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Theorem lnopl 31676
Description: Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnopl (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))

Proof of Theorem lnopl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnop 31620 . . . . . 6 (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
21simprbi 496 . . . . 5 (𝑇 ∈ LinOp → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
3 oveq1 7412 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · 𝑦) = (𝐴 · 𝑦))
43fvoveq1d 7427 . . . . . . 7 (𝑥 = 𝐴 → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝑦) + 𝑧)))
5 oveq1 7412 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 · (𝑇𝑦)) = (𝐴 · (𝑇𝑦)))
65oveq1d 7420 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)))
74, 6eqeq12d 2742 . . . . . 6 (𝑥 = 𝐴 → ((𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧))))
8 oveq2 7413 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
98fvoveq1d 7427 . . . . . . 7 (𝑦 = 𝐵 → (𝑇‘((𝐴 · 𝑦) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝑧)))
10 fveq2 6885 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
1110oveq2d 7421 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 · (𝑇𝑦)) = (𝐴 · (𝑇𝐵)))
1211oveq1d 7420 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)))
139, 12eqeq12d 2742 . . . . . 6 (𝑦 = 𝐵 → ((𝑇‘((𝐴 · 𝑦) + 𝑧)) = ((𝐴 · (𝑇𝑦)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧))))
14 oveq2 7413 . . . . . . . 8 (𝑧 = 𝐶 → ((𝐴 · 𝐵) + 𝑧) = ((𝐴 · 𝐵) + 𝐶))
1514fveq2d 6889 . . . . . . 7 (𝑧 = 𝐶 → (𝑇‘((𝐴 · 𝐵) + 𝑧)) = (𝑇‘((𝐴 · 𝐵) + 𝐶)))
16 fveq2 6885 . . . . . . . 8 (𝑧 = 𝐶 → (𝑇𝑧) = (𝑇𝐶))
1716oveq2d 7421 . . . . . . 7 (𝑧 = 𝐶 → ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
1815, 17eqeq12d 2742 . . . . . 6 (𝑧 = 𝐶 → ((𝑇‘((𝐴 · 𝐵) + 𝑧)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝑧)) ↔ (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
197, 13, 18rspc3v 3622 . . . . 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 ∈ LinOp → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
21203expb 1117 . . 3 ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇 ∈ LinOp → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶))))
2221impcom 407 . 2 ((𝑇 ∈ LinOp ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ))) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
2322anassrs 467 1 (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wf 6533  cfv 6537  (class class class)co 7405  cc 11110  chba 30681   + cva 30682   · csm 30683  LinOpclo 30709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-hilex 30761
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-lnop 31603
This theorem is referenced by:  lnop0  31728  lnopmul  31729  lnopli  31730
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