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Theorem diafval 41014
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 41013 . . . 4 (𝐾𝑉 → (DIsoA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})))
65fveq1d 6909 . . 3 (𝐾𝑉 → ((DIsoA‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))‘𝑊))
71, 6eqtrid 2787 . 2 (𝐾𝑉𝐼 = ((𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}))‘𝑊))
8 breq2 5152 . . . . 5 (𝑤 = 𝑊 → (𝑦 𝑤𝑦 𝑊))
98rabbidv 3441 . . . 4 (𝑤 = 𝑊 → {𝑦𝐵𝑦 𝑤} = {𝑦𝐵𝑦 𝑊})
10 fveq2 6907 . . . . . 6 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
11 diaval.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
1210, 11eqtr4di 2793 . . . . 5 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
13 fveq2 6907 . . . . . . . 8 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = ((trL‘𝐾)‘𝑊))
14 diaval.r . . . . . . . 8 𝑅 = ((trL‘𝐾)‘𝑊)
1513, 14eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = 𝑅)
1615fveq1d 6909 . . . . . 6 (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘𝑓) = (𝑅𝑓))
1716breq1d 5158 . . . . 5 (𝑤 = 𝑊 → ((((trL‘𝐾)‘𝑤)‘𝑓) 𝑥 ↔ (𝑅𝑓) 𝑥))
1812, 17rabeqbidv 3452 . . . 4 (𝑤 = 𝑊 → {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥} = {𝑓𝑇 ∣ (𝑅𝑓) 𝑥})
199, 18mpteq12dv 5239 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥}) = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
20 eqid 2735 . . 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 6921 . . . 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 2795 1 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433   class class class wbr 5148  cmpt 5231  cfv 6563  Basecbs 17245  lecple 17305  LHypclh 39967  LTrncltrn 40084  trLctrl 40141  DIsoAcdia 41011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-disoa 41012
This theorem is referenced by:  diaval  41015  diafn  41017
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