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

Theorem dicfval 37753
Description: The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Dec-2013.)
Hypotheses
Ref Expression
dicval.l = (le‘𝐾)
dicval.a 𝐴 = (Atoms‘𝐾)
dicval.h 𝐻 = (LHyp‘𝐾)
dicval.p 𝑃 = ((oc‘𝐾)‘𝑊)
dicval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicval.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dicval.i 𝐼 = ((DIsoC‘𝐾)‘𝑊)
Assertion
Ref Expression
dicfval ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
Distinct variable groups:   𝐴,𝑟   𝑓,𝑔,𝑞,𝑟,𝑠,𝐾   ,𝑞   𝐴,𝑞   𝑇,𝑔   𝑓,𝑊,𝑔,𝑞,𝑟,𝑠
Allowed substitution hints:   𝐴(𝑓,𝑔,𝑠)   𝑃(𝑓,𝑔,𝑠,𝑟,𝑞)   𝑇(𝑓,𝑠,𝑟,𝑞)   𝐸(𝑓,𝑔,𝑠,𝑟,𝑞)   𝐻(𝑓,𝑔,𝑠,𝑟,𝑞)   𝐼(𝑓,𝑔,𝑠,𝑟,𝑞)   (𝑓,𝑔,𝑠,𝑟)   𝑉(𝑓,𝑔,𝑠,𝑟,𝑞)

Proof of Theorem dicfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dicval.i . . 3 𝐼 = ((DIsoC‘𝐾)‘𝑊)
2 dicval.l . . . . 5 = (le‘𝐾)
3 dicval.a . . . . 5 𝐴 = (Atoms‘𝐾)
4 dicval.h . . . . 5 𝐻 = (LHyp‘𝐾)
52, 3, 4dicffval 37752 . . . 4 (𝐾𝑉 → (DIsoC‘𝐾) = (𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))
65fveq1d 6501 . . 3 (𝐾𝑉 → ((DIsoC‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))}))‘𝑊))
71, 6syl5eq 2827 . 2 (𝐾𝑉𝐼 = ((𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))}))‘𝑊))
8 breq2 4933 . . . . . 6 (𝑤 = 𝑊 → (𝑟 𝑤𝑟 𝑊))
98notbid 310 . . . . 5 (𝑤 = 𝑊 → (¬ 𝑟 𝑤 ↔ ¬ 𝑟 𝑊))
109rabbidv 3404 . . . 4 (𝑤 = 𝑊 → {𝑟𝐴 ∣ ¬ 𝑟 𝑤} = {𝑟𝐴 ∣ ¬ 𝑟 𝑊})
11 fveq2 6499 . . . . . . . . . 10 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12 dicval.t . . . . . . . . . 10 𝑇 = ((LTrn‘𝐾)‘𝑊)
1311, 12syl6eqr 2833 . . . . . . . . 9 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
14 fveq2 6499 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((oc‘𝐾)‘𝑤) = ((oc‘𝐾)‘𝑊))
15 dicval.p . . . . . . . . . . 11 𝑃 = ((oc‘𝐾)‘𝑊)
1614, 15syl6eqr 2833 . . . . . . . . . 10 (𝑤 = 𝑊 → ((oc‘𝐾)‘𝑤) = 𝑃)
1716fveqeq2d 6507 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞 ↔ (𝑔𝑃) = 𝑞))
1813, 17riotaeqbidv 6940 . . . . . . . 8 (𝑤 = 𝑊 → (𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞) = (𝑔𝑇 (𝑔𝑃) = 𝑞))
1918fveq2d 6503 . . . . . . 7 (𝑤 = 𝑊 → (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)))
2019eqeq2d 2789 . . . . . 6 (𝑤 = 𝑊 → (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ↔ 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞))))
21 fveq2 6499 . . . . . . . 8 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
22 dicval.e . . . . . . . 8 𝐸 = ((TEndo‘𝐾)‘𝑊)
2321, 22syl6eqr 2833 . . . . . . 7 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
2423eleq2d 2852 . . . . . 6 (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤) ↔ 𝑠𝐸))
2520, 24anbi12d 621 . . . . 5 (𝑤 = 𝑊 → ((𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤)) ↔ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)))
2625opabbidv 4995 . . . 4 (𝑤 = 𝑊 → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})
2710, 26mpteq12dv 5012 . . 3 (𝑤 = 𝑊 → (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))}) = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
28 eqid 2779 . . 3 (𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})) = (𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))}))
293fvexi 6513 . . . 4 𝐴 ∈ V
3029mptrabex 6814 . . 3 (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}) ∈ V
3127, 28, 30fvmpt 6595 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))}))‘𝑊) = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
327, 31sylan9eq 2835 1 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1507  wcel 2050  {crab 3093   class class class wbr 4929  {copab 4991  cmpt 5008  cfv 6188  crio 6936  lecple 16428  occoc 16429  Atomscatm 35841  LHypclh 36562  LTrncltrn 36679  TEndoctendo 37330  DIsoCcdic 37750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-dic 37751
This theorem is referenced by:  dicval  37754  dicfnN  37761
  Copyright terms: Public domain W3C validator