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Theorem lnoval 26991
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 6103 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 lnoval.1 . . . . . 6 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2662 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54oveq2d 6565 . . . 4 (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑𝑚 𝑋))
6 fveq2 6103 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
7 lnoval.3 . . . . . . . . . . 11 𝐺 = ( +𝑣𝑈)
86, 7syl6eqr 2662 . . . . . . . . . 10 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
9 fveq2 6103 . . . . . . . . . . . 12 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
10 lnoval.5 . . . . . . . . . . . 12 𝑅 = ( ·𝑠OLD𝑈)
119, 10syl6eqr 2662 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑅)
1211oveqd 6566 . . . . . . . . . 10 (𝑢 = 𝑈 → (𝑥( ·𝑠OLD𝑢)𝑦) = (𝑥𝑅𝑦))
13 eqidd 2611 . . . . . . . . . 10 (𝑢 = 𝑈𝑧 = 𝑧)
148, 12, 13oveq123d 6570 . . . . . . . . 9 (𝑢 = 𝑈 → ((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧) = ((𝑥𝑅𝑦)𝐺𝑧))
1514fveq2d 6107 . . . . . . . 8 (𝑢 = 𝑈 → (𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)))
1615eqeq1d 2612 . . . . . . 7 (𝑢 = 𝑈 → ((𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))))
174, 16raleqbidv 3129 . . . . . 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))))
184, 17raleqbidv 3129 . . . . 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))))
1918ralbidv 2969 . . . 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))))
205, 19rabeqbidv 3168 . . 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))})
21 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
22 lnoval.2 . . . . . 6 𝑌 = (BaseSet‘𝑊)
2321, 22syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
2423oveq1d 6564 . . . 4 (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑𝑚 𝑋) = (𝑌𝑚 𝑋))
25 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑊 → ( +𝑣𝑤) = ( +𝑣𝑊))
26 lnoval.4 . . . . . . . . 9 𝐻 = ( +𝑣𝑊)
2725, 26syl6eqr 2662 . . . . . . . 8 (𝑤 = 𝑊 → ( +𝑣𝑤) = 𝐻)
28 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑊))
29 lnoval.6 . . . . . . . . . 10 𝑆 = ( ·𝑠OLD𝑊)
3028, 29syl6eqr 2662 . . . . . . . . 9 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = 𝑆)
3130oveqd 6566 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥( ·𝑠OLD𝑤)(𝑡𝑦)) = (𝑥𝑆(𝑡𝑦)))
32 eqidd 2611 . . . . . . . 8 (𝑤 = 𝑊 → (𝑡𝑧) = (𝑡𝑧))
3327, 31, 32oveq123d 6570 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧)))
3433eqeq2d 2620 . . . . . 6 (𝑤 = 𝑊 → ((𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))))
35342ralbidv 2972 . . . . 5 (𝑤 = 𝑊 → (∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))))
3635ralbidv 2969 . . . 4 (𝑤 = 𝑊 → (∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))))
3724, 36rabeqbidv 3168 . . 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))})
38 df-lno 26983 . . 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))})
39 ovex 6577 . . . 4 (𝑌𝑚 𝑋) ∈ V
4039rabex 4740 . . 3 {𝑡 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))} ∈ V
4120, 37, 38, 40ovmpt2 6694 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 LnOp 𝑊) = {𝑡 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))})
421, 41syl5eq 2656 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 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  cc 9813  NrmCVeccnv 26823   +𝑣 cpv 26824  BaseSetcba 26825   ·𝑠OLD cns 26826   LnOp clno 26979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-lno 26983
This theorem is referenced by:  islno  26992  hhlnoi  28143
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