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Theorem islno 29695
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 29694 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑤 ∈ (𝑌m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))})
98eleq2d 2823 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿𝑇 ∈ {𝑤 ∈ (𝑌m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))}))
10 fveq1 6841 . . . . . . 7 (𝑤 = 𝑇 → (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)))
11 fveq1 6841 . . . . . . . . 9 (𝑤 = 𝑇 → (𝑤𝑦) = (𝑇𝑦))
1211oveq2d 7373 . . . . . . . 8 (𝑤 = 𝑇 → (𝑥𝑆(𝑤𝑦)) = (𝑥𝑆(𝑇𝑦)))
13 fveq1 6841 . . . . . . . 8 (𝑤 = 𝑇 → (𝑤𝑧) = (𝑇𝑧))
1412, 13oveq12d 7375 . . . . . . 7 (𝑤 = 𝑇 → ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧)))
1510, 14eqeq12d 2752 . . . . . 6 (𝑤 = 𝑇 → ((𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
16152ralbidv 3212 . . . . 5 (𝑤 = 𝑇 → (∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
1716ralbidv 3174 . . . 4 (𝑤 = 𝑇 → (∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
1817elrab 3645 . . 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 6856 . . . . 5 𝑌 ∈ V
201fvexi 6856 . . . . 5 𝑋 ∈ V
2119, 20elmap 8809 . . . 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 286 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 1541  wcel 2106  wral 3064  {crab 3407  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  cc 11049  NrmCVeccnv 29526   +𝑣 cpv 29527  BaseSetcba 29528   ·𝑠OLD cns 29529   LnOp clno 29682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-lno 29686
This theorem is referenced by:  lnolin  29696  lnof  29697  lnocoi  29699  0lno  29732  ipblnfi  29797
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