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Theorem lnoval 27495
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 6158 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 lnoval.1 . . . . . 6 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2673 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54oveq2d 6631 . . . 4 (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑𝑚 𝑋))
6 fveq2 6158 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
7 lnoval.3 . . . . . . . . . . 11 𝐺 = ( +𝑣𝑈)
86, 7syl6eqr 2673 . . . . . . . . . 10 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
9 fveq2 6158 . . . . . . . . . . . 12 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
10 lnoval.5 . . . . . . . . . . . 12 𝑅 = ( ·𝑠OLD𝑈)
119, 10syl6eqr 2673 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑅)
1211oveqd 6632 . . . . . . . . . 10 (𝑢 = 𝑈 → (𝑥( ·𝑠OLD𝑢)𝑦) = (𝑥𝑅𝑦))
13 eqidd 2622 . . . . . . . . . 10 (𝑢 = 𝑈𝑧 = 𝑧)
148, 12, 13oveq123d 6636 . . . . . . . . 9 (𝑢 = 𝑈 → ((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧) = ((𝑥𝑅𝑦)𝐺𝑧))
1514fveq2d 6162 . . . . . . . 8 (𝑢 = 𝑈 → (𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)))
1615eqeq1d 2623 . . . . . . 7 (𝑢 = 𝑈 → ((𝑡‘((𝑥( ·𝑠OLD𝑢)𝑦)( +𝑣𝑢)𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧))))
174, 16raleqbidv 3145 . . . . . 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 3145 . . . . 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 2982 . . . 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 3185 . . 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 6158 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
22 lnoval.2 . . . . . 6 𝑌 = (BaseSet‘𝑊)
2321, 22syl6eqr 2673 . . . . 5 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
2423oveq1d 6630 . . . 4 (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑𝑚 𝑋) = (𝑌𝑚 𝑋))
25 fveq2 6158 . . . . . . . . 9 (𝑤 = 𝑊 → ( +𝑣𝑤) = ( +𝑣𝑊))
26 lnoval.4 . . . . . . . . 9 𝐻 = ( +𝑣𝑊)
2725, 26syl6eqr 2673 . . . . . . . 8 (𝑤 = 𝑊 → ( +𝑣𝑤) = 𝐻)
28 fveq2 6158 . . . . . . . . . 10 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑊))
29 lnoval.6 . . . . . . . . . 10 𝑆 = ( ·𝑠OLD𝑊)
3028, 29syl6eqr 2673 . . . . . . . . 9 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = 𝑆)
3130oveqd 6632 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥( ·𝑠OLD𝑤)(𝑡𝑦)) = (𝑥𝑆(𝑡𝑦)))
32 eqidd 2622 . . . . . . . 8 (𝑤 = 𝑊 → (𝑡𝑧) = (𝑡𝑧))
3327, 31, 32oveq123d 6636 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧)))
3433eqeq2d 2631 . . . . . 6 (𝑤 = 𝑊 → ((𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))))
35342ralbidv 2985 . . . . 5 (𝑤 = 𝑊 → (∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥( ·𝑠OLD𝑤)(𝑡𝑦))( +𝑣𝑤)(𝑡𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))))
3635ralbidv 2982 . . . 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 3185 . . 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 27487 . . 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 6643 . . . 4 (𝑌𝑚 𝑋) ∈ V
4039rabex 4783 . . 3 {𝑡 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))} ∈ V
4120, 37, 38, 40ovmpt2 6761 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 LnOp 𝑊) = {𝑡 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑡‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑡𝑦))𝐻(𝑡𝑧))})
421, 41syl5eq 2667 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 384   = wceq 1480  wcel 1987  wral 2908  {crab 2912  cfv 5857  (class class class)co 6615  𝑚 cmap 7817  cc 9894  NrmCVeccnv 27327   +𝑣 cpv 27328  BaseSetcba 27329   ·𝑠OLD cns 27330   LnOp clno 27483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-lno 27487
This theorem is referenced by:  islno  27496  hhlnoi  28647
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