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Theorem phllmhm 18965
 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 2402 . . . . 5
5 eqid 2402 . . . . 5
6 eqid 2402 . . . . 5
71, 2, 3, 4, 5, 6isphl 18961 . . . 4 LMHom ringLMod
87simp3bi 1014 . . 3 LMHom ringLMod
9 simp1 997 . . . 4 LMHom ringLMod LMHom ringLMod
109ralimi 2797 . . 3 LMHom ringLMod LMHom ringLMod
118, 10syl 17 . 2 LMHom ringLMod
12 oveq2 6286 . . . . . 6
1312mpteq2dv 4482 . . . . 5
14 phllmhm.g . . . . 5
1513, 14syl6eqr 2461 . . . 4
1615eleq1d 2471 . . 3 LMHom ringLMod LMHom ringLMod
1716rspccva 3159 . 2 LMHom ringLMod LMHom ringLMod
1811, 17sylan 469 1 LMHom ringLMod
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   w3a 974   wceq 1405   wcel 1842  wral 2754   cmpt 4453  cfv 5569  (class class class)co 6278  cbs 14841  cstv 14911  Scalarcsca 14912  cip 14914  c0g 15054  csr 17813   LMHom clmhm 17985  clvec 18068  ringLModcrglmod 18135  cphl 18957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-iota 5533  df-fv 5577  df-ov 6281  df-phl 18959 This theorem is referenced by:  ipcl  18966  ip0l  18969  ipdir  18972  ipass  18978
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