Theorem islno 30958

Description: The predicate "is 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
islno	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐿 ↔ (𝑇:𝑋⟶𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧)))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑈   𝑥,𝑊,𝑦,𝑧   𝑦,𝑋,𝑧   𝑥,𝑇,𝑦,𝑧

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)   𝑆(𝑥,𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)   𝐻(𝑥,𝑦,𝑧)   𝐿(𝑥,𝑦,𝑧)   𝑋(𝑥)   𝑌(𝑥,𝑦,𝑧)

Proof of Theorem islno

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnoval.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
2		lnoval.2	. . . 4 ⊢ 𝑌 = (BaseSet‘𝑊)
3		lnoval.3	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4		lnoval.4	. . . 4 ⊢ 𝐻 = ( +𝑣 ‘𝑊)
5		lnoval.5	. . . 4 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈)
6		lnoval.6	. . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊)
7		lnoval.7	. . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
8	1, 2, 3, 4, 5, 6, 7	lnoval 30957	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑤 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧))})
9	8	eleq2d 2850	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐿 ↔ 𝑇 ∈ {𝑤 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧))}))
10		fveq1 6868	. . . . . . 7 ⊢ (𝑤 = 𝑇 → (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)))
11		fveq1 6868	. . . . . . . . 9 ⊢ (𝑤 = 𝑇 → (𝑤‘𝑦) = (𝑇‘𝑦))
12	11	oveq2d 7414	. . . . . . . 8 ⊢ (𝑤 = 𝑇 → (𝑥𝑆(𝑤‘𝑦)) = (𝑥𝑆(𝑇‘𝑦)))
13		fveq1 6868	. . . . . . . 8 ⊢ (𝑤 = 𝑇 → (𝑤‘𝑧) = (𝑇‘𝑧))
14	12, 13	oveq12d 7416	. . . . . . 7 ⊢ (𝑤 = 𝑇 → ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧)))
15	10, 14	eqeq12d 2780	. . . . . 6 ⊢ (𝑤 = 𝑇 → ((𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧)) ↔ (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧))))
16	15	2ralbidv 3228	. . . . 5 ⊢ (𝑤 = 𝑇 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧))))
17	16	ralbidv 3187	. . . 4 ⊢ (𝑤 = 𝑇 → (∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧))))
18	17	elrab 3652	. . 3 ⊢ (𝑇 ∈ {𝑤 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧))} ↔ (𝑇 ∈ (𝑌 ↑m 𝑋) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧))))
19	2	fvexi 6883	. . . . 5 ⊢ 𝑌 ∈ V
20	1	fvexi 6883	. . . . 5 ⊢ 𝑋 ∈ V
21	19, 20	elmap 8855	. . . 4 ⊢ (𝑇 ∈ (𝑌 ↑m 𝑋) ↔ 𝑇:𝑋⟶𝑌)
22	21	anbi1i 633	. . 3 ⊢ ((𝑇 ∈ (𝑌 ↑m 𝑋) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧))) ↔ (𝑇:𝑋⟶𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧))))
23	18, 22	bitri 277	. 2 ⊢ (𝑇 ∈ {𝑤 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧))} ↔ (𝑇:𝑋⟶𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧))))
24	9, 23	bitrdi 289	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐿 ↔ (𝑇:𝑋⟶𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398   ↑_m cmap 8810  ℂcc 11073  NrmCVeccnv 30789   +_𝑣 cpv 30790  BaseSetcba 30791   ·_𝑠OLD cns 30792   LnOp clno 30945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-map 8812  df-lno 30949

This theorem is referenced by:  lnolin  30959  lnof  30960  lnocoi  30962  0lno  30995  ipblnfi  31060