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Theorem diaelrnN 38217
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 2820 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2820 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 diaelrn.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 diaelrn.i . . . . 5 𝐼 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diafn 38206 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
6 fvelrnb 6702 . . . 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 5045 . . . . . 6 (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊𝑥(le‘𝐾)𝑊))
98elrab 3660 . . . . 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 38214 . . . . . . 7 (((𝐾𝑉𝑊𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐼𝑥) ⊆ 𝑇)
1211ex 415 . . . . . 6 ((𝐾𝑉𝑊𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐼𝑥) ⊆ 𝑇))
13 sseq1 3971 . . . . . . 7 ((𝐼𝑥) = 𝑆 → ((𝐼𝑥) ⊆ 𝑇𝑆𝑇))
1413biimpcd 251 . . . . . 6 ((𝐼𝑥) ⊆ 𝑇 → ((𝐼𝑥) = 𝑆𝑆𝑇))
1512, 14syl6 35 . . . . 5 ((𝐾𝑉𝑊𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐼𝑥) = 𝑆𝑆𝑇)))
169, 15syl5bi 244 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} → ((𝐼𝑥) = 𝑆𝑆𝑇)))
1716rexlimdv 3270 . . 3 ((𝐾𝑉𝑊𝐻) → (∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼𝑥) = 𝑆𝑆𝑇))
187, 17sylbid 242 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆 ∈ ran 𝐼𝑆𝑇))
1918imp 409 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  wrex 3126  {crab 3129  wss 3913   class class class wbr 5042  ran crn 5532   Fn wfn 6326  cfv 6331  Basecbs 16462  lecple 16551  LHypclh 37156  LTrncltrn 37273  DIsoAcdia 38200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-disoa 38201
This theorem is referenced by:  dvadiaN  38300  djaclN  38308  djajN  38309
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