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Theorem phlsrng 19198
 Description: The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypothesis
Ref Expression
phlsrng.f Scalar
Assertion
Ref Expression
phlsrng

Proof of Theorem phlsrng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2451 . . 3
2 phlsrng.f . . 3 Scalar
3 eqid 2451 . . 3
4 eqid 2451 . . 3
5 eqid 2451 . . 3
6 eqid 2451 . . 3
71, 2, 3, 4, 5, 6isphl 19195 . 2 LMHom ringLMod
87simp2bi 1024 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   w3a 985   wceq 1444   wcel 1887  wral 2737   cmpt 4461  cfv 5582  (class class class)co 6290  cbs 15121  cstv 15192  Scalarcsca 15193  cip 15195  c0g 15338  csr 18072   LMHom clmhm 18242  clvec 18325  ringLModcrglmod 18392  cphl 19191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-nul 4534 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-iota 5546  df-fv 5590  df-ov 6293  df-phl 19193 This theorem is referenced by:  iporthcom  19202  ip0r  19204  ipdi  19207  ip2di  19208  ipassr  19213  ipassr2  19214  cphcjcl  22161
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