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Theorem dicval 35984
Description: The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)
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
dicval (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
Distinct variable groups:   𝑓,𝑔,𝑠,𝐾   𝑇,𝑔   𝑓,𝑊,𝑔,𝑠   𝑓,𝐸,𝑠   𝑃,𝑓   𝑄,𝑓,𝑔,𝑠   𝑇,𝑓
Allowed substitution hints:   𝐴(𝑓,𝑔,𝑠)   𝑃(𝑔,𝑠)   𝑇(𝑠)   𝐸(𝑔)   𝐻(𝑓,𝑔,𝑠)   𝐼(𝑓,𝑔,𝑠)   (𝑓,𝑔,𝑠)   𝑉(𝑓,𝑔,𝑠)

Proof of Theorem dicval
Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicval.l . . . . 5 = (le‘𝐾)
2 dicval.a . . . . 5 𝐴 = (Atoms‘𝐾)
3 dicval.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 dicval.p . . . . 5 𝑃 = ((oc‘𝐾)‘𝑊)
5 dicval.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
6 dicval.e . . . . 5 𝐸 = ((TEndo‘𝐾)‘𝑊)
7 dicval.i . . . . 5 𝐼 = ((DIsoC‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dicfval 35983 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
98adantr 481 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
109fveq1d 6160 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = ((𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})‘𝑄))
11 simpr 477 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
12 breq1 4626 . . . . . 6 (𝑟 = 𝑄 → (𝑟 𝑊𝑄 𝑊))
1312notbid 308 . . . . 5 (𝑟 = 𝑄 → (¬ 𝑟 𝑊 ↔ ¬ 𝑄 𝑊))
1413elrab 3351 . . . 4 (𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↔ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
1511, 14sylibr 224 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊})
16 eqeq2 2632 . . . . . . . . 9 (𝑞 = 𝑄 → ((𝑔𝑃) = 𝑞 ↔ (𝑔𝑃) = 𝑄))
1716riotabidv 6578 . . . . . . . 8 (𝑞 = 𝑄 → (𝑔𝑇 (𝑔𝑃) = 𝑞) = (𝑔𝑇 (𝑔𝑃) = 𝑄))
1817fveq2d 6162 . . . . . . 7 (𝑞 = 𝑄 → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1918eqeq2d 2631 . . . . . 6 (𝑞 = 𝑄 → (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ↔ 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
2019anbi1d 740 . . . . 5 (𝑞 = 𝑄 → ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸) ↔ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)))
2120opabbidv 4688 . . . 4 (𝑞 = 𝑄 → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
22 eqid 2621 . . . 4 (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}) = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})
23 fvex 6168 . . . . . . . . . . 11 ((TEndo‘𝐾)‘𝑊) ∈ V
246, 23eqeltri 2694 . . . . . . . . . 10 𝐸 ∈ V
2524uniex 6918 . . . . . . . . 9 𝐸 ∈ V
2625rnex 7062 . . . . . . . 8 ran 𝐸 ∈ V
2726uniex 6918 . . . . . . 7 ran 𝐸 ∈ V
2827pwex 4818 . . . . . 6 𝒫 ran 𝐸 ∈ V
2928, 24xpex 6927 . . . . 5 (𝒫 ran 𝐸 × 𝐸) ∈ V
30 simpl 473 . . . . . . . . 9 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
31 fvssunirn 6184 . . . . . . . . . . 11 (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝑠
32 elssuni 4440 . . . . . . . . . . . . 13 (𝑠𝐸𝑠 𝐸)
3332adantl 482 . . . . . . . . . . . 12 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑠 𝐸)
34 rnss 5324 . . . . . . . . . . . 12 (𝑠 𝐸 → ran 𝑠 ⊆ ran 𝐸)
35 uniss 4431 . . . . . . . . . . . 12 (ran 𝑠 ⊆ ran 𝐸 ran 𝑠 ran 𝐸)
3633, 34, 353syl 18 . . . . . . . . . . 11 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → ran 𝑠 ran 𝐸)
3731, 36syl5ss 3599 . . . . . . . . . 10 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝐸)
3827elpw2 4798 . . . . . . . . . 10 ((𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∈ 𝒫 ran 𝐸 ↔ (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝐸)
3937, 38sylibr 224 . . . . . . . . 9 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∈ 𝒫 ran 𝐸)
4030, 39eqeltrd 2698 . . . . . . . 8 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑓 ∈ 𝒫 ran 𝐸)
41 simpr 477 . . . . . . . 8 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑠𝐸)
4240, 41jca 554 . . . . . . 7 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸))
4342ssopab2i 4973 . . . . . 6 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ⊆ {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸)}
44 df-xp 5090 . . . . . 6 (𝒫 ran 𝐸 × 𝐸) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸)}
4543, 44sseqtr4i 3623 . . . . 5 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ⊆ (𝒫 ran 𝐸 × 𝐸)
4629, 45ssexi 4773 . . . 4 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ∈ V
4721, 22, 46fvmpt 6249 . . 3 (𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} → ((𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
4815, 47syl 17 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
4910, 48eqtrd 2655 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2912  Vcvv 3190  wss 3560  𝒫 cpw 4136   cuni 4409   class class class wbr 4623  {copab 4682  cmpt 4683   × cxp 5082  ran crn 5085  cfv 5857  crio 6575  lecple 15888  occoc 15889  Atomscatm 34069  LHypclh 34789  LTrncltrn 34906  TEndoctendo 35559  DIsoCcdic 35980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-dic 35981
This theorem is referenced by:  dicopelval  35985  dicelvalN  35986  dicval2  35987  dicfnN  35991  dicvalrelN  35993  dicssdvh  35994  dicelval1sta  35995  dihpN  36144
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