Theorem lnolin 30512

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 30511	. . . 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 1465	. . 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 7411	. . . . 5 ⊢ (𝑢 = 𝐴 → (𝑢𝑅𝑤) = (𝐴𝑅𝑤))
12	11	fvoveq1d 7426	. . . 4 ⊢ (𝑢 = 𝐴 → (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)))
13		oveq1 7411	. . . . 5 ⊢ (𝑢 = 𝐴 → (𝑢𝑆(𝑇‘𝑤)) = (𝐴𝑆(𝑇‘𝑤)))
14	13	oveq1d 7419	. . . 4 ⊢ (𝑢 = 𝐴 → ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) = ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)))
15	12, 14	eqeq12d 2742	. . 3 ⊢ (𝑢 = 𝐴 → ((𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) ↔ (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡))))
16		oveq2 7412	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴𝑅𝑤) = (𝐴𝑅𝐵))
17	16	fvoveq1d 7426	. . . 4 ⊢ (𝑤 = 𝐵 → (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)))
18		fveq2 6884	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑇‘𝑤) = (𝑇‘𝐵))
19	18	oveq2d 7420	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴𝑆(𝑇‘𝑤)) = (𝐴𝑆(𝑇‘𝐵)))
20	19	oveq1d 7419	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡)))
21	17, 20	eqeq12d 2742	. . 3 ⊢ (𝑤 = 𝐵 → ((𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡))))
22		oveq2 7412	. . . . 5 ⊢ (𝑡 = 𝐶 → ((𝐴𝑅𝐵)𝐺𝑡) = ((𝐴𝑅𝐵)𝐺𝐶))
23	22	fveq2d 6888	. . . 4 ⊢ (𝑡 = 𝐶 → (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)))
24		fveq2 6884	. . . . 5 ⊢ (𝑡 = 𝐶 → (𝑇‘𝑡) = (𝑇‘𝐶))
25	24	oveq2d 7420	. . . 4 ⊢ (𝑡 = 𝐶 → ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶)))
26	23, 25	eqeq12d 2742	. . 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 3622	. 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 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404  ℂcc 11107  NrmCVeccnv 30342   +_𝑣 cpv 30343  BaseSetcba 30344   ·_𝑠OLD cns 30345   LnOp clno 30498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-lno 30502

This theorem is referenced by:  lno0  30514  lnocoi  30515  lnoadd  30516  lnosub  30517  lnomul  30518