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Theorem lnfnl 29124
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 29076 . . . . . 6 (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
21simprbi 478 . . . . 5 (𝑇 ∈ LinFn → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
3 oveq1 6799 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
43fvoveq1d 6814 . . . . . . 7 (𝑥 = 𝐴 → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝑦) + 𝑧)))
5 oveq1 6799 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · (𝑇𝑦)) = (𝐴 · (𝑇𝑦)))
65oveq1d 6807 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)))
74, 6eqeq12d 2785 . . . . . 6 (𝑥 = 𝐴 → ((𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧))))
8 oveq2 6800 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
98fvoveq1d 6814 . . . . . . 7 (𝑦 = 𝐵 → (𝑇‘((𝐴 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝑧)))
10 fveq2 6332 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
1110oveq2d 6808 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · (𝑇𝑦)) = (𝐴 · (𝑇𝐵)))
1211oveq1d 6807 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)))
139, 12eqeq12d 2785 . . . . . 6 (𝑦 = 𝐵 → ((𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧))))
14 oveq2 6800 . . . . . . . 8 (𝑧 = 𝐶 → ((𝐴 · 𝐵) + 𝑧) = ((𝐴 · 𝐵) + 𝐶))
1514fveq2d 6336 . . . . . . 7 (𝑧 = 𝐶 → (𝑇‘((𝐴 · 𝐵) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝐶)))
16 fveq2 6332 . . . . . . . 8 (𝑧 = 𝐶 → (𝑇𝑧) = (𝑇𝐶))
1716oveq2d 6808 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
1815, 17eqeq12d 2785 . . . . . 6 (𝑧 = 𝐶 → ((𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
197, 13, 18rspc3v 3473 . . . . 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 1112 . . 3 ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
2221impcom 394 . 2 ((𝑇 ∈ LinFn ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ))) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
2322anassrs 458 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 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  wf 6027  cfv 6031  (class class class)co 6792  cc 10135   + caddc 10140   · cmul 10142  chil 28110   + cva 28111   · csm 28112  LinFnclf 28145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-hilex 28190
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-lnfn 29041
This theorem is referenced by:  lnfnli  29233
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