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Theorem lnfnl 29758
 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 29710 . . . . . 6 (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
21simprbi 500 . . . . 5 (𝑇 ∈ LinFn → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
3 oveq1 7152 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
43fvoveq1d 7167 . . . . . . 7 (𝑥 = 𝐴 → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝑦) + 𝑧)))
5 oveq1 7152 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · (𝑇𝑦)) = (𝐴 · (𝑇𝑦)))
65oveq1d 7160 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)))
74, 6eqeq12d 2814 . . . . . 6 (𝑥 = 𝐴 → ((𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧))))
8 oveq2 7153 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
98fvoveq1d 7167 . . . . . . 7 (𝑦 = 𝐵 → (𝑇‘((𝐴 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝑧)))
10 fveq2 6655 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
1110oveq2d 7161 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · (𝑇𝑦)) = (𝐴 · (𝑇𝐵)))
1211oveq1d 7160 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)))
139, 12eqeq12d 2814 . . . . . 6 (𝑦 = 𝐵 → ((𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧))))
14 oveq2 7153 . . . . . . . 8 (𝑧 = 𝐶 → ((𝐴 · 𝐵) + 𝑧) = ((𝐴 · 𝐵) + 𝐶))
1514fveq2d 6659 . . . . . . 7 (𝑧 = 𝐶 → (𝑇‘((𝐴 · 𝐵) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝐶)))
16 fveq2 6655 . . . . . . . 8 (𝑧 = 𝐶 → (𝑇𝑧) = (𝑇𝐶))
1716oveq2d 7161 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
1815, 17eqeq12d 2814 . . . . . 6 (𝑧 = 𝐶 → ((𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
197, 13, 18rspc3v 3585 . . . . 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))))
202, 19syl5 34 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
21203expb 1117 . . 3 ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
2221impcom 411 . 2 ((𝑇 ∈ LinFn ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ))) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
2322anassrs 471 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  ℂcc 10542   + caddc 10547   · cmul 10549   ℋchba 28746   +ℎ cva 28747   ·ℎ csm 28748  LinFnclf 28781 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 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-hilex 28826 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 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8409  df-lnfn 29675 This theorem is referenced by:  lnfnli  29867
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