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Theorem diaelrnN 41047
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 = (LHyp‘𝐾)
diaelrn.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
diaelrn.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
Assertion
Ref Expression
diaelrnN (((𝐾𝑉𝑊𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆𝑇)

Proof of Theorem diaelrnN
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2737 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 diaelrn.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 diaelrn.i . . . . 5 𝐼 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diafn 41036 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
6 fvelrnb 6969 . . . 4 (𝐼 Fn {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} → (𝑆 ∈ ran 𝐼 ↔ ∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼𝑥) = 𝑆))
75, 6syl 17 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆 ∈ ran 𝐼 ↔ ∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼𝑥) = 𝑆))
8 breq1 5146 . . . . . 6 (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊𝑥(le‘𝐾)𝑊))
98elrab 3692 . . . . 5 (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊))
10 diaelrn.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
111, 2, 3, 10, 4diass 41044 . . . . . . 7 (((𝐾𝑉𝑊𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐼𝑥) ⊆ 𝑇)
1211ex 412 . . . . . 6 ((𝐾𝑉𝑊𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐼𝑥) ⊆ 𝑇))
13 sseq1 4009 . . . . . . 7 ((𝐼𝑥) = 𝑆 → ((𝐼𝑥) ⊆ 𝑇𝑆𝑇))
1413biimpcd 249 . . . . . 6 ((𝐼𝑥) ⊆ 𝑇 → ((𝐼𝑥) = 𝑆𝑆𝑇))
1512, 14syl6 35 . . . . 5 ((𝐾𝑉𝑊𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐼𝑥) = 𝑆𝑆𝑇)))
169, 15biimtrid 242 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} → ((𝐼𝑥) = 𝑆𝑆𝑇)))
1716rexlimdv 3153 . . 3 ((𝐾𝑉𝑊𝐻) → (∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼𝑥) = 𝑆𝑆𝑇))
187, 17sylbid 240 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆 ∈ ran 𝐼𝑆𝑇))
1918imp 406 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  wss 3951   class class class wbr 5143  ran crn 5686   Fn wfn 6556  cfv 6561  Basecbs 17247  lecple 17304  LHypclh 39986  LTrncltrn 40103  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:  dvadiaN  41130  djaclN  41138  djajN  41139
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