Theorem lnolin 27609

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 27608	. . . 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 1432	. . 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 479	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ∀𝑢 ∈ ℂ ∀𝑤 ∈ 𝑋 ∀𝑡 ∈ 𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)))
11		oveq1 6657	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢𝑅𝑤) = (𝐴𝑅𝑤))
12	11	oveq1d 6665	. . . . 5 ⊢ (𝑢 = 𝐴 → ((𝑢𝑅𝑤)𝐺𝑡) = ((𝐴𝑅𝑤)𝐺𝑡))
13	12	fveq2d 6195	. . . 4 ⊢ (𝑢 = 𝐴 → (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)))
14		oveq1 6657	. . . . 5 ⊢ (𝑢 = 𝐴 → (𝑢𝑆(𝑇‘𝑤)) = (𝐴𝑆(𝑇‘𝑤)))
15	14	oveq1d 6665	. . . 4 ⊢ (𝑢 = 𝐴 → ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) = ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)))
16	13, 15	eqeq12d 2637	. . 3 ⊢ (𝑢 = 𝐴 → ((𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) ↔ (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡))))
17		oveq2 6658	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝐴𝑅𝑤) = (𝐴𝑅𝐵))
18	17	oveq1d 6665	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝐴𝑅𝑤)𝐺𝑡) = ((𝐴𝑅𝐵)𝐺𝑡))
19	18	fveq2d 6195	. . . 4 ⊢ (𝑤 = 𝐵 → (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)))
20		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑇‘𝑤) = (𝑇‘𝐵))
21	20	oveq2d 6666	. . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴𝑆(𝑇‘𝑤)) = (𝐴𝑆(𝑇‘𝐵)))
22	21	oveq1d 6665	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡)))
23	19, 22	eqeq12d 2637	. . 3 ⊢ (𝑤 = 𝐵 → ((𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝑤))𝐻(𝑇‘𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡))))
24		oveq2 6658	. . . . 5 ⊢ (𝑡 = 𝐶 → ((𝐴𝑅𝐵)𝐺𝑡) = ((𝐴𝑅𝐵)𝐺𝐶))
25	24	fveq2d 6195	. . . 4 ⊢ (𝑡 = 𝐶 → (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)))
26		fveq2 6191	. . . . 5 ⊢ (𝑡 = 𝐶 → (𝑇‘𝑡) = (𝑇‘𝐶))
27	26	oveq2d 6666	. . . 4 ⊢ (𝑡 = 𝐶 → ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶)))
28	25, 27	eqeq12d 2637	. . 3 ⊢ (𝑡 = 𝐶 → ((𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇‘𝐵))𝐻(𝑇‘𝐶))))
29	16, 23, 28	rspc3v 3325	. 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))))
30	10, 29	mpan9 486	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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  NrmCVeccnv 27439   +_𝑣 cpv 27440  BaseSetcba 27441   ·_𝑠OLD cns 27442   LnOp clno 27595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-lno 27599

This theorem is referenced by:  lno0  27611  lnocoi  27612  lnoadd  27613  lnosub  27614  lnomul  27615