HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  lnopcoi Structured version   Visualization version   GIF version

Theorem lnopcoi 29707
Description: The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnopco.1 𝑆 ∈ LinOp
lnopco.2 𝑇 ∈ LinOp
Assertion
Ref Expression
lnopcoi (𝑆𝑇) ∈ LinOp

Proof of Theorem lnopcoi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnopco.1 . . . 4 𝑆 ∈ LinOp
21lnopfi 29673 . . 3 𝑆: ℋ⟶ ℋ
3 lnopco.2 . . . 4 𝑇 ∈ LinOp
43lnopfi 29673 . . 3 𝑇: ℋ⟶ ℋ
52, 4hocofi 29470 . 2 (𝑆𝑇): ℋ⟶ ℋ
63lnopli 29672 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
76fveq2d 6667 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
8 id 22 . . . . . . . 8 (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
94ffvelrni 6842 . . . . . . . 8 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
104ffvelrni 6842 . . . . . . . 8 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
111lnopli 29672 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
128, 9, 10, 11syl3an 1152 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
137, 12eqtrd 2853 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
14133expa 1110 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
15 hvmulcl 28717 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
16 hvaddcl 28716 . . . . . . 7 (((𝑥 · 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
1715, 16sylan 580 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
182, 4hocoi 29468 . . . . . 6 (((𝑥 · 𝑦) + 𝑧) ∈ ℋ → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))))
1917, 18syl 17 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))))
202, 4hocoi 29468 . . . . . . . 8 (𝑦 ∈ ℋ → ((𝑆𝑇)‘𝑦) = (𝑆‘(𝑇𝑦)))
2120oveq2d 7161 . . . . . . 7 (𝑦 ∈ ℋ → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
2221adantl 482 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
232, 4hocoi 29468 . . . . . 6 (𝑧 ∈ ℋ → ((𝑆𝑇)‘𝑧) = (𝑆‘(𝑇𝑧)))
2422, 23oveqan12d 7164 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
2514, 19, 243eqtr4d 2863 . . . 4 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
26253impa 1102 . . 3 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
2726rgen3 3201 . 2 𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))
28 ellnop 29562 . 2 ((𝑆𝑇) ∈ LinOp ↔ ((𝑆𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))))
295, 27, 28mpbir2an 707 1 (𝑆𝑇) ∈ LinOp
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  chba 28623   + cva 28624   · csm 28625  LinOpclo 28651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-hilex 28703  ax-hfvadd 28704  ax-hfvmul 28709
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-lnop 29545
This theorem is referenced by:  lnopco0i  29708  nmopcoi  29799  bdopcoi  29802  nmopcoadj0i  29807
  Copyright terms: Public domain W3C validator