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Theorem diaval 40505
Description: The partial isomorphism A for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
Hypotheses
Ref Expression
diaval.b 𝐡 = (Baseβ€˜πΎ)
diaval.l ≀ = (leβ€˜πΎ)
diaval.h 𝐻 = (LHypβ€˜πΎ)
diaval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diaval.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
diaval.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diaval (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
Distinct variable groups:   𝑓,𝐾   𝑇,𝑓   𝑓,π‘Š   𝑓,𝑋
Allowed substitution hints:   𝐡(𝑓)   𝑅(𝑓)   𝐻(𝑓)   𝐼(𝑓)   ≀ (𝑓)   𝑉(𝑓)

Proof of Theorem diaval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 diaval.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 diaval.l . . . . 5 ≀ = (leβ€˜πΎ)
3 diaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
4 diaval.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 diaval.r . . . . 5 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
6 diaval.i . . . . 5 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
71, 2, 3, 4, 5, 6diafval 40504 . . . 4 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
87adantr 480 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ 𝐼 = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
98fveq1d 6899 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = ((π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})β€˜π‘‹))
10 simpr 484 . . . 4 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
11 breq1 5151 . . . . 5 (𝑦 = 𝑋 β†’ (𝑦 ≀ π‘Š ↔ 𝑋 ≀ π‘Š))
1211elrab 3682 . . . 4 (𝑋 ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↔ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
1310, 12sylibr 233 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ 𝑋 ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š})
14 breq2 5152 . . . . 5 (π‘₯ = 𝑋 β†’ ((π‘…β€˜π‘“) ≀ π‘₯ ↔ (π‘…β€˜π‘“) ≀ 𝑋))
1514rabbidv 3437 . . . 4 (π‘₯ = 𝑋 β†’ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯} = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
16 eqid 2728 . . . 4 (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}) = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})
174fvexi 6911 . . . . 5 𝑇 ∈ V
1817rabex 5334 . . . 4 {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋} ∈ V
1915, 16, 18fvmpt 7005 . . 3 (𝑋 ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} β†’ ((π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})β€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
2013, 19syl 17 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ ((π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯})β€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
219, 20eqtrd 2768 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3429   class class class wbr 5148   ↦ cmpt 5231  β€˜cfv 6548  Basecbs 17180  lecple 17240  LHypclh 39457  LTrncltrn 39574  trLctrl 39631  DIsoAcdia 40501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-disoa 40502
This theorem is referenced by:  diaelval  40506  diass  40515  diaord  40520  dia0  40525  dia1N  40526  diassdvaN  40533  dia1dim  40534  cdlemm10N  40591  dibval3N  40619  dihwN  40762
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