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Theorem lnoval 30575
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 = (BaseSetβ€˜π‘ˆ)
lnoval.2 π‘Œ = (BaseSetβ€˜π‘Š)
lnoval.3 𝐺 = ( +𝑣 β€˜π‘ˆ)
lnoval.4 𝐻 = ( +𝑣 β€˜π‘Š)
lnoval.5 𝑅 = ( ·𝑠OLD β€˜π‘ˆ)
lnoval.6 𝑆 = ( ·𝑠OLD β€˜π‘Š)
lnoval.7 𝐿 = (π‘ˆ LnOp π‘Š)
Assertion
Ref Expression
lnoval ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝐿 = {𝑑 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‘β€˜π‘¦))𝐻(π‘‘β€˜π‘§))})
Distinct variable groups:   π‘₯,𝑑,𝑦,𝑧,π‘ˆ   𝑑,π‘Š,π‘₯,𝑦,𝑧   𝑑,𝑋,𝑦,𝑧   𝑑,π‘Œ   𝑑,𝐺   𝑑,𝑅   𝑑,𝐻   𝑑,𝑆
Allowed substitution hints:   𝑅(π‘₯,𝑦,𝑧)   𝑆(π‘₯,𝑦,𝑧)   𝐺(π‘₯,𝑦,𝑧)   𝐻(π‘₯,𝑦,𝑧)   𝐿(π‘₯,𝑦,𝑧,𝑑)   𝑋(π‘₯)   π‘Œ(π‘₯,𝑦,𝑧)

Proof of Theorem lnoval
Dummy variables 𝑒 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnoval.7 . 2 𝐿 = (π‘ˆ LnOp π‘Š)
2 fveq2 6897 . . . . . 6 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = (BaseSetβ€˜π‘ˆ))
3 lnoval.1 . . . . . 6 𝑋 = (BaseSetβ€˜π‘ˆ)
42, 3eqtr4di 2786 . . . . 5 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = 𝑋)
54oveq2d 7436 . . . 4 (𝑒 = π‘ˆ β†’ ((BaseSetβ€˜π‘€) ↑m (BaseSetβ€˜π‘’)) = ((BaseSetβ€˜π‘€) ↑m 𝑋))
6 fveq2 6897 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ ( +𝑣 β€˜π‘’) = ( +𝑣 β€˜π‘ˆ))
7 lnoval.3 . . . . . . . . . 10 𝐺 = ( +𝑣 β€˜π‘ˆ)
86, 7eqtr4di 2786 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ ( +𝑣 β€˜π‘’) = 𝐺)
9 fveq2 6897 . . . . . . . . . . 11 (𝑒 = π‘ˆ β†’ ( ·𝑠OLD β€˜π‘’) = ( ·𝑠OLD β€˜π‘ˆ))
10 lnoval.5 . . . . . . . . . . 11 𝑅 = ( ·𝑠OLD β€˜π‘ˆ)
119, 10eqtr4di 2786 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ ( ·𝑠OLD β€˜π‘’) = 𝑅)
1211oveqd 7437 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (π‘₯( ·𝑠OLD β€˜π‘’)𝑦) = (π‘₯𝑅𝑦))
13 eqidd 2729 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ 𝑧 = 𝑧)
148, 12, 13oveq123d 7441 . . . . . . . 8 (𝑒 = π‘ˆ β†’ ((π‘₯( ·𝑠OLD β€˜π‘’)𝑦)( +𝑣 β€˜π‘’)𝑧) = ((π‘₯𝑅𝑦)𝐺𝑧))
1514fveqeq2d 6905 . . . . . . 7 (𝑒 = π‘ˆ β†’ ((π‘‘β€˜((π‘₯( ·𝑠OLD β€˜π‘’)𝑦)( +𝑣 β€˜π‘’)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§)) ↔ (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§))))
164, 15raleqbidv 3339 . . . . . 6 (𝑒 = π‘ˆ β†’ (βˆ€π‘§ ∈ (BaseSetβ€˜π‘’)(π‘‘β€˜((π‘₯( ·𝑠OLD β€˜π‘’)𝑦)( +𝑣 β€˜π‘’)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§)) ↔ βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§))))
174, 16raleqbidv 3339 . . . . 5 (𝑒 = π‘ˆ β†’ (βˆ€π‘¦ ∈ (BaseSetβ€˜π‘’)βˆ€π‘§ ∈ (BaseSetβ€˜π‘’)(π‘‘β€˜((π‘₯( ·𝑠OLD β€˜π‘’)𝑦)( +𝑣 β€˜π‘’)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§)) ↔ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§))))
1817ralbidv 3174 . . . 4 (𝑒 = π‘ˆ β†’ (βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (BaseSetβ€˜π‘’)βˆ€π‘§ ∈ (BaseSetβ€˜π‘’)(π‘‘β€˜((π‘₯( ·𝑠OLD β€˜π‘’)𝑦)( +𝑣 β€˜π‘’)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§)) ↔ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§))))
195, 18rabeqbidv 3446 . . 3 (𝑒 = π‘ˆ β†’ {𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m (BaseSetβ€˜π‘’)) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (BaseSetβ€˜π‘’)βˆ€π‘§ ∈ (BaseSetβ€˜π‘’)(π‘‘β€˜((π‘₯( ·𝑠OLD β€˜π‘’)𝑦)( +𝑣 β€˜π‘’)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§))} = {𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§))})
