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Theorem islno 27909
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 27908 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑤 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))})
98eleq2d 2817 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿𝑇 ∈ {𝑤 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))}))
10 fveq1 6343 . . . . . . 7 (𝑤 = 𝑇 → (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)))
11 fveq1 6343 . . . . . . . . 9 (𝑤 = 𝑇 → (𝑤𝑦) = (𝑇𝑦))
1211oveq2d 6821 . . . . . . . 8 (𝑤 = 𝑇 → (𝑥𝑆(𝑤𝑦)) = (𝑥𝑆(𝑇𝑦)))
13 fveq1 6343 . . . . . . . 8 (𝑤 = 𝑇 → (𝑤𝑧) = (𝑇𝑧))
1412, 13oveq12d 6823 . . . . . . 7 (𝑤 = 𝑇 → ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧)))
1510, 14eqeq12d 2767 . . . . . 6 (𝑤 = 𝑇 → ((𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
16152ralbidv 3119 . . . . 5 (𝑤 = 𝑇 → (∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
1716ralbidv 3116 . . . 4 (𝑤 = 𝑇 → (∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
1817elrab 3496 . . 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))))
19 fvex 6354 . . . . . 6 (BaseSet‘𝑊) ∈ V
202, 19eqeltri 2827 . . . . 5 𝑌 ∈ V
21 fvex 6354 . . . . . 6 (BaseSet‘𝑈) ∈ V
221, 21eqeltri 2827 . . . . 5 𝑋 ∈ V
2320, 22elmap 8044 . . . 4 (𝑇 ∈ (𝑌𝑚 𝑋) ↔ 𝑇:𝑋𝑌)
2423anbi1i 733 . . 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))))
2518, 24bitri 264 . 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))))
269, 25syl6bb 276 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 196  wa 383   = wceq 1624  wcel 2131  wral 3042  {crab 3046  Vcvv 3332  wf 6037  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  cc 10118  NrmCVeccnv 27740   +𝑣 cpv 27741  BaseSetcba 27742   ·𝑠OLD cns 27743   LnOp clno 27896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-map 8017  df-lno 27900
This theorem is referenced by:  lnolin  27910  lnof  27911  lnocoi  27913  0lno  27946  ipblnfi  28012
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