Theorem lnfnl 31897

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

Proof of Theorem lnfnl

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

Step	Hyp	Ref	Expression
1		ellnfn 31849	. . . . . 6 ⊢ (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧))))
2	1	simprbi 496	. . . . 5 ⊢ (𝑇 ∈ LinFn → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))
3		oveq1 7421	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝐴 ·ℎ 𝑦))
4	3	fvoveq1d 7436	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)))
5		oveq1 7421	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 · (𝑇‘𝑦)) = (𝐴 · (𝑇‘𝑦)))
6	5	oveq1d 7429	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)) = ((𝐴 · (𝑇‘𝑦)) + (𝑇‘𝑧)))
7	4, 6	eqeq12d 2750	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)) ↔ (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝐴 · (𝑇‘𝑦)) + (𝑇‘𝑧))))
8		oveq2 7422	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ 𝑦) = (𝐴 ·ℎ 𝐵))
9	8	fvoveq1d 7436	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)))
10		fveq2 6887	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
11	10	oveq2d 7430	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · (𝑇‘𝑦)) = (𝐴 · (𝑇‘𝐵)))
12	11	oveq1d 7429	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 · (𝑇‘𝑦)) + (𝑇‘𝑧)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝑧)))
13	9, 12	eqeq12d 2750	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝐴 · (𝑇‘𝑦)) + (𝑇‘𝑧)) ↔ (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝑧))))
14		oveq2 7422	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → ((𝐴 ·ℎ 𝐵) +ℎ 𝑧) = ((𝐴 ·ℎ 𝐵) +ℎ 𝐶))
15	14	fveq2d 6891	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)))
16		fveq2 6887	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑇‘𝑧) = (𝑇‘𝐶))
17	16	oveq2d 7430	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝑧)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶)))
18	15, 17	eqeq12d 2750	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝑧)) ↔ (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))))
19	7, 13, 18	rspc3v 3622	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))))
20	2, 19	syl5 34	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))))
21	20	3expb 1120	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))))
22	21	impcom 407	. 2 ⊢ ((𝑇 ∈ LinFn ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ))) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶)))
23	22	anassrs 467	1 ⊢ (((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  ℂcc 11136   + caddc 11141   · cmul 11143   ℋchba 30885   +_ℎ cva 30886   ·_ℎ csm 30887  LinFnclf 30920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-cnex 11194  ax-hilex 30965

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8851  df-lnfn 31814

This theorem is referenced by:  lnfnli  32006