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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 = ( +𝑣𝑈)
lnoval.4 𝐻 = ( +𝑣𝑊)
lnoval.5 𝑅 = ( ·𝑠OLD𝑈)
lnoval.6 𝑆 = ( ·𝑠OLD𝑊)
lnoval.7 𝐿 = (𝑈 LnOp 𝑊)
Assertion
Ref Expression
lnoval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑡 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))})
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.
StepHypRef 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)
42, 3syl6eqr 2849 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54oveq2d 7037 . . . 4 (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑𝑚 𝑋))
6 fveq2 6543 . . . . . . . . . 10 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
7 lnoval.3 . . . . . . . . . 10 𝐺 = ( +𝑣𝑈)
86, 7syl6eqr 2849 . . . . . . . . 9 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
9 fveq2 6543 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
10 lnoval.5 . . . . . . . . . . 11 𝑅 = ( ·𝑠OLD𝑈)
119, 10syl6eqr 2849 . . . . . . . . . 10 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑅)
1211oveqd 7038 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑥( ·𝑠OLD𝑢)𝑦) = (𝑥𝑅𝑦))
13 eqidd 2796 . . . . . . . . 9 (𝑢 = 𝑈𝑧 = 𝑧)
148, 12, 13oveq123d 7042 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧) = ((𝑥𝑅𝑦)𝐺𝑧))
1514fveqeq2d 6551 . . . . . . 7 (𝑢 = 𝑈 → ((𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))))
164, 15raleqbidv 3361 . . . . . 6 (𝑢 = 𝑈 → (∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ ∀𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))))
174, 16raleqbidv 3361 . . . . 5 (𝑢 = 𝑈 → (∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))))
1817ralbidv 3164 . . . 4 (𝑢 = 𝑈 → (∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))))
195, 18rabeqbidv 3430 . . 3 (𝑢 = 𝑈 → {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))} = {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))})
20 fveq2 6543 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
21 lnoval.2 . . . . . 6 𝑌 = (BaseSet‘𝑊)
2220, 21syl6eqr 2849 . . . . 5 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
2322oveq1d 7036 . . . 4 (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑𝑚 𝑋) = (𝑌𝑚 𝑋))
24 fveq2 6543 . . . . . . . . 9 (𝑤 = 𝑊 → ( +𝑣𝑤) = ( +𝑣𝑊))
25 lnoval.4 . . . . . . . . 9 𝐻 = ( +𝑣𝑊)
2624, 25syl6eqr 2849 . . . . . . . 8 (𝑤 = 𝑊 → ( +𝑣𝑤) = 𝐻)
27 fveq2 6543 . . . . . . . . . 10 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑊))
28 lnoval.6 . . . . . . . . . 10 𝑆 = ( ·𝑠OLD𝑊)
2927, 28syl6eqr 2849 . . . . . . . . 9 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = 𝑆)
3029oveqd 7038 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥( ·𝑠OLD𝑤)(𝑡𝑦)) = (𝑥𝑆(𝑡𝑦)))
31 eqidd 2796 . . . . . . . 8 (𝑤 = 𝑊 → (𝑡𝑧) = (𝑡𝑧))
3226, 30, 31oveq123d 7042 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧)))
3332eqeq2d 2805 . . . . . 6 (𝑤 = 𝑊 → ((𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))))
34332ralbidv 3166 . . . . 5 (𝑤 = 𝑊 → (∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))))
3534ralbidv 3164 . . . 4 (𝑤 = 𝑊 → (∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))))
3623, 35rabeqbidv 3430 . . 3 (𝑤 = 𝑊 → {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))} = {𝑡 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))})
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( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))})
38 ovex 7053 . . . 4 (𝑌𝑚 𝑋) ∈ V
3938rabex 5131 . . 3 {𝑡 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))} ∈ V
4019, 36, 37, 39ovmpo 7171 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 LnOp 𝑊) = {𝑡 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))})
411, 40syl5eq 2843 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑡 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  wral 3105  {crab 3109  cfv 6230  (class class class)co 7021  𝑚 cmap 8261  cc 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
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