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Theorem lfli 38425
Description: Property of a linear functional. (lnfnli 31765 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 38424 . . . 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 499 . . 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 1129 . 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 7409 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ Β· π‘₯) = (𝑅 Β· π‘₯))
1312fvoveq1d 7424 . . . . 5 (π‘Ÿ = 𝑅 β†’ (πΊβ€˜((π‘Ÿ Β· π‘₯) + 𝑦)) = (πΊβ€˜((𝑅 Β· π‘₯) + 𝑦)))
14 oveq1 7409 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ Γ— (πΊβ€˜π‘₯)) = (𝑅 Γ— (πΊβ€˜π‘₯)))
1514oveq1d 7417 . . . . 5 (π‘Ÿ = 𝑅 β†’ ((π‘Ÿ Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) = ((𝑅 Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)))
1613, 15eqeq12d 2740 . . . 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 7410 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑅 Β· π‘₯) = (𝑅 Β· 𝑋))
1817fvoveq1d 7424 . . . . 5 (π‘₯ = 𝑋 β†’ (πΊβ€˜((𝑅 Β· π‘₯) + 𝑦)) = (πΊβ€˜((𝑅 Β· 𝑋) + 𝑦)))
19 fveq2 6882 . . . . . . 7 (π‘₯ = 𝑋 β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘‹))
2019oveq2d 7418 . . . . . 6 (π‘₯ = 𝑋 β†’ (𝑅 Γ— (πΊβ€˜π‘₯)) = (𝑅 Γ— (πΊβ€˜π‘‹)))
2120oveq1d 7417 . . . . 5 (π‘₯ = 𝑋 β†’ ((𝑅 Γ— (πΊβ€˜π‘₯)) ⨣ (πΊβ€˜π‘¦)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘¦)))
2218, 21eqeq12d 2740 . . . 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 7410 . . . . . 6 (𝑦 = π‘Œ β†’ ((𝑅 Β· 𝑋) + 𝑦) = ((𝑅 Β· 𝑋) + π‘Œ))
2423fveq2d 6886 . . . . 5 (𝑦 = π‘Œ β†’ (πΊβ€˜((𝑅 Β· 𝑋) + 𝑦)) = (πΊβ€˜((𝑅 Β· 𝑋) + π‘Œ)))
25 fveq2 6882 . . . . . 6 (𝑦 = π‘Œ β†’ (πΊβ€˜π‘¦) = (πΊβ€˜π‘Œ))
2625oveq2d 7418 . . . . 5 (𝑦 = π‘Œ β†’ ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘¦)) = ((𝑅 Γ— (πΊβ€˜π‘‹)) ⨣ (πΊβ€˜π‘Œ)))
2724, 26eqeq12d 2740 . . . 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 3620 . . 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 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‘¦)) 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 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  Basecbs 17145  +gcplusg 17198  .rcmulr 17199  Scalarcsca 17201   ·𝑠 cvsca 17202  LFnlclfn 38421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-lfl 38422
This theorem is referenced by:  lfl0  38429  lfladd  38430  lflsub  38431  lflmul  38432  lflnegcl  38439  lflvscl  38441  lkrlss  38459  hdmapln1  41271
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