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Theorem diaelrnN 36160
 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 2621 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 diaelrn.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 diaelrn.i . . . . 5 𝐼 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diafn 36149 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
6 fvelrnb 6241 . . . 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 4654 . . . . . 6 (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊𝑥(le‘𝐾)𝑊))
98elrab 3361 . . . . 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 36157 . . . . . . 7 (((𝐾𝑉𝑊𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐼𝑥) ⊆ 𝑇)
1211ex 450 . . . . . 6 ((𝐾𝑉𝑊𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐼𝑥) ⊆ 𝑇))
13 sseq1 3624 . . . . . . 7 ((𝐼𝑥) = 𝑆 → ((𝐼𝑥) ⊆ 𝑇𝑆𝑇))
1413biimpcd 239 . . . . . 6 ((𝐼𝑥) ⊆ 𝑇 → ((𝐼𝑥) = 𝑆𝑆𝑇))
1512, 14syl6 35 . . . . 5 ((𝐾𝑉𝑊𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐼𝑥) = 𝑆𝑆𝑇)))
169, 15syl5bi 232 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} → ((𝐼𝑥) = 𝑆𝑆𝑇)))
1716rexlimdv 3028 . . 3 ((𝐾𝑉𝑊𝐻) → (∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼𝑥) = 𝑆𝑆𝑇))
187, 17sylbid 230 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆 ∈ ran 𝐼𝑆𝑇))
1918imp 445 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1482   ∈ wcel 1989  ∃wrex 2912  {crab 2915   ⊆ wss 3572   class class class wbr 4651  ran crn 5113   Fn wfn 5881  ‘cfv 5886  Basecbs 15851  lecple 15942  LHypclh 35096  LTrncltrn 35213  DIsoAcdia 36143 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-disoa 36144 This theorem is referenced by:  dvadiaN  36243  djaclN  36251  djajN  36252
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