20 fveq2 6897 . . . . . 6 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = (BaseSetβ€˜π‘Š))
21 lnoval.2 . . . . . 6 π‘Œ = (BaseSetβ€˜π‘Š)
2220, 21eqtr4di 2786 . . . . 5 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = π‘Œ)
2322oveq1d 7435 . . . 4 (𝑀 = π‘Š β†’ ((BaseSetβ€˜π‘€) ↑m 𝑋) = (π‘Œ ↑m 𝑋))
24 fveq2 6897 . . . . . . . . 9 (𝑀 = π‘Š β†’ ( +𝑣 β€˜π‘€) = ( +𝑣 β€˜π‘Š))
25 lnoval.4 . . . . . . . . 9 𝐻 = ( +𝑣 β€˜π‘Š)
2624, 25eqtr4di 2786 . . . . . . . 8 (𝑀 = π‘Š β†’ ( +𝑣 β€˜π‘€) = 𝐻)
27 fveq2 6897 . . . . . . . . . 10 (𝑀 = π‘Š β†’ ( ·𝑠OLD β€˜π‘€) = ( ·𝑠OLD β€˜π‘Š))
28 lnoval.6 . . . . . . . . . 10 𝑆 = ( ·𝑠OLD β€˜π‘Š)
2927, 28eqtr4di 2786 . . . . . . . . 9 (𝑀 = π‘Š β†’ ( ·𝑠OLD β€˜π‘€) = 𝑆)
3029oveqd 7437 . . . . . . . 8 (𝑀 = π‘Š β†’ (π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦)) = (π‘₯𝑆(π‘‘β€˜π‘¦)))
31 eqidd 2729 . . . . . . . 8 (𝑀 = π‘Š β†’ (π‘‘β€˜π‘§) = (π‘‘β€˜π‘§))
3226, 30, 31oveq123d 7441 . . . . . . 7 (𝑀 = π‘Š β†’ ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§)) = ((π‘₯𝑆(π‘‘β€˜π‘¦))𝐻(π‘‘β€˜π‘§)))
3332eqeq2d 2739 . . . . . 6 (𝑀 = π‘Š β†’ ((π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§)) ↔ (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‘β€˜π‘¦))𝐻(π‘‘β€˜π‘§))))
34332ralbidv 3215 . . . . 5 (𝑀 = π‘Š β†’ (βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§)) ↔ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‘β€˜π‘¦))𝐻(π‘‘β€˜π‘§))))
3534ralbidv 3174 . . . 4 (𝑀 = π‘Š β†’ (βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§)) ↔ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‘β€˜π‘¦))𝐻(π‘‘β€˜π‘§))))
3623, 35rabeqbidv 3446 . . 3 (𝑀 = π‘Š β†’ {𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§))} = {𝑑 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‘β€˜π‘¦))𝐻(π‘‘β€˜π‘§))})
37 df-lno 30567 . . 3 LnOp = (𝑒 ∈ NrmCVec, 𝑀 ∈ NrmCVec ↦ {𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m (BaseSetβ€˜π‘’)) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (BaseSetβ€˜π‘’)βˆ€π‘§ ∈ (BaseSetβ€˜π‘’)(π‘‘β€˜((π‘₯( ·𝑠OLD β€˜π‘’)𝑦)( +𝑣 β€˜π‘’)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘€)(π‘‘β€˜π‘¦))( +𝑣 β€˜π‘€)(π‘‘β€˜π‘§))})
38 ovex 7453 . . . 4 (π‘Œ ↑m 𝑋) ∈ V
3938rabex 5334 . . 3 {𝑑 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‘β€˜π‘¦))𝐻(π‘‘β€˜π‘§))} ∈ V
4019, 36, 37, 39ovmpo 7581 . 2 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ (π‘ˆ LnOp π‘Š) = {𝑑 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‘β€˜π‘¦))𝐻(π‘‘β€˜π‘§))})
411, 40eqtrid 2780 1 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝐿 = {𝑑 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‘β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‘β€˜π‘¦))𝐻(π‘‘β€˜π‘§))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  {crab 3429  β€˜cfv 6548  (class class class)co 7420   ↑m cmap 8845  β„‚cc 11137  NrmCVeccnv 30407   +𝑣 cpv 30408  BaseSetcba 30409   ·𝑠OLD cns 30410   LnOp clno 30563
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 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-lno 30567
This theorem is referenced by:  islno  30576  hhlnoi  31723
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