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Theorem phllmhm 21059
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 2733 . . . . 5 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
5 eqid 2733 . . . . 5 (*π‘Ÿβ€˜πΉ) = (*π‘Ÿβ€˜πΉ)
6 eqid 2733 . . . . 5 (0gβ€˜πΉ) = (0gβ€˜πΉ)
71, 2, 3, 4, 5, 6isphl 21055 . . . 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 1148 . . 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 1137 . . . 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 7369 . . . . . 6 (𝑦 = 𝐴 β†’ (π‘₯ , 𝑦) = (π‘₯ , 𝐴))
1312mpteq2dv 5211 . . . . 5 (𝑦 = 𝐴 β†’ (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) = (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝐴)))
14 phllmhm.g . . . . 5 𝐺 = (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝐴))
1513, 14eqtr4di 2791 . . . 4 (𝑦 = 𝐴 β†’ (π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) = 𝐺)
1615eleq1d 2819 . . 3 (𝑦 = 𝐴 β†’ ((π‘₯ ∈ 𝑉 ↦ (π‘₯ , 𝑦)) ∈ (π‘Š LMHom (ringLModβ€˜πΉ)) ↔ 𝐺 ∈ (π‘Š LMHom (ringLModβ€˜πΉ))))
1716rspccva 3582 . 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 581 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   ↦ cmpt 5192  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  *π‘Ÿcstv 17143  Scalarcsca 17144  Β·π‘–cip 17146  0gc0g 17329  *-Ringcsr 20346   LMHom clmhm 20524  LVecclvec 20607  ringLModcrglmod 20675  PreHilcphl 21051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-iota 6452  df-fv 6508  df-ov 7364  df-phl 21053
This theorem is referenced by:  ipcl  21060  ip0l  21063  ipdir  21066  ipass  21072
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