Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islfld Structured version   Visualization version   GIF version

Theorem islfld 37980
Description: Properties that determine a linear functional. TODO: use this in place of islfl 37978 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
islfld.v (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘Š))
islfld.a (πœ‘ β†’ + = (+gβ€˜π‘Š))
islfld.d (πœ‘ β†’ 𝐷 = (Scalarβ€˜π‘Š))
islfld.s (πœ‘ β†’ Β· = ( ·𝑠 β€˜π‘Š))
islfld.k (πœ‘ β†’ 𝐾 = (Baseβ€˜π·))
islfld.p (πœ‘ β†’ ⨣ = (+gβ€˜π·))
islfld.t (πœ‘ β†’ Γ— = (.rβ€˜π·))
islfld.f (πœ‘ β†’ 𝐹 = (LFnlβ€˜π‘Š))
islfld.u (πœ‘ β†’ 𝐺:π‘‰βŸΆπΎ)
islfld.l ((πœ‘ ∧ (π‘Ÿ ∈ 𝐾 ∧ π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)))
islfld.w (πœ‘ β†’ π‘Š ∈ 𝑋)
Assertion
Ref Expression
islfld (πœ‘ β†’ 𝐺 ∈ 𝐹)
Distinct variable groups:   π‘₯,π‘Ÿ,𝑦,𝐺   𝐾,π‘Ÿ,π‘₯,𝑦   π‘₯,𝑉,𝑦   π‘Š,π‘Ÿ,π‘₯,𝑦   πœ‘,π‘Ÿ,π‘₯,𝑦
Allowed substitution hints:   𝐷(π‘₯,𝑦,π‘Ÿ)   + (π‘₯,𝑦,π‘Ÿ)   ⨣ (π‘₯,𝑦,π‘Ÿ)   Β· (π‘₯,𝑦,π‘Ÿ)   Γ— (π‘₯,𝑦,π‘Ÿ)   𝐹(π‘₯,𝑦,π‘Ÿ)   𝑉(π‘Ÿ)   𝑋(π‘₯,𝑦,π‘Ÿ)

Proof of Theorem islfld
StepHypRef Expression
1 islfld.w . . 3 (πœ‘ β†’ π‘Š ∈ 𝑋)
2 islfld.u . . . 4 (πœ‘ β†’ 𝐺:π‘‰βŸΆπΎ)
3 islfld.v . . . . 5 (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘Š))
4 islfld.k . . . . . 6 (πœ‘ β†’ 𝐾 = (Baseβ€˜π·))
5 islfld.d . . . . . . 7 (πœ‘ β†’ 𝐷 = (Scalarβ€˜π‘Š))
65fveq2d 6896 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π·) = (Baseβ€˜(Scalarβ€˜π‘Š)))
74, 6eqtrd 2773 . . . . 5 (πœ‘ β†’ 𝐾 = (Baseβ€˜(Scalarβ€˜π‘Š)))
83, 7feq23d 6713 . . . 4 (πœ‘ β†’ (𝐺:π‘‰βŸΆπΎ ↔ 𝐺:(Baseβ€˜π‘Š)⟢(Baseβ€˜(Scalarβ€˜π‘Š))))
92, 8mpbid 231 . . 3 (πœ‘ β†’ 𝐺:(Baseβ€˜π‘Š)⟢(Baseβ€˜(Scalarβ€˜π‘Š)))
10 islfld.l . . . . 5 ((πœ‘ ∧ (π‘Ÿ ∈ 𝐾 ∧ π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)))
1110ralrimivvva 3204 . . . 4 (πœ‘ β†’ βˆ€π‘Ÿ ∈ 𝐾 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)))
12 islfld.a . . . . . . . . . 10 (πœ‘ β†’ + = (+gβ€˜π‘Š))
13 islfld.s . . . . . . . . . . 11 (πœ‘ β†’ Β· = ( ·𝑠 β€˜π‘Š))
1413oveqd 7426 . . . . . . . . . 10 (πœ‘ β†’ (π‘Ÿ Β· π‘₯) = (π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯))
15 eqidd 2734 . . . . . . . . . 10 (πœ‘ β†’ 𝑦 = 𝑦)
1612, 14, 15oveq123d 7430 . . . . . . . . 9 (πœ‘ β†’ ((π‘Ÿ Β· π‘₯) + 𝑦) = ((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦))
1716fveq2d 6896 . . . . . . . 8 (πœ‘ β†’ (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = (πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)))
18 islfld.p . . . . . . . . . 10 (πœ‘ β†’ ⨣ = (+gβ€˜π·))
195fveq2d 6896 . . . . . . . . . 10 (πœ‘ β†’ (+gβ€˜π·) = (+gβ€˜(Scalarβ€˜π‘Š)))
2018, 19eqtrd 2773 . . . . . . . . 9 (πœ‘ β†’ ⨣ = (+gβ€˜(Scalarβ€˜π‘Š)))
21 islfld.t . . . . . . . . . . 11 (πœ‘ β†’ Γ— = (.rβ€˜π·))
