28 fveq1 5871 . . . . . . . . 9
2928oveq1d 6310 . . . . . . . 8
30 fveq1 5871 . . . . . . . . 9
3130oveq1d 6310 . . . . . . . 8
3229, 31oveq12d 6313 . . . . . . 7
3332oveq2d 6311 . . . . . 6
3427, 33eqeq12d 2489 . . . . 5
35342ralbidv 2911 . . . 4
3615, 22, 35eloprabg 6385 . . 3
3736anbi2d 703 . 2
382, 37syl5bb 257 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973   wceq 1379   wcel 1767  wral 2817  cop 4039   crn 5006  cfv 5594  (class class class)co 6295  coprab 6296  c1 9505   caddc 9507   cmul 9509  cneg 9818  c2 10597  cexp 12146  cnv 25300  ccphlo 25550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-cnv 5013  df-dm 5015  df-rn 5016  df-iota 5557  df-fv 5602  df-ov 6298  df-oprab 6299  df-ph 25551 This theorem is referenced by:  cncph  25557  isph  25560  phpar  25562  hhph  25918
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