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

Theorem isline 36755
Description: The predicate "is a line". (Contributed by NM, 19-Sep-2011.)
Hypotheses
Ref Expression
isline.l = (le‘𝐾)
isline.j = (join‘𝐾)
isline.a 𝐴 = (Atoms‘𝐾)
isline.n 𝑁 = (Lines‘𝐾)
Assertion
Ref Expression
isline (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
Distinct variable groups:   𝑞,𝑝,𝑟,𝐴   𝐾,𝑝,𝑞,𝑟   𝑋,𝑞,𝑟
Allowed substitution hints:   𝐷(𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   (𝑟,𝑞,𝑝)   𝑁(𝑟,𝑞,𝑝)   𝑋(𝑝)

Proof of Theorem isline
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isline.l . . . 4 = (le‘𝐾)
2 isline.j . . . 4 = (join‘𝐾)
3 isline.a . . . 4 𝐴 = (Atoms‘𝐾)
4 isline.n . . . 4 𝑁 = (Lines‘𝐾)
51, 2, 3, 4lineset 36754 . . 3 (𝐾𝐷𝑁 = {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})})
65eleq2d 2895 . 2 (𝐾𝐷 → (𝑋𝑁𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})}))
73fvexi 6677 . . . . . . . 8 𝐴 ∈ V
87rabex 5226 . . . . . . 7 {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V
9 eleq1 2897 . . . . . . 7 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → (𝑋 ∈ V ↔ {𝑝𝐴𝑝 (𝑞 𝑟)} ∈ V))
108, 9mpbiri 259 . . . . . 6 (𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)} → 𝑋 ∈ V)
1110adantl 482 . . . . 5 ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
1211a1i 11 . . . 4 ((𝑞𝐴𝑟𝐴) → ((𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V))
1312rexlimivv 3289 . . 3 (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}) → 𝑋 ∈ V)
14 eqeq1 2822 . . . . 5 (𝑥 = 𝑋 → (𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)} ↔ 𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
1514anbi2d 628 . . . 4 (𝑥 = 𝑋 → ((𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
16152rexbidv 3297 . . 3 (𝑥 = 𝑋 → (∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)}) ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
1713, 16elab3 3671 . 2 (𝑋 ∈ {𝑥 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑥 = {𝑝𝐴𝑝 (𝑞 𝑟)})} ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)}))
186, 17syl6bb 288 1 (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wrex 3136  {crab 3139  Vcvv 3492   class class class wbr 5057  cfv 6348  (class class class)co 7145  lecple 16560  joincjn 17542  Atomscatm 36279  Linesclines 36510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-lines 36517
This theorem is referenced by:  islinei  36756  linepsubN  36768  isline2  36790
  Copyright terms: Public domain W3C validator