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Theorem diafval 41033
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 41032 . . . 4 (𝐾𝑉 → (DIsoA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})))
65fveq1d 6908 . . 3 (𝐾𝑉 → ((DIsoA‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))‘𝑊))
71, 6eqtrid 2789 . 2 (𝐾𝑉𝐼 = ((𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))‘𝑊))
8 breq2 5147 . . . . 5 (𝑤 = 𝑊 → (𝑦 𝑤𝑦 𝑊))
98rabbidv 3444 . . . 4 (𝑤 = 𝑊 → {𝑦𝐵𝑦 𝑤} = {𝑦𝐵𝑦 𝑊})
10 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
11 diaval.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
1210, 11eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
13 fveq2 6906 . . . . . . . 8 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = ((trL‘𝐾)‘𝑊))
14 diaval.r . . . . . . . 8 𝑅 = ((trL‘𝐾)‘𝑊)
1513, 14eqtr4di 2795 . . . . . . 7 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = 𝑅)
1615fveq1d 6908 . . . . . 6 (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘𝑓) = (𝑅𝑓))
1716breq1d 5153 . . . . 5 (𝑤 = 𝑊 → ((((trL‘𝐾)‘𝑤)‘𝑓) 𝑥 ↔ (𝑅𝑓) 𝑥))
1812, 17rabeqbidv 3455 . . . 4 (𝑤 = 𝑊 → {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥} = {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})
199, 18mpteq12dv 5233 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}) = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
20 eqid 2737 . . 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}))
212fvexi 6920 . . . 4 𝐵 ∈ V
2221mptrabex 7245 . . 3 (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}) ∈ V
2319, 20, 22fvmpt 7016 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))‘𝑊) = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
247, 23sylan9eq 2797 1 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436   class class class wbr 5143  cmpt 5225  cfv 6561  Basecbs 17247  lecple 17304  LHypclh 39986  LTrncltrn 40103  trLctrl 40160  DIsoAcdia 41030
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-pr 5432
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-disoa 41031
This theorem is referenced by:  diaval  41034  diafn  41036
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