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Theorem phllmhm 16863
 Description: The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
phlsrng.f Scalar
phllmhm.h
phllmhm.v
phllmhm.g
Assertion
Ref Expression
phllmhm LMHom ringLMod
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem phllmhm
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 phllmhm.v . . . . 5
2 phlsrng.f . . . . 5 Scalar
3 phllmhm.h . . . . 5
4 eqid 2436 . . . . 5
5 eqid 2436 . . . . 5
6 eqid 2436 . . . . 5
71, 2, 3, 4, 5, 6isphl 16859 . . . 4 LMHom ringLMod
87simp3bi 974 . . 3 LMHom ringLMod
9 simp1 957 . . . 4 LMHom ringLMod LMHom ringLMod
109ralimi 2781 . . 3 LMHom ringLMod LMHom ringLMod
118, 10syl 16 . 2 LMHom ringLMod
12 oveq2 6089 . . . . . 6
1312mpteq2dv 4296 . . . . 5
14 phllmhm.g . . . . 5
1513, 14syl6eqr 2486 . . . 4
1615eleq1d 2502 . . 3 LMHom ringLMod LMHom ringLMod
1716rspccva 3051 . 2 LMHom ringLMod LMHom ringLMod
1811, 17sylan 458 1 LMHom ringLMod
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705   cmpt 4266  cfv 5454  (class class class)co 6081  cbs 13469  cstv 13531  Scalarcsca 13532  cip 13534  c0g 13723  csr 15932   LMHom clmhm 16095  clvec 16174  ringLModcrglmod 16241  cphl 16855 This theorem is referenced by:  ipcl  16864  ip0l  16867  ipdir  16870  ipass  16876 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338 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 2285  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-sbc 3162  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-mpt 4268  df-iota 5418  df-fv 5462  df-ov 6084  df-phl 16857
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