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Theorem islno 29115
Description: The predicate "is a linear operator." (Contributed by NM, 4-Dec-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
islno ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿 ↔ (𝑇:𝑋𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧)))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑈   𝑥,𝑊,𝑦,𝑧   𝑦,𝑋,𝑧   𝑥,𝑇,𝑦,𝑧
Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)   𝑆(𝑥,𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)   𝐻(𝑥,𝑦,𝑧)   𝐿(𝑥,𝑦,𝑧)   𝑋(𝑥)   𝑌(𝑥,𝑦,𝑧)

Proof of Theorem islno
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 lnoval.1 . . . 4 𝑋 = (BaseSet‘𝑈)
2 lnoval.2 . . . 4 𝑌 = (BaseSet‘𝑊)
3 lnoval.3 . . . 4 𝐺 = ( +𝑣𝑈)
4 lnoval.4 . . . 4 𝐻 = ( +𝑣𝑊)
5 lnoval.5 . . . 4 𝑅 = ( ·𝑠OLD𝑈)
6 lnoval.6 . . . 4 𝑆 = ( ·𝑠OLD𝑊)
7 lnoval.7 . . . 4 𝐿 = (𝑈 LnOp 𝑊)
81, 2, 3, 4, 5, 6, 7lnoval 29114 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑤 ∈ (𝑌m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))})
98eleq2d 2824 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿𝑇 ∈ {𝑤 ∈ (𝑌m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))}))
10 fveq1 6773 . . . . . . 7 (𝑤 = 𝑇 → (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)))
11 fveq1 6773 . . . . . . . . 9 (𝑤 = 𝑇 → (𝑤𝑦) = (𝑇𝑦))
1211oveq2d 7291 . . . . . . . 8 (𝑤 = 𝑇 → (𝑥𝑆(𝑤𝑦)) = (𝑥𝑆(𝑇𝑦)))
13 fveq1 6773 . . . . . . . 8 (𝑤 = 𝑇 → (𝑤𝑧) = (𝑇𝑧))
1412, 13oveq12d 7293 . . . . . . 7 (𝑤 = 𝑇 → ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧)))
1510, 14eqeq12d 2754 . . . . . 6 (𝑤 = 𝑇 → ((𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
16152ralbidv 3129 . . . . 5 (𝑤 = 𝑇 → (∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
1716ralbidv 3112 . . . 4 (𝑤 = 𝑇 → (∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
1817elrab 3624 . . 3 (𝑇 ∈ {𝑤 ∈ (𝑌m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))} ↔ (𝑇 ∈ (𝑌m 𝑋) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
192fvexi 6788 . . . . 5 𝑌 ∈ V
201fvexi 6788 . . . . 5 𝑋 ∈ V
2119, 20elmap 8659 . . . 4 (𝑇 ∈ (𝑌m 𝑋) ↔ 𝑇:𝑋𝑌)
2221anbi1i 624 . . 3 ((𝑇 ∈ (𝑌m 𝑋) ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))) ↔ (𝑇:𝑋𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
2318, 22bitri 274 . 2 (𝑇 ∈ {𝑤 ∈ (𝑌m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))} ↔ (𝑇:𝑋𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
249, 23bitrdi 287 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿 ↔ (𝑇:𝑋𝑌 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  {crab 3068  wf 6429  cfv 6433  (class class class)co 7275  m cmap 8615  cc 10869  NrmCVeccnv 28946   +𝑣 cpv 28947  BaseSetcba 28948   ·𝑠OLD cns 28949   LnOp clno 29102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-lno 29106
This theorem is referenced by:  lnolin  29116  lnof  29117  lnocoi  29119  0lno  29152  ipblnfi  29217
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