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Theorem phlsrng 16863
 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 2437 . . 3
2 phlsrng.f . . 3 Scalar
3 eqid 2437 . . 3
4 eqid 2437 . . 3
5 eqid 2437 . . 3
6 eqid 2437 . . 3
71, 2, 3, 4, 5, 6isphl 16860 . 2 LMHom ringLMod
87simp2bi 974 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 937   wceq 1653   wcel 1726  wral 2706   cmpt 4267  cfv 5455  (class class class)co 6082  cbs 13470  cstv 13532  Scalarcsca 13533  cip 13535  c0g 13724  csr 15933   LMHom clmhm 16096  clvec 16175  ringLModcrglmod 16242  cphl 16856 This theorem is referenced by:  iporthcom  16867  ip0r  16869  ipdi  16872  ip2di  16873  ipassr  16878  ipassr2  16879  cphcjcl  19147 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-nul 4339 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-iota 5419  df-fv 5463  df-ov 6085  df-phl 16858
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