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Theorem diaelrnN 40518
Description: Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diaelrn.h 𝐻 = (LHypβ€˜πΎ)
diaelrn.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diaelrn.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diaelrnN (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ ran 𝐼) β†’ 𝑆 βŠ† 𝑇)

Proof of Theorem diaelrnN
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2728 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
3 diaelrn.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
4 diaelrn.i . . . . 5 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
51, 2, 3, 4diafn 40507 . . . 4 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {𝑦 ∈ (Baseβ€˜πΎ) ∣ 𝑦(leβ€˜πΎ)π‘Š})
6 fvelrnb 6959 . . . 4 (𝐼 Fn {𝑦 ∈ (Baseβ€˜πΎ) ∣ 𝑦(leβ€˜πΎ)π‘Š} β†’ (𝑆 ∈ ran 𝐼 ↔ βˆƒπ‘₯ ∈ {𝑦 ∈ (Baseβ€˜πΎ) ∣ 𝑦(leβ€˜πΎ)π‘Š} (πΌβ€˜π‘₯) = 𝑆))
75, 6syl 17 . . 3 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ ran 𝐼 ↔ βˆƒπ‘₯ ∈ {𝑦 ∈ (Baseβ€˜πΎ) ∣ 𝑦(leβ€˜πΎ)π‘Š} (πΌβ€˜π‘₯) = 𝑆))
8 breq1 5151 . . . . . 6 (𝑦 = π‘₯ β†’ (𝑦(leβ€˜πΎ)π‘Š ↔ π‘₯(leβ€˜πΎ)π‘Š))
98elrab 3682 . . . . 5 (π‘₯ ∈ {𝑦 ∈ (Baseβ€˜πΎ) ∣ 𝑦(leβ€˜πΎ)π‘Š} ↔ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š))
10 diaelrn.t . . . . . . . 8 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
111, 2, 3, 10, 4diass 40515 . . . . . . 7 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š)) β†’ (πΌβ€˜π‘₯) βŠ† 𝑇)
1211ex 412 . . . . . 6 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š) β†’ (πΌβ€˜π‘₯) βŠ† 𝑇))
13 sseq1 4005 . . . . . . 7 ((πΌβ€˜π‘₯) = 𝑆 β†’ ((πΌβ€˜π‘₯) βŠ† 𝑇 ↔ 𝑆 βŠ† 𝑇))
1413biimpcd 248 . . . . . 6 ((πΌβ€˜π‘₯) βŠ† 𝑇 β†’ ((πΌβ€˜π‘₯) = 𝑆 β†’ 𝑆 βŠ† 𝑇))
1512, 14syl6 35 . . . . 5 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š) β†’ ((πΌβ€˜π‘₯) = 𝑆 β†’ 𝑆 βŠ† 𝑇)))
169, 15biimtrid 241 . . . 4 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (π‘₯ ∈ {𝑦 ∈ (Baseβ€˜πΎ) ∣ 𝑦(leβ€˜πΎ)π‘Š} β†’ ((πΌβ€˜π‘₯) = 𝑆 β†’ 𝑆 βŠ† 𝑇)))
1716rexlimdv 3150 . . 3 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (βˆƒπ‘₯ ∈ {𝑦 ∈ (Baseβ€˜πΎ) ∣ 𝑦(leβ€˜πΎ)π‘Š} (πΌβ€˜π‘₯) = 𝑆 β†’ 𝑆 βŠ† 𝑇))
187, 17sylbid 239 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ ran 𝐼 β†’ 𝑆 βŠ† 𝑇))
1918imp 406 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ ran 𝐼) β†’ 𝑆 βŠ† 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3067  {crab 3429   βŠ† wss 3947   class class class wbr 5148  ran crn 5679   Fn wfn 6543  β€˜cfv 6548  Basecbs 17179  lecple 17239  LHypclh 39457  LTrncltrn 39574  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:  dvadiaN  40601  djaclN  40609  djajN  40610
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