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Theorem lfli 36354
 Description: Property of a linear functional. (lnfnli 29823 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 36353 . . . 4 (𝑊𝑍 → (𝐺𝐹 ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))))
109simplbda 503 . . 3 ((𝑊𝑍𝐺𝐹) → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
11103adant3 1129 . 2 ((𝑊𝑍𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
12 oveq1 7142 . . . . . 6 (𝑟 = 𝑅 → (𝑟 · 𝑥) = (𝑅 · 𝑥))
1312fvoveq1d 7157 . . . . 5 (𝑟 = 𝑅 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑥) + 𝑦)))
14 oveq1 7142 . . . . . 6 (𝑟 = 𝑅 → (𝑟 × (𝐺𝑥)) = (𝑅 × (𝐺𝑥)))
1514oveq1d 7150 . . . . 5 (𝑟 = 𝑅 → ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑅 × (𝐺𝑥)) (𝐺𝑦)))
1613, 15eqeq12d 2814 . . . 4 (𝑟 = 𝑅 → ((𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) ↔ (𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺𝑥)) (𝐺𝑦))))
17 oveq2 7143 . . . . . 6 (𝑥 = 𝑋 → (𝑅 · 𝑥) = (𝑅 · 𝑋))
1817fvoveq1d 7157 . . . . 5 (𝑥 = 𝑋 → (𝐺‘((𝑅 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑦)))
19 fveq2 6645 . . . . . . 7 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
2019oveq2d 7151 . . . . . 6 (𝑥 = 𝑋 → (𝑅 × (𝐺𝑥)) = (𝑅 × (𝐺𝑋)))
2120oveq1d 7150 . . . . 5 (𝑥 = 𝑋 → ((𝑅 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑦)))
2218, 21eqeq12d 2814 . . . 4 (𝑥 = 𝑋 → ((𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺𝑥)) (𝐺𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑦))))
23 oveq2 7143 . . . . . 6 (𝑦 = 𝑌 → ((𝑅 · 𝑋) + 𝑦) = ((𝑅 · 𝑋) + 𝑌))
2423fveq2d 6649 . . . . 5 (𝑦 = 𝑌 → (𝐺‘((𝑅 · 𝑋) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑌)))
25 fveq2 6645 . . . . . 6 (𝑦 = 𝑌 → (𝐺𝑦) = (𝐺𝑌))
2625oveq2d 7151 . . . . 5 (𝑦 = 𝑌 → ((𝑅 × (𝐺𝑋)) (𝐺𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌)))
2724, 26eqeq12d 2814 . . . 4 (𝑦 = 𝑌 → ((𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌))))
2816, 22, 27rspc3v 3584 . . 3 ((𝑅𝐾𝑋𝑉𝑌𝑉) → (∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌))))
29283ad2ant3 1132 . 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 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  Scalarcsca 16560   ·𝑠 cvsca 16561  LFnlclfn 36350 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-lfl 36351 This theorem is referenced by:  lfl0  36358  lfladd  36359  lflsub  36360  lflmul  36361  lflnegcl  36368  lflvscl  36370  lkrlss  36388  hdmapln1  39199
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