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Theorem diaval 40414
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 40413 . . . 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 6886 . 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 5144 . . . . 5 (𝑦 = 𝑋 β†’ (𝑦 ≀ π‘Š ↔ 𝑋 ≀ π‘Š))
1211elrab 3678 . . . 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 5145 . . . . 5 (π‘₯ = 𝑋 β†’ ((π‘…β€˜π‘“) ≀ π‘₯ ↔ (π‘…β€˜π‘“) ≀ 𝑋))
1514rabbidv 3434 . . . 4 (π‘₯ = 𝑋 β†’ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯} = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
16 eqid 2726 . . . 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 6898 . . . . 5 𝑇 ∈ V
1817rabex 5325 . . . 4 {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋} ∈ V
1915, 16, 18fvmpt 6991 . . 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 2766 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 1533   ∈ wcel 2098  {crab 3426   class class class wbr 5141   ↦ cmpt 5224  β€˜cfv 6536  Basecbs 17151  lecple 17211  LHypclh 39366  LTrncltrn 39483  trLctrl 39540  DIsoAcdia 40410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-disoa 40411
This theorem is referenced by:  diaelval  40415  diass  40424  diaord  40429  dia0  40434  dia1N  40435  diassdvaN  40442  dia1dim  40443  cdlemm10N  40500  dibval3N  40528  dihwN  40671
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