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Theorem dicval 41178
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 41177 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
98adantr 480 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
109fveq1d 6908 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = ((𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)})‘𝑄))
11 simpr 484 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
12 breq1 5146 . . . . . 6 (𝑟 = 𝑄 → (𝑟 𝑊𝑄 𝑊))
1312notbid 318 . . . . 5 (𝑟 = 𝑄 → (¬ 𝑟 𝑊 ↔ ¬ 𝑄 𝑊))
1413elrab 3692 . . . 4 (𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↔ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
1511, 14sylibr 234 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑄 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊})
16 eqeq2 2749 . . . . . . . . 9 (𝑞 = 𝑄 → ((𝑔𝑃) = 𝑞 ↔ (𝑔𝑃) = 𝑄))
1716riotabidv 7390 . . . . . . . 8 (𝑞 = 𝑄 → (𝑔𝑇 (𝑔𝑃) = 𝑞) = (𝑔𝑇 (𝑔𝑃) = 𝑄))
1817fveq2d 6910 . . . . . . 7 (𝑞 = 𝑄 → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1918eqeq2d 2748 . . . . . 6 (𝑞 = 𝑄 → (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ↔ 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
2019anbi1d 631 . . . . 5 (𝑞 = 𝑄 → ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸) ↔ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)))
2120opabbidv 5209 . . . 4 (𝑞 = 𝑄 → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
22 eqid 2737 . . . 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)})
236fvexi 6920 . . . . . . . . . 10 𝐸 ∈ V
2423uniex 7761 . . . . . . . . 9 𝐸 ∈ V
2524rnex 7932 . . . . . . . 8 ran 𝐸 ∈ V
2625uniex 7761 . . . . . . 7 ran 𝐸 ∈ V
2726pwex 5380 . . . . . 6 𝒫 ran 𝐸 ∈ V
2827, 23xpex 7773 . . . . 5 (𝒫 ran 𝐸 × 𝐸) ∈ V
29 simpl 482 . . . . . . . . 9 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
30 fvssunirn 6939 . . . . . . . . . . 11 (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝑠
31 elssuni 4937 . . . . . . . . . . . . 13 (𝑠𝐸𝑠 𝐸)
3231adantl 481 . . . . . . . . . . . 12 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑠 𝐸)
33 rnss 5950 . . . . . . . . . . . 12 (𝑠 𝐸 → ran 𝑠 ⊆ ran 𝐸)
34 uniss 4915 . . . . . . . . . . . 12 (ran 𝑠 ⊆ ran 𝐸 ran 𝑠 ran 𝐸)
3532, 33, 343syl 18 . . . . . . . . . . 11 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → ran 𝑠 ran 𝐸)
3630, 35sstrid 3995 . . . . . . . . . 10 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝐸)
3726elpw2 5334 . . . . . . . . . 10 ((𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∈ 𝒫 ran 𝐸 ↔ (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ⊆ ran 𝐸)
3836, 37sylibr 234 . . . . . . . . 9 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∈ 𝒫 ran 𝐸)
3929, 38eqeltrd 2841 . . . . . . . 8 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑓 ∈ 𝒫 ran 𝐸)
40 simpr 484 . . . . . . . 8 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → 𝑠𝐸)
4139, 40jca 511 . . . . . . 7 ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) → (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸))
4241ssopab2i 5555 . . . . . 6 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ⊆ {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸)}
43 df-xp 5691 . . . . . 6 (𝒫 ran 𝐸 × 𝐸) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 ∈ 𝒫 ran 𝐸𝑠𝐸)}
4442, 43sseqtrri 4033 . . . . 5 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ⊆ (𝒫 ran 𝐸 × 𝐸)
4528, 44ssexi 5322 . . . 4 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ∈ V
4621, 22, 45fvmpt 7016 . . 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)})
4715, 46syl 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)})
4810, 47eqtrd 2777 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 395   = wceq 1540  wcel 2108  {crab 3436  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  {copab 5205  cmpt 5225   × cxp 5683  ran crn 5686  cfv 6561  crio 7387  lecple 17304  occoc 17305  Atomscatm 39264  LHypclh 39986  LTrncltrn 40103  TEndoctendo 40754  DIsoCcdic 41174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-dic 41175
This theorem is referenced by:  dicopelval  41179  dicelvalN  41180  dicval2  41181  dicfnN  41185  dicvalrelN  41187  dicssdvh  41188  dicelval1sta  41189  dihpN  41338
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