Theorem lnopl 31816

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 31760	. . . . . 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 7376	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝐴 ·ℎ 𝑦))
4	3	fvoveq1d 7391	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)))
5		oveq1 7376	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ (𝑇‘𝑦)) = (𝐴 ·ℎ (𝑇‘𝑦)))
6	5	oveq1d 7384	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)) = ((𝐴 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
7	4, 6	eqeq12d 2745	. . . . . 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 7377	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ 𝑦) = (𝐴 ·ℎ 𝐵))
9	8	fvoveq1d 7391	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)))
10		fveq2 6840	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
11	10	oveq2d 7385	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ (𝑇‘𝑦)) = (𝐴 ·ℎ (𝑇‘𝐵)))
12	11	oveq1d 7384	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝑧)))
13	9, 12	eqeq12d 2745	. . . . . 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 7377	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → ((𝐴 ·ℎ 𝐵) +ℎ 𝑧) = ((𝐴 ·ℎ 𝐵) +ℎ 𝐶))
15	14	fveq2d 6844	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)))
16		fveq2 6840	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑇‘𝑧) = (𝑇‘𝐶))
17	16	oveq2d 7385	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝑧)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶)))
18	15, 17	eqeq12d 2745	. . . . . 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 3601	. . . . 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 1120	. . 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 1540   ∈ wcel 2109  ∀wral 3044  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  ℂcc 11042   ℋchba 30821   +_ℎ cva 30822   ·_ℎ csm 30823  LinOpclo 30849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-hilex 30901

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-lnop 31743

This theorem is referenced by:  lnop0  31868  lnopmul  31869  lnopli  31870