MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  islno Structured version   Visualization version   GIF version

Theorem islno 28457
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 28456 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐿 = {𝑤 ∈ (𝑌m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))})
98eleq2d 2895 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿𝑇 ∈ {𝑤 ∈ (𝑌m 𝑋) ∣ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧))}))
10 fveq1 6662 . . . . . . 7 (𝑤 = 𝑇 → (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)))
11 fveq1 6662 . . . . . . . . 9 (𝑤 = 𝑇 → (𝑤𝑦) = (𝑇𝑦))
1211oveq2d 7161 . . . . . . . 8 (𝑤 = 𝑇 → (𝑥𝑆(𝑤𝑦)) = (𝑥𝑆(𝑇𝑦)))
13 fveq1 6662 . . . . . . . 8 (𝑤 = 𝑇 → (𝑤𝑧) = (𝑇𝑧))
1412, 13oveq12d 7163 . . . . . . 7 (𝑤 = 𝑇 → ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧)))
1510, 14eqeq12d 2834 . . . . . 6 (𝑤 = 𝑇 → ((𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
16152ralbidv 3196 . . . . 5 (𝑤 = 𝑇 → (∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
1716ralbidv 3194 . . . 4 (𝑤 = 𝑇 → (∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑤‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑤𝑦))𝐻(𝑤𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦𝑋𝑧𝑋 (𝑇‘((𝑥𝑅𝑦)𝐺𝑧)) = ((𝑥𝑆(𝑇𝑦))𝐻(𝑇𝑧))))
1817elrab 3677 . . 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 6677 . . . . 5 𝑌 ∈ V
201fvexi 6677 . . . . 5 𝑋 ∈ V
2119, 20elmap 8424 . . . 4 (𝑇 ∈ (𝑌m 𝑋) ↔ 𝑇:𝑋𝑌)
2221anbi1i 623 . . 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 276 . 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, 23syl6bb 288 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  cc 10523  NrmCVeccnv 28288   +𝑣 cpv 28289  BaseSetcba 28290   ·𝑠OLD cns 28291   LnOp clno 28444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-lno 28448
This theorem is referenced by:  lnolin  28458  lnof  28459  lnocoi  28461  0lno  28494  ipblnfi  28559
  Copyright terms: Public domain W3C validator