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

Theorem diael 40404
Description: A member of the value of the partial isomorphism A is a translation, i.e., a vector. (Contributed by NM, 17-Jan-2014.)
Hypotheses
Ref Expression
diass.b 𝐡 = (Baseβ€˜πΎ)
diass.l ≀ = (leβ€˜πΎ)
diass.h 𝐻 = (LHypβ€˜πΎ)
diass.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diass.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diael (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ 𝐹 ∈ 𝑇)

Proof of Theorem diael
StepHypRef Expression
1 diass.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 diass.l . . . 4 ≀ = (leβ€˜πΎ)
3 diass.h . . . 4 𝐻 = (LHypβ€˜πΎ)
4 diass.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 diass.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5diass 40403 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) βŠ† 𝑇)
76sseld 3973 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) β†’ 𝐹 ∈ 𝑇))
873impia 1114 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ 𝐹 ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5138  β€˜cfv 6533  Basecbs 17143  lecple 17203  LHypclh 39345  LTrncltrn 39462  DIsoAcdia 40389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-disoa 40390
This theorem is referenced by:  dialss  40407  dibelval1st1  40511  diblsmopel  40532
  Copyright terms: Public domain W3C validator