Theorem lnopl 31943

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 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶)))

Proof of Theorem lnopl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ellnop 31887	. . . . . 6 ⊢ (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))))
2	1	simprbi 496	. . . . 5 ⊢ (𝑇 ∈ LinOp → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
3		oveq1 7438	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝐴 ·ℎ 𝑦))
4	3	fvoveq1d 7453	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)))
5		oveq1 7438	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ (𝑇‘𝑦)) = (𝐴 ·ℎ (𝑇‘𝑦)))
6	5	oveq1d 7446	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)) = ((𝐴 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
7	4, 6	eqeq12d 2751	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)) ↔ (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝐴 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧))))
8		oveq2 7439	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ 𝑦) = (𝐴 ·ℎ 𝐵))
9	8	fvoveq1d 7453	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)))
10		fveq2 6907	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
11	10	oveq2d 7447	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ (𝑇‘𝑦)) = (𝐴 ·ℎ (𝑇‘𝐵)))
12	11	oveq1d 7446	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝑧)))
13	9, 12	eqeq12d 2751	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝐴 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)) ↔ (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝑧))))
14		oveq2 7439	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → ((𝐴 ·ℎ 𝐵) +ℎ 𝑧) = ((𝐴 ·ℎ 𝐵) +ℎ 𝐶))
15	14	fveq2d 6911	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)))
16		fveq2 6907	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑇‘𝑧) = (𝑇‘𝐶))
17	16	oveq2d 7447	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝑧)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶)))
18	15, 17	eqeq12d 2751	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝑧)) ↔ (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶))))
19	7, 13, 18	rspc3v 3638	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶))))
20	2, 19	syl5 34	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇 ∈ LinOp → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶))))
21	20	3expb 1119	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇 ∈ LinOp → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶))))
22	21	impcom 407	. 2 ⊢ ((𝑇 ∈ LinOp ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ))) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶)))
23	22	anassrs 467	1 ⊢ (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℂcc 11151   ℋchba 30948   +_ℎ cva 30949   ·_ℎ csm 30950  LinOpclo 30976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-hilex 31028

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-lnop 31870

This theorem is referenced by:  lnop0  31995  lnopmul  31996  lnopli  31997