Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diaelrnN Structured version   Visualization version   GIF version

Theorem diaelrnN 37066
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 2799 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2799 . . . . 5 (le‘𝐾) = (le‘𝐾)
3 diaelrn.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 diaelrn.i . . . . 5 𝐼 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diafn 37055 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
6 fvelrnb 6468 . . . 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 4846 . . . . . 6 (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊𝑥(le‘𝐾)𝑊))
98elrab 3556 . . . . 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 37063 . . . . . . 7 (((𝐾𝑉𝑊𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐼𝑥) ⊆ 𝑇)
1211ex 402 . . . . . 6 ((𝐾𝑉𝑊𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐼𝑥) ⊆ 𝑇))
13 sseq1 3822 . . . . . . 7 ((𝐼𝑥) = 𝑆 → ((𝐼𝑥) ⊆ 𝑇𝑆𝑇))
1413biimpcd 241 . . . . . 6 ((𝐼𝑥) ⊆ 𝑇 → ((𝐼𝑥) = 𝑆𝑆𝑇))
1512, 14syl6 35 . . . . 5 ((𝐾𝑉𝑊𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐼𝑥) = 𝑆𝑆𝑇)))
169, 15syl5bi 234 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} → ((𝐼𝑥) = 𝑆𝑆𝑇)))
1716rexlimdv 3211 . . 3 ((𝐾𝑉𝑊𝐻) → (∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼𝑥) = 𝑆𝑆𝑇))
187, 17sylbid 232 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆 ∈ ran 𝐼𝑆𝑇))
1918imp 396 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wrex 3090  {crab 3093  wss 3769   class class class wbr 4843  ran crn 5313   Fn wfn 6096  cfv 6101  Basecbs 16184  lecple 16274  LHypclh 36005  LTrncltrn 36122  DIsoAcdia 37049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-disoa 37050
This theorem is referenced by:  dvadiaN  37149  djaclN  37157  djajN  37158
  Copyright terms: Public domain W3C validator