Theorem lnfnl 32192

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 32144	. . . . . 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 502	. . . . 5 ⊢ (𝑇 ∈ LinFn → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))
3		oveq1 7407	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝐴 ·ℎ 𝑦))
4	3	fvoveq1d 7422	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)))
5		oveq1 7407	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 · (𝑇‘𝑦)) = (𝐴 · (𝑇‘𝑦)))
6	5	oveq1d 7415	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)) = ((𝐴 · (𝑇‘𝑦)) + (𝑇‘𝑧)))
7	4, 6	eqeq12d 2781	. . . . . 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 7408	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ 𝑦) = (𝐴 ·ℎ 𝐵))
9	8	fvoveq1d 7422	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)))
10		fveq2 6871	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
11	10	oveq2d 7416	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · (𝑇‘𝑦)) = (𝐴 · (𝑇‘𝐵)))
12	11	oveq1d 7415	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 · (𝑇‘𝑦)) + (𝑇‘𝑧)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝑧)))
13	9, 12	eqeq12d 2781	. . . . . 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 7408	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → ((𝐴 ·ℎ 𝐵) +ℎ 𝑧) = ((𝐴 ·ℎ 𝐵) +ℎ 𝐶))
15	14	fveq2d 6875	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)))
16		fveq2 6871	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑇‘𝑧) = (𝑇‘𝐶))
17	16	oveq2d 7416	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝑧)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶)))
18	15, 17	eqeq12d 2781	. . . . . 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 3600	. . . . 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 35	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))))
21	20	3expb 1136	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))))
22	21	impcom 412	. 2 ⊢ ((𝑇 ∈ LinFn ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ))) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶)))
23	22	anassrs 472	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 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  ℂcc 11086   + caddc 11091   · cmul 11093   ℋchba 31180   +_ℎ cva 31181   ·_ℎ csm 31182  LinFnclf 31215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-hilex 31260

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-map 8814  df-lnfn 32109

This theorem is referenced by:  lnfnli  32301