MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phllmhm Structured version   Visualization version   GIF version
Theorem phllmhm 21184
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 𝐹 = (Scalarβ€˜π‘Š)
phllmhm.h , = (Β·π‘–β€˜π‘Š)
phllmhm.v 𝑉 = (Baseβ€˜π‘Š)
phllmhm.g 𝐺 = (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝐴))
Assertion
Ref Expression
phllmhm ((π‘Š ∈ PreHil ∧ 𝐴 ∈ 𝑉) β†’ 𝐺 ∈ (π‘Š LMHom (ringLModβ€˜πΉ)))
Distinct variable groups:   π‘₯,𝐴   π‘₯, ,   π‘₯,𝑉   π‘₯,π‘Š
Allowed substitution hints:   𝐹(π‘₯)   𝐺(π‘₯)
Proof of Theorem phllmhm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 phllmhm.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
2 phlsrng.f . . . . 5 𝐹 = (Scalarβ€˜π‘Š)
3 phllmhm.h . . . . 5 , = (Β·π‘–β€˜π‘Š)
4 eqid 2732 . . . . 5 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
5 eqid 2732 . . . . 5 (*π‘Ÿβ€˜πΉ) = (*π‘Ÿβ€˜πΉ)
6 eqid 2732 . . . . 5 (0gβ€˜πΉ) = (0gβ€˜πΉ)
71, 2, 3, 4, 5, 6isphl 21180 . . . 4 (π‘Š ∈ PreHil ↔ (π‘Š ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ βˆ€π‘¦ ∈ 𝑉 ((π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((𝑦 , 𝑦) = (0gβ€˜πΉ) β†’ 𝑦 = (0gβ€˜π‘Š)) ∧ βˆ€π‘₯ ∈ 𝑉 ((*π‘Ÿβ€˜πΉ)β€˜(𝑦 , π‘₯)) = (π‘₯ , 𝑦))))
87simp3bi 1147 . . 3 (π‘Š ∈ PreHil β†’ βˆ€π‘¦ ∈ 𝑉 ((π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((𝑦 , 𝑦) = (0gβ€˜πΉ) β†’ 𝑦 = (0gβ€˜π‘Š)) ∧ βˆ€π‘₯ ∈ 𝑉 ((*π‘Ÿβ€˜πΉ)β€˜(𝑦 , π‘₯)) = (π‘₯ , 𝑦)))
9 simp1 1136 . . . 4 (((π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((𝑦 , 𝑦) = (0gβ€˜πΉ) β†’ 𝑦 = (0gβ€˜π‘Š)) ∧ βˆ€π‘₯ ∈ 𝑉 ((*π‘Ÿβ€˜πΉ)β€˜(𝑦 , π‘₯)) = (π‘₯ , 𝑦)) β†’ (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)))
109ralimi 3083 . . 3 (βˆ€π‘¦ ∈ 𝑉 ((π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ ((𝑦 , 𝑦) = (0gβ€˜πΉ) β†’ 𝑦 = (0gβ€˜π‘Š)) ∧ βˆ€π‘₯ ∈ 𝑉 ((*π‘Ÿβ€˜πΉ)β€˜(𝑦 , π‘₯)) = (π‘₯ , 𝑦)) β†’ βˆ€π‘¦ ∈ 𝑉 (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)))
118, 10syl 17 . 2 (π‘Š ∈ PreHil β†’ βˆ€π‘¦ ∈ 𝑉 (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)))
12 oveq2 7416 . . . . . 6 (𝑦 = 𝐴 β†’ (π‘₯ , 𝑦) = (π‘₯ , 𝐴))
1312mpteq2dv 5250 . . . . 5 (𝑦 = 𝐴 β†’ (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) = (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝐴)))
14 phllmhm.g . . . . 5 𝐺 = (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝐴))
1513, 14eqtr4di 2790 . . . 4 (𝑦 = 𝐴 β†’ (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) = 𝐺)
1615eleq1d 2818 . . 3 (𝑦 = 𝐴 β†’ ((π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ↔ 𝐺 ∈ (π‘Š LMHom (ringLModβ€˜πΉ))))
1716rspccva 3611 . 2 ((βˆ€π‘¦ ∈ 𝑉 (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ∧ 𝐴 ∈ 𝑉) β†’ 𝐺 ∈ (π‘Š LMHom (ringLModβ€˜πΉ)))
1811, 17sylan 580 1 ((π‘Š ∈ PreHil ∧ 𝐴 ∈ 𝑉) β†’ 𝐺 ∈ (π‘Š LMHom (ringLModβ€˜πΉ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  *π‘Ÿcstv 17198  Scalarcsca 17199  Β·π‘–cip 17201  0gc0g 17384  *-Ringcsr 20451   LMHom clmhm 20629  LVecclvec 20712  ringLModcrglmod 20781  PreHilcphl 21176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6495  df-fv 6551  df-ov 7411  df-phl 21178
This theorem is referenced by:  ipcl  21185  ip0l  21188  ipdir  21191  ipass  21197
  Copyright terms: Public domain W3C validator