Theorem lnfnl 32534

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 32486	. . . . . 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 503	. . . . 5 ⊢ (𝑇 ∈ LinFn → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))
3		oveq1 7427	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝐴 ·ℎ 𝑦))
4	3	fvoveq1d 7442	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)))
5		oveq1 7427	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 · (𝑇‘𝑦)) = (𝐴 · (𝑇‘𝑦)))
6	5	oveq1d 7435	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)) = ((𝐴 · (𝑇‘𝑦)) + (𝑇‘𝑧)))
7	4, 6	eqeq12d 2776	. . . . . 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 7428	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ 𝑦) = (𝐴 ·ℎ 𝐵))
9	8	fvoveq1d 7442	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑇‘((𝐴 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)))
10		fveq2 6885	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
11	10	oveq2d 7436	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · (𝑇‘𝑦)) = (𝐴 · (𝑇‘𝐵)))
12	11	oveq1d 7435	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 · (𝑇‘𝑦)) + (𝑇‘𝑧)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝑧)))
13	9, 12	eqeq12d 2776	. . . . . 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 7428	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → ((𝐴 ·ℎ 𝐵) +ℎ 𝑧) = ((𝐴 ·ℎ 𝐵) +ℎ 𝐶))
15	14	fveq2d 6889	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝑧)) = (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)))
16		fveq2 6885	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑇‘𝑧) = (𝑇‘𝐶))
17	16	oveq2d 7436	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝑧)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶)))
18	15, 17	eqeq12d 2776	. . . . . 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 3591	. . . . 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 1138	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇 ∈ LinFn → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))))
22	21	impcom 413	. 2 ⊢ ((𝑇 ∈ LinFn ∧ (𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ))) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶)))
23	22	anassrs 473	1 ⊢ (((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  ℂcc 11199   + caddc 11204   · cmul 11206   ℋchba 31522   +_ℎ cva 31523   ·_ℎ csm 31524  LinFnclf 31557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11257  ax-hilex 31602

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8850  df-lnfn 32451

This theorem is used by:  lnfnli  32643