Theorem lnoval 28225

Description: The set of linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-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
lnoval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑡 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))})

Distinct variable groups:   𝑥,𝑡,𝑦,𝑧,𝑈   𝑡,𝑊,𝑥,𝑦,𝑧   𝑡,𝑋,𝑦,𝑧   𝑡,𝑌   𝑡,𝐺   𝑡,𝑅   𝑡,𝐻   𝑡,𝑆

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

Proof of Theorem lnoval

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

Step	Hyp	Ref	Expression
1		lnoval.7	. 2 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
2		fveq2 6543	. . . . . 6 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		lnoval.1	. . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	syl6eqr 2849	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5	4	oveq2d 7037	. . . 4 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑𝑚 𝑋))
6		fveq2 6543	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
7		lnoval.3	. . . . . . . . . 10 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
8	6, 7	syl6eqr 2849	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺)
9		fveq2 6543	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = ( ·𝑠OLD ‘𝑈))
10		lnoval.5	. . . . . . . . . . 11 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑈)
11	9, 10	syl6eqr 2849	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = 𝑅)
12	11	oveqd 7038	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑥( ·𝑠OLD ‘𝑢)𝑦) = (𝑥𝑅𝑦))
13		eqidd 2796	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → 𝑧 = 𝑧)
14	8, 12, 13	oveq123d 7042	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((𝑥( ·𝑠OLD ‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧) = ((𝑥𝑅𝑦)𝐺𝑧))
15	14	fveqeq2d 6551	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑡‘((𝑥( ·𝑠OLD ‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))))
16	4, 15	raleqbidv 3361	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD ‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))))
17	4, 16	raleqbidv 3361	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD ‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))))
18	17	ralbidv 3164	. . . 4 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD ‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))))
19	5, 18	rabeqbidv 3430	. . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD ‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))} = {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))})
20		fveq2 6543	. . . . . 6 ⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
21		lnoval.2	. . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊)
22	20, 21	syl6eqr 2849	. . . . 5 ⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
23	22	oveq1d 7036	. . . 4 ⊢ (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑𝑚 𝑋) = (𝑌 ↑𝑚 𝑋))
24		fveq2 6543	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑊))
25		lnoval.4	. . . . . . . . 9 ⊢ 𝐻 = ( +𝑣 ‘𝑊)
26	24, 25	syl6eqr 2849	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐻)
27		fveq2 6543	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑊))
28		lnoval.6	. . . . . . . . . 10 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑊)
29	27, 28	syl6eqr 2849	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = 𝑆)
30	29	oveqd 7038	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦)) = (𝑥𝑆(𝑡‘𝑦)))
31		eqidd 2796	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑡‘𝑧) = (𝑡‘𝑧))
32	26, 30, 31	oveq123d 7042	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧)))
33	32	eqeq2d 2805	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))))
34	33	2ralbidv 3166	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))))
35	34	ralbidv 3164	. . . 4 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))))
36	23, 35	rabeqbidv 3430	. . 3 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))} = {𝑡 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))})
37		df-lno 28217	. . 3 ⊢ LnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD ‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD ‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))})
38		ovex 7053	. . . 4 ⊢ (𝑌 ↑𝑚 𝑋) ∈ V
39	38	rabex 5131	. . 3 ⊢ {𝑡 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))} ∈ V
40	19, 36, 37, 39	ovmpo 7171	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 LnOp 𝑊) = {𝑡 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))})
41	1, 40	syl5eq 2843	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑡 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡‘𝑦))𝐻(𝑡‘𝑧))})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105  {crab 3109  ‘cfv 6230  (class class class)co 7021   ↑_𝑚 cmap 8261  ℂcc 10386  NrmCVeccnv 28057   +_𝑣 cpv 28058  BaseSetcba 28059   ·_𝑠OLD cns 28060   LnOp clno 28213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pr 5226

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-iota 6194  df-fun 6232  df-fv 6238  df-ov 7024  df-oprab 7025  df-mpo 7026  df-lno 28217

This theorem is referenced by:  islno  28226  hhlnoi  29373