Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lineval12a Unicode version

Theorem lineval12a 26084
 Description: The line passing through two distinct points and is a set of points . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
lineval12a.1 PPoints
lineval12a.3
lineval12a.4 Ig
lineval12a.5
lineval12a.6
Assertion
Ref Expression
lineval12a

Proof of Theorem lineval12a
StepHypRef Expression
1 lineval12a.1 . . . . 5 PPoints
2 lineval12a.3 . . . . 5
3 lineval12a.4 . . . . 5 Ig
4 lineval12a.6 . . . . 5
51, 2, 3, 4lineval3a 26083 . . . 4
64snssd 3760 . . . 4
75, 6eqsstrd 3212 . . 3
8 oveq1 5865 . . . 4
98sseq1d 3205 . . 3
107, 9syl5ibr 212 . 2
11 eqid 2283 . . . 4 PLines PLines
123adantl 452 . . . 4 Ig
13 lineval12a.5 . . . . . 6
1413adantl 452 . . . . 5
154adantl 452 . . . . 5
16 simpl 443 . . . . 5
171, 11, 2, 12, 14, 15, 16lineval12 26081 . . . 4 PLines
181, 11, 12, 17isig12 26064 . . 3
1918ex 423 . 2
2010, 19pm2.61ine 2522 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684   wne 2446   wss 3152  csn 3640  cfv 5255  (class class class)co 5858  PPointscpoints 26056  PLinescplines 26058  Igcig 26060  cline 26076 This theorem is referenced by:  lineval5a  26088  lineval6a  26089 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-ig2 26061  df-li 26077
 Copyright terms: Public domain W3C validator