Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diafval Structured version   Visualization version   GIF version

Theorem diafval 35338
 Description: The partial isomorphism A for a lattice 𝐾. (Contributed by NM, 15-Oct-2013.)
Hypotheses
Ref Expression
diaval.b 𝐵 = (Base‘𝐾)
diaval.l = (le‘𝐾)
diaval.h 𝐻 = (LHyp‘𝐾)
diaval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
diaval.r 𝑅 = ((trL‘𝐾)‘𝑊)
diaval.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
Assertion
Ref Expression
diafval ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
Distinct variable groups:   𝑥,𝑦,   𝑥,𝐵,𝑦   𝑥,𝑓,𝑦,𝐾   𝑥,𝑅   𝑇,𝑓,𝑥   𝑓,𝑊,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑓)   𝑅(𝑦,𝑓)   𝑇(𝑦)   𝐻(𝑥,𝑦,𝑓)   𝐼(𝑥,𝑦,𝑓)   (𝑓)   𝑉(𝑥,𝑦,𝑓)

Proof of Theorem diafval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 diaval.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
2 diaval.b . . . . 5 𝐵 = (Base‘𝐾)
3 diaval.l . . . . 5 = (le‘𝐾)
4 diaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
52, 3, 4diaffval 35337 . . . 4 (𝐾𝑉 → (DIsoA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})))
65fveq1d 6105 . . 3 (𝐾𝑉 → ((DIsoA‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))‘𝑊))
71, 6syl5eq 2656 . 2 (𝐾𝑉𝐼 = ((𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))‘𝑊))
8 breq2 4587 . . . . 5 (𝑤 = 𝑊 → (𝑦 𝑤𝑦 𝑊))
98rabbidv 3164 . . . 4 (𝑤 = 𝑊 → {𝑦𝐵𝑦 𝑤} = {𝑦𝐵𝑦 𝑊})
10 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
11 diaval.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
1210, 11syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
13 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = ((trL‘𝐾)‘𝑊))
14 diaval.r . . . . . . . 8 𝑅 = ((trL‘𝐾)‘𝑊)
1513, 14syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = 𝑅)
1615fveq1d 6105 . . . . . 6 (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘𝑓) = (𝑅𝑓))
1716breq1d 4593 . . . . 5 (𝑤 = 𝑊 → ((((trL‘𝐾)‘𝑤)‘𝑓) 𝑥 ↔ (𝑅𝑓) 𝑥))
1812, 17rabeqbidv 3168 . . . 4 (𝑤 = 𝑊 → {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥} = {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})
199, 18mpteq12dv 4663 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}) = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
20 eqid 2610 . . 3 (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})) = (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))
21 fvex 6113 . . . . 5 (Base‘𝐾) ∈ V
222, 21eqeltri 2684 . . . 4 𝐵 ∈ V
2322mptrabex 6392 . . 3 (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}) ∈ V
2419, 20, 23fvmpt 6191 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))‘𝑊) = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
257, 24sylan9eq 2664 1 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  Basecbs 15695  lecple 15775  LHypclh 34288  LTrncltrn 34405  trLctrl 34463  DIsoAcdia 35335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-disoa 35336 This theorem is referenced by:  diaval  35339  diafn  35341
 Copyright terms: Public domain W3C validator