225fveq2d 6896 . . . . . . . . . . 11 (πœ‘ β†’ (.rβ€˜π·) = (.rβ€˜(Scalarβ€˜π‘Š)))
2321, 22eqtrd 2773 . . . . . . . . . 10 (πœ‘ β†’ Γ— = (.rβ€˜(Scalarβ€˜π‘Š)))
2423oveqd 7426 . . . . . . . . 9 (πœ‘ β†’ (π‘Ÿ Γ— (πΊβ€˜π‘₯)) = (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯)))
25 eqidd 2734 . . . . . . . . 9 (πœ‘ β†’ (πΊβ€˜π‘¦) = (πΊβ€˜π‘¦))
2620, 24, 25oveq123d 7430 . . . . . . . 8 (πœ‘ β†’ ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦)))
2717, 26eqeq12d 2749 . . . . . . 7 (πœ‘ β†’ ((πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) ↔ (πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦))))
283, 27raleqbidv 3343 . . . . . 6 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝑉 (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) ↔ βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦))))
293, 28raleqbidv 3343 . . . . 5 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) ↔ βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦))))
307, 29raleqbidv 3343 . . . 4 (πœ‘ β†’ (βˆ€π‘Ÿ ∈ 𝐾 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) ↔ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦))))
3111, 30mpbid 231 . . 3 (πœ‘ β†’ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦)))
32 eqid 2733 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
33 eqid 2733 . . . . 5 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
34 eqid 2733 . . . . 5 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
35 eqid 2733 . . . . 5 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
36 eqid 2733 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
37 eqid 2733 . . . . 5 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
38 eqid 2733 . . . . 5 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
39 eqid 2733 . . . . 5 (LFnlβ€˜π‘Š) = (LFnlβ€˜π‘Š)
4032, 33, 34, 35, 36, 37, 38, 39islfl 37978 . . . 4 (π‘Š ∈ 𝑋 β†’ (𝐺 ∈ (LFnlβ€˜π‘Š) ↔ (𝐺:(Baseβ€˜π‘Š)⟢(Baseβ€˜(Scalarβ€˜π‘Š)) ∧ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦)))))
4140biimpar 479 . . 3 ((π‘Š ∈ 𝑋 ∧ (𝐺:(Baseβ€˜π‘Š)⟢(Baseβ€˜(Scalarβ€˜π‘Š)) ∧ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(πΊβ€˜((π‘Ÿ( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)) = ((π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘₯))(+gβ€˜(Scalarβ€˜π‘Š))(πΊβ€˜π‘¦)))) β†’ 𝐺 ∈ (LFnlβ€˜π‘Š))
421, 9, 31, 41syl12anc 836 . 2 (πœ‘ β†’ 𝐺 ∈ (LFnlβ€˜π‘Š))
43 islfld.f . 2 (πœ‘ β†’ 𝐹 = (LFnlβ€˜π‘Š))
4442, 43eleqtrrd 2837 1 (πœ‘ β†’ 𝐺 ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201  LFnlclfn 37975
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-lfl 37976
This theorem is referenced by:  lflvscl  37995
  Copyright terms: Public domain W3C validator