Theorem islno 30556

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 30555	. . 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 2814	. 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 6890	. . . . . . 7 ⊢ (𝑤 = 𝑇 → (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)))
11		fveq1 6890	. . . . . . . . 9 ⊢ (𝑤 = 𝑇 → (𝑤‘𝑦) = (𝑇‘𝑦))
12	11	oveq2d 7430	. . . . . . . 8 ⊢ (𝑤 = 𝑇 → (𝑥𝑆(𝑤‘𝑦)) = (𝑥𝑆(𝑇‘𝑦)))
13		fveq1 6890	. . . . . . . 8 ⊢ (𝑤 = 𝑇 → (𝑤‘𝑧) = (𝑇‘𝑧))
14	12, 13	oveq12d 7432	. . . . . . 7 ⊢ (𝑤 = 𝑇 → ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧)))
15	10, 14	eqeq12d 2743	. . . . . 6 ⊢ (𝑤 = 𝑇 → ((𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤‘𝑦))𝐻(𝑤‘𝑧)) ↔ (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇‘𝑦))𝐻(𝑇‘𝑧))))
16	15	2ralbidv 3213	. . . . 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 3172	. . . 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 3680	. . 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 6905	. . . . 5 ⊢ 𝑌 ∈ V
20	1	fvexi 6905	. . . . 5 ⊢ 𝑋 ∈ V
21	19, 20	elmap 8883	. . . 4 ⊢ (𝑇 ∈ (𝑌 ↑m 𝑋) ↔ 𝑇:𝑋⟶𝑌)
22	21	anbi1i 623	. . 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 275	. 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 287	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 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056  {crab 3427  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ↑_m cmap 8838  ℂcc 11130  NrmCVeccnv 30387   +_𝑣 cpv 30388  BaseSetcba 30389   ·_𝑠OLD cns 30390   LnOp clno 30543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-lno 30547

This theorem is referenced by:  lnolin  30557  lnof  30558  lnocoi  30560  0lno  30593  ipblnfi  30658