Theorem lnolin 30773

Description: Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnoval.1	⊢ 𝑋 = (BaseSet‘𝑈)
lnoval.2	⊢ 𝑌 = (BaseSet‘𝑊)
lnoval.3	⊢ 𝐺 = ( +𝑣 ‘𝑈)
lnoval.4	⊢ 𝐻 = ( +𝑣 ‘𝑊)
lnoval.5	⊢ 𝑅 = ( ·𝑠OLD ‘𝑈)
lnoval.6	⊢ 𝑆 = ( ·𝑠OLD ‘𝑊)
lnoval.7	⊢ 𝐿 = (𝑈 LnOp 𝑊)

Assertion

Ref	Expression
lnolin	⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶)))

Proof of Theorem lnolin

Dummy variables 𝑢 𝑡 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnoval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
2		lnoval.2	. . . . 5 ⊢ 𝑌 = (BaseSet‘𝑊)
3		lnoval.3	. . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4		lnoval.4	. . . . 5 ⊢ 𝐻 = ( +𝑣 ‘𝑊)
5		lnoval.5	. . . . 5 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈)
6		lnoval.6	. . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊)
7		lnoval.7	. . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
8	1, 2, 3, 4, 5, 6, 7	islno 30772	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐿 ↔ (𝑇:𝑋⟶𝑌 ∧ ∀𝑢 ∈ ℂ ∀𝑤 ∈ 𝑋 ∀𝑡 ∈ 𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)))))
9	8	biimp3a 1471	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇:𝑋⟶𝑌 ∧ ∀𝑢 ∈ ℂ ∀𝑤 ∈ 𝑋 ∀𝑡 ∈ 𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡))))
10	9	simprd 495	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ∀𝑢 ∈ ℂ ∀𝑤 ∈ 𝑋 ∀𝑡 ∈ 𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)))
11		oveq1 7438	. . . . 5 ⊢ (𝑢 = 𝐴 → (𝑢𝑅𝑤) = (𝐴𝑅𝑤))
12	11	fvoveq1d 7453	. . . 4 ⊢ (𝑢 = 𝐴 → (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)))
13		oveq1 7438	. . . . 5 ⊢ (𝑢 = 𝐴 → (𝑢𝑆(𝑇‘𝑤)) = (𝐴𝑆(𝑇‘𝑤)))
14	13	oveq1d 7446	. . . 4 ⊢ (𝑢 = 𝐴 → ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) = ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)))
15	12, 14	eqeq12d 2753	. . 3 ⊢ (𝑢 = 𝐴 → ((𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) ↔ (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡))))
16		oveq2 7439	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴𝑅𝑤) = (𝐴𝑅𝐵))
17	16	fvoveq1d 7453	. . . 4 ⊢ (𝑤 = 𝐵 → (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)))
18		fveq2 6906	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑇‘𝑤) = (𝑇‘𝐵))
19	18	oveq2d 7447	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴𝑆(𝑇‘𝑤)) = (𝐴𝑆(𝑇‘𝐵)))
20	19	oveq1d 7446	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡)))
21	17, 20	eqeq12d 2753	. . 3 ⊢ (𝑤 = 𝐵 → ((𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡))))
22		oveq2 7439	. . . . 5 ⊢ (𝑡 = 𝐶 → ((𝐴𝑅𝐵)𝐺𝑡) = ((𝐴𝑅𝐵)𝐺𝐶))
23	22	fveq2d 6910	. . . 4 ⊢ (𝑡 = 𝐶 → (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)))
24		fveq2 6906	. . . . 5 ⊢ (𝑡 = 𝐶 → (𝑇‘𝑡) = (𝑇‘𝐶))
25	24	oveq2d 7447	. . . 4 ⊢ (𝑡 = 𝐶 → ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶)))
26	23, 25	eqeq12d 2753	. . 3 ⊢ (𝑡 = 𝐶 → ((𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶))))
27	15, 21, 26	rspc3v 3638	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑢 ∈ ℂ ∀𝑤 ∈ 𝑋 ∀𝑡 ∈ 𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶))))
28	10, 27	mpan9 506	1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  NrmCVeccnv 30603   +_𝑣 cpv 30604  BaseSetcba 30605   ·_𝑠OLD cns 30606   LnOp clno 30759

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-lno 30763

This theorem is referenced by:  lno0  30775  lnocoi  30776  lnoadd  30777  lnosub  30778  lnomul  30779