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Theorem lfli 37919
Description: Property of a linear functional. (lnfnli 31280 analog.) (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lflset.v 𝑉 = (Baseβ€˜π‘Š)
lflset.a + = (+gβ€˜π‘Š)
lflset.d 𝐷 = (Scalarβ€˜π‘Š)
lflset.s Β· = ( ·𝑠 β€˜π‘Š)
lflset.k 𝐾 = (Baseβ€˜π·)
lflset.p ⨣ = (+gβ€˜π·)
lflset.t Γ— = (.rβ€˜π·)
lflset.f 𝐹 = (LFnlβ€˜π‘Š)
Assertion
Ref Expression
lfli ((π‘Š ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (πΊβ€˜((𝑅 Β· 𝑋) + π‘Œ)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘Œ)))
Proof of Theorem lfli
Dummy variables π‘Ÿ π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lflset.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
2 lflset.a . . . . 5 + = (+gβ€˜π‘Š)
3 lflset.d . . . . 5 𝐷 = (Scalarβ€˜π‘Š)
4 lflset.s . . . . 5 Β· = ( ·𝑠 β€˜π‘Š)
5 lflset.k . . . . 5 𝐾 = (Baseβ€˜π·)
6 lflset.p . . . . 5 ⨣ = (+gβ€˜π·)
7 lflset.t . . . . 5 Γ— = (.rβ€˜π·)
8 lflset.f . . . . 5 𝐹 = (LFnlβ€˜π‘Š)
91, 2, 3, 4, 5, 6, 7, 8islfl 37918 . . . 4 (π‘Š ∈ 𝑍 β†’ (𝐺 ∈ 𝐹 ↔ (𝐺:π‘‰βŸΆπΎ ∧ 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‘¦)))))
109simplbda 500 . . 3 ((π‘Š ∈ 𝑍 ∧ 𝐺 ∈ 𝐹) 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‘¦)))
11103adant3 1132 . 2 ((π‘Š ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) 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 oveq1 7412 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ Β· π‘₯) = (𝑅 Β· π‘₯))
1312fvoveq1d 7427 . . . . 5 (π‘Ÿ = 𝑅 β†’ (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = (πΊβ€˜((𝑅 Β· π‘₯) + 𝑦)))
14 oveq1 7412 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ Γ— (πΊβ€˜π‘₯)) = (𝑅 Γ— (πΊβ€˜π‘₯)))
1514oveq1d 7420 . . . . 5 (π‘Ÿ = 𝑅 β†’ ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) = ((𝑅 Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)))
1613, 15eqeq12d 2748 . . . 4 (π‘Ÿ = 𝑅 β†’ ((πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) ↔ (πΊβ€˜((𝑅 Β· π‘₯) + 𝑦)) = ((𝑅 Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦))))
17 oveq2 7413 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑅 Β· π‘₯) = (𝑅 Β· 𝑋))
1817fvoveq1d 7427 . . . . 5 (π‘₯ = 𝑋 β†’ (πΊβ€˜((𝑅 Β· π‘₯) + 𝑦)) = (πΊβ€˜((𝑅 Β· 𝑋) + 𝑦)))
19 fveq2 6888 . . . . . . 7 (π‘₯ = 𝑋 β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘‹))
2019oveq2d 7421 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑅 Γ— (πΊβ€˜π‘₯)) = (𝑅 Γ— (πΊβ€˜π‘‹)))
2120oveq1d 7420 . . . . 5 (π‘₯ = 𝑋 β†’ ((𝑅 Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘¦)))
2218, 21eqeq12d 2748 . . . 4 (π‘₯ = 𝑋 β†’ ((πΊβ€˜((𝑅 Β· π‘₯) + 𝑦)) = ((𝑅 Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) ↔ (πΊβ€˜((𝑅 Β· 𝑋) + 𝑦)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘¦))))
23 oveq2 7413 . . . . . 6 (𝑦 = π‘Œ β†’ ((𝑅 Β· 𝑋) + 𝑦) = ((𝑅 Β· 𝑋) + π‘Œ))
2423fveq2d 6892 . . . . 5 (𝑦 = π‘Œ β†’ (πΊβ€˜((𝑅 Β· 𝑋) + 𝑦)) = (πΊβ€˜((𝑅 Β· 𝑋) + π‘Œ)))
25 fveq2 6888 . . . . . 6 (𝑦 = π‘Œ β†’ (πΊβ€˜π‘¦) = (πΊβ€˜π‘Œ))
2625oveq2d 7421 . . . . 5 (𝑦 = π‘Œ β†’ ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘¦)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘Œ)))
2724, 26eqeq12d 2748 . . . 4 (𝑦 = π‘Œ β†’ ((πΊβ€˜((𝑅 Β· 𝑋) + 𝑦)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘¦)) ↔ (πΊβ€˜((𝑅 Β· 𝑋) + π‘Œ)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘Œ))))
2816, 22, 27rspc3v 3626 . . 3 ((𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) 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‘¦)) β†’ (πΊβ€˜((𝑅 Β· 𝑋) + π‘Œ)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘Œ))))
29283ad2ant3 1135 . 2 ((π‘Š ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) 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‘¦)) β†’ (πΊβ€˜((𝑅 Β· 𝑋) + π‘Œ)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘Œ))))
3011, 29mpd 15 1 ((π‘Š ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (πΊβ€˜((𝑅 Β· 𝑋) + π‘Œ)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  +gcplusg 17193  .rcmulr 17194  Scalarcsca 17196   ·𝑠 cvsca 17197  LFnlclfn 37915
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-lfl 37916
This theorem is referenced by:  lfl0  37923  lfladd  37924  lflsub  37925  lflmul  37926  lflnegcl  37933  lflvscl  37935  lkrlss  37953  hdmapln1  40765
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