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Theorem lfli 39043
Description: Property of a linear functional. (lnfnli 32069 analog.) (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lflset.v 𝑉 = (Base‘𝑊)
lflset.a + = (+g𝑊)
lflset.d 𝐷 = (Scalar‘𝑊)
lflset.s · = ( ·𝑠𝑊)
lflset.k 𝐾 = (Base‘𝐷)
lflset.p = (+g𝐷)
lflset.t × = (.r𝐷)
lflset.f 𝐹 = (LFnl‘𝑊)
Assertion
Ref Expression
lfli ((𝑊𝑍𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌)))

Proof of Theorem lfli
Dummy variables 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lflset.v . . . . 5 𝑉 = (Base‘𝑊)
2 lflset.a . . . . 5 + = (+g𝑊)
3 lflset.d . . . . 5 𝐷 = (Scalar‘𝑊)
4 lflset.s . . . . 5 · = ( ·𝑠𝑊)
5 lflset.k . . . . 5 𝐾 = (Base‘𝐷)
6 lflset.p . . . . 5 = (+g𝐷)
7 lflset.t . . . . 5 × = (.r𝐷)
8 lflset.f . . . . 5 𝐹 = (LFnl‘𝑊)
91, 2, 3, 4, 5, 6, 7, 8islfl 39042 . . . 4 (𝑊𝑍 → (𝐺𝐹 ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))))
109simplbda 499 . . 3 ((𝑊𝑍𝐺𝐹) → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
11103adant3 1131 . 2 ((𝑊𝑍𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
12 oveq1 7438 . . . . . 6 (𝑟 = 𝑅 → (𝑟 · 𝑥) = (𝑅 · 𝑥))
1312fvoveq1d 7453 . . . . 5 (𝑟 = 𝑅 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑥) + 𝑦)))
14 oveq1 7438 . . . . . 6 (𝑟 = 𝑅 → (𝑟 × (𝐺𝑥)) = (𝑅 × (𝐺𝑥)))
1514oveq1d 7446 . . . . 5 (𝑟 = 𝑅 → ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑅 × (𝐺𝑥)) (𝐺𝑦)))
1613, 15eqeq12d 2751 . . . 4 (𝑟 = 𝑅 → ((𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) ↔ (𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺𝑥)) (𝐺𝑦))))
17 oveq2 7439 . . . . . 6 (𝑥 = 𝑋 → (𝑅 · 𝑥) = (𝑅 · 𝑋))
1817fvoveq1d 7453 . . . . 5 (𝑥 = 𝑋 → (𝐺‘((𝑅 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑦)))
19 fveq2 6907 . . . . . . 7 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
2019oveq2d 7447 . . . . . 6 (𝑥 = 𝑋 → (𝑅 × (𝐺𝑥)) = (𝑅 × (𝐺𝑋)))
2120oveq1d 7446 . . . . 5 (𝑥 = 𝑋 → ((𝑅 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑦)))
2218, 21eqeq12d 2751 . . . 4 (𝑥 = 𝑋 → ((𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺𝑥)) (𝐺𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑦))))
23 oveq2 7439 . . . . . 6 (𝑦 = 𝑌 → ((𝑅 · 𝑋) + 𝑦) = ((𝑅 · 𝑋) + 𝑌))
2423fveq2d 6911 . . . . 5 (𝑦 = 𝑌 → (𝐺‘((𝑅 · 𝑋) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑌)))
25 fveq2 6907 . . . . . 6 (𝑦 = 𝑌 → (𝐺𝑦) = (𝐺𝑌))
2625oveq2d 7447 . . . . 5 (𝑦 = 𝑌 → ((𝑅 × (𝐺𝑋)) (𝐺𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌)))
2724, 26eqeq12d 2751 . . . 4 (𝑦 = 𝑌 → ((𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌))))
2816, 22, 27rspc3v 3638 . . 3 ((𝑅𝐾𝑋𝑉𝑌𝑉) → (∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌))))
29283ad2ant3 1134 . 2 ((𝑊𝑍𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌))))
3011, 29mpd 15 1 ((𝑊𝑍𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  LFnlclfn 39039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-lfl 39040
This theorem is referenced by:  lfl0  39047  lfladd  39048  lflsub  39049  lflmul  39050  lflnegcl  39057  lflvscl  39059  lkrlss  39077  hdmapln1  41889
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