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Theorem lfli 37552
Description: Property of a linear functional. (lnfnli 31024 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 37551 . . . 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 501 . . 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 1133 . 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 7369 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ Β· π‘₯) = (𝑅 Β· π‘₯))
1312fvoveq1d 7384 . . . . 5 (π‘Ÿ = 𝑅 β†’ (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = (πΊβ€˜((𝑅 Β· π‘₯) + 𝑦)))
14 oveq1 7369 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ Γ— (πΊβ€˜π‘₯)) = (𝑅 Γ— (πΊβ€˜π‘₯)))
1514oveq1d 7377 . . . . 5 (π‘Ÿ = 𝑅 β†’ ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) = ((𝑅 Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)))
1613, 15eqeq12d 2753 . . . 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 7370 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑅 Β· π‘₯) = (𝑅 Β· 𝑋))
1817fvoveq1d 7384 . . . . 5 (π‘₯ = 𝑋 β†’ (πΊβ€˜((𝑅 Β· π‘₯) + 𝑦)) = (πΊβ€˜((𝑅 Β· 𝑋) + 𝑦)))
19 fveq2 6847 . . . . . . 7 (π‘₯ = 𝑋 β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘‹))
2019oveq2d 7378 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑅 Γ— (πΊβ€˜π‘₯)) = (𝑅 Γ— (πΊβ€˜π‘‹)))
2120oveq1d 7377 . . . . 5 (π‘₯ = 𝑋 β†’ ((𝑅 Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘¦)))
2218, 21eqeq12d 2753 . . . 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 7370 . . . . . 6 (𝑦 = π‘Œ β†’ ((𝑅 Β· 𝑋) + 𝑦) = ((𝑅 Β· 𝑋) + π‘Œ))
2423fveq2d 6851 . . . . 5 (𝑦 = π‘Œ β†’ (πΊβ€˜((𝑅 Β· 𝑋) + 𝑦)) = (πΊβ€˜((𝑅 Β· 𝑋) + π‘Œ)))
25 fveq2 6847 . . . . . 6 (𝑦 = π‘Œ β†’ (πΊβ€˜π‘¦) = (πΊβ€˜π‘Œ))
2625oveq2d 7378 . . . . 5 (𝑦 = π‘Œ β†’ ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘¦)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘Œ)))
2724, 26eqeq12d 2753 . . . 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 3596 . . 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 1136 . 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 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  +gcplusg 17140  .rcmulr 17141  Scalarcsca 17143   ·𝑠 cvsca 17144  LFnlclfn 37548
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-lfl 37549
This theorem is referenced by:  lfl0  37556  lfladd  37557  lflsub  37558  lflmul  37559  lflnegcl  37566  lflvscl  37568  lkrlss  37586  hdmapln1  40398
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