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Theorem lnfnl 28174
 Description: Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnfnl (((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))

Proof of Theorem lnfnl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnfn 28126 . . . . . 6 (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
21simprbi 479 . . . . 5 (𝑇 ∈ LinFn → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
3 oveq1 6556 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
43oveq1d 6564 . . . . . . . 8 (𝑥 = 𝐴 → ((𝑥 · 𝑦) + 𝑧) = ((𝐴 · 𝑦) + 𝑧))
54fveq2d 6107 . . . . . . 7 (𝑥 = 𝐴 → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝑦) + 𝑧)))
6 oveq1 6556 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · (𝑇𝑦)) = (𝐴 · (𝑇𝑦)))
76oveq1d 6564 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)))
85, 7eqeq12d 2625 . . . . . 6 (𝑥 = 𝐴 → ((𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧))))
9 oveq2 6557 . . . . . . . . 9 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
109oveq1d 6564 . . . . . . . 8 (𝑦 = 𝐵 → ((𝐴 · 𝑦) + 𝑧) = ((𝐴 · 𝐵) + 𝑧))
1110fveq2d 6107 . . . . . . 7 (𝑦 = 𝐵 → (𝑇‘((𝐴 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝑧)))
12 fveq2 6103 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
1312oveq2d 6565 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · (𝑇𝑦)) = (𝐴 · (𝑇𝐵)))
1413oveq1d 6564 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)))
1511, 14eqeq12d 2625 . . . . . 6 (𝑦 = 𝐵 → ((𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧))))
16 oveq2 6557 . . . . . . . 8 (𝑧 = 𝐶 → ((𝐴 · 𝐵) + 𝑧) = ((𝐴 · 𝐵) + 𝐶))
1716fveq2d 6107 . . . . . . 7 (𝑧 = 𝐶 → (𝑇‘((𝐴 · 𝐵) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝐶)))
18 fveq2 6103 . . . . . . . 8 (𝑧 = 𝐶 → (𝑇𝑧) = (𝑇𝐶))
1918oveq2d 6565 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
2017, 19eqeq12d 2625 . . . . . 6 (𝑧 = 𝐶 → ((𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
218, 15, 20rspc3v 3296 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
222, 21syl5 33 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
23223expb 1258 . . 3 ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
2423impcom 445 . 2 ((𝑇 ∈ LinFn ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ))) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
2524anassrs 678 1 (((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   + caddc 9818   · cmul 9820   ℋchil 27160   +ℎ cva 27161   ·ℎ csm 27162  LinFnclf 27195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-lnfn 28091 This theorem is referenced by:  lnfnli  28283
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