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 39558
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 2733 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2733 . . . . 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 39547 . . . 4 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {𝑦 ∈ (Baseβ€˜πΎ) ∣ 𝑦(leβ€˜πΎ)π‘Š})
6 fvelrnb 6907 . . . 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 5112 . . . . . 6 (𝑦 = π‘₯ β†’ (𝑦(leβ€˜πΎ)π‘Š ↔ π‘₯(leβ€˜πΎ)π‘Š))
98elrab 3649 . . . . 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 39555 . . . . . . 7 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š)) β†’ (πΌβ€˜π‘₯) βŠ† 𝑇)
1211ex 414 . . . . . 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 3973 . . . . . . 7 ((πΌβ€˜π‘₯) = 𝑆 β†’ ((πΌβ€˜π‘₯) βŠ† 𝑇 ↔ 𝑆 βŠ† 𝑇))
1413biimpcd 249 . . . . . 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 3147 . . 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 408 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070  {crab 3406   βŠ† wss 3914   class class class wbr 5109  ran crn 5638   Fn wfn 6495  β€˜cfv 6500  Basecbs 17091  lecple 17148  LHypclh 38497  LTrncltrn 38614  DIsoAcdia 39541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-disoa 39542
This theorem is referenced by:  dvadiaN  39641  djaclN  39649  djajN  39650
  Copyright terms: Public domain W3C validator