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Theorem lfli 33867
 Description: Property of a linear functional. (lnfnli 28787 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 33866 . . . 4 (𝑊𝑍 → (𝐺𝐹 ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))))
109simplbda 653 . . 3 ((𝑊𝑍𝐺𝐹) → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
11103adant3 1079 . 2 ((𝑊𝑍𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)))
12 oveq1 6622 . . . . . . 7 (𝑟 = 𝑅 → (𝑟 · 𝑥) = (𝑅 · 𝑥))
1312oveq1d 6630 . . . . . 6 (𝑟 = 𝑅 → ((𝑟 · 𝑥) + 𝑦) = ((𝑅 · 𝑥) + 𝑦))
1413fveq2d 6162 . . . . 5 (𝑟 = 𝑅 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑥) + 𝑦)))
15 oveq1 6622 . . . . . 6 (𝑟 = 𝑅 → (𝑟 × (𝐺𝑥)) = (𝑅 × (𝐺𝑥)))
1615oveq1d 6630 . . . . 5 (𝑟 = 𝑅 → ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑅 × (𝐺𝑥)) (𝐺𝑦)))
1714, 16eqeq12d 2636 . . . 4 (𝑟 = 𝑅 → ((𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) ↔ (𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺𝑥)) (𝐺𝑦))))
18 oveq2 6623 . . . . . . 7 (𝑥 = 𝑋 → (𝑅 · 𝑥) = (𝑅 · 𝑋))
1918oveq1d 6630 . . . . . 6 (𝑥 = 𝑋 → ((𝑅 · 𝑥) + 𝑦) = ((𝑅 · 𝑋) + 𝑦))
2019fveq2d 6162 . . . . 5 (𝑥 = 𝑋 → (𝐺‘((𝑅 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑦)))
21 fveq2 6158 . . . . . . 7 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
2221oveq2d 6631 . . . . . 6 (𝑥 = 𝑋 → (𝑅 × (𝐺𝑥)) = (𝑅 × (𝐺𝑋)))
2322oveq1d 6630 . . . . 5 (𝑥 = 𝑋 → ((𝑅 × (𝐺𝑥)) (𝐺𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑦)))
2420, 23eqeq12d 2636 . . . 4 (𝑥 = 𝑋 → ((𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺𝑥)) (𝐺𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑦))))
25 oveq2 6623 . . . . . 6 (𝑦 = 𝑌 → ((𝑅 · 𝑋) + 𝑦) = ((𝑅 · 𝑋) + 𝑌))
2625fveq2d 6162 . . . . 5 (𝑦 = 𝑌 → (𝐺‘((𝑅 · 𝑋) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑌)))
27 fveq2 6158 . . . . . 6 (𝑦 = 𝑌 → (𝐺𝑦) = (𝐺𝑌))
2827oveq2d 6631 . . . . 5 (𝑦 = 𝑌 → ((𝑅 × (𝐺𝑋)) (𝐺𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌)))
2926, 28eqeq12d 2636 . . . 4 (𝑦 = 𝑌 → ((𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌))))
3017, 24, 29rspc3v 3314 . . 3 ((𝑅𝐾𝑋𝑉𝑌𝑉) → (∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌))))
31303ad2ant3 1082 . 2 ((𝑊𝑍𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺𝑥)) (𝐺𝑦)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌))))
3211, 31mpd 15 1 ((𝑊𝑍𝐺𝐹 ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺𝑋)) (𝐺𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2908  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  .rcmulr 15882  Scalarcsca 15884   ·𝑠 cvsca 15885  LFnlclfn 33863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-lfl 33864 This theorem is referenced by:  lfl0  33871  lfladd  33872  lflsub  33873  lflmul  33874  lflnegcl  33881  lflvscl  33883  lkrlss  33901  hdmapln1  36717
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