Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lineval222 Unicode version

Theorem lineval222 26079
 Description: The line passing through two distinct points and . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
lineval2.1 PPoints
lineval2.2 PLines
lineval2.3
lineval2.4 Ig
lineval2.5
lineval2.6
lineval2.7
Assertion
Ref Expression
lineval222
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem lineval222
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lineval2.3 . . . 4
21a1i 10 . . 3
32oveqd 5875 . 2
4 lineval2.1 . . . 4 PPoints
5 lineval2.2 . . . 4 PLines
6 eqid 2283 . . . 4
7 lineval2.4 . . . 4 Ig
84, 5, 6, 7linevala2 26078 . . 3
98oveqd 5875 . 2
10 lineval2.5 . . . 4
11 lineval2.6 . . . 4
12 riotaex 6308 . . . . . 6
13 snex 4216 . . . . . 6
1412, 13ifex 3623 . . . . 5
1514a1i 10 . . . 4
16 simpl 443 . . . . . . 7
17 simpr 447 . . . . . . 7
1816, 17neeq12d 2461 . . . . . 6
1916eleq1d 2349 . . . . . . . 8
2017eleq1d 2349 . . . . . . . 8
2119, 20anbi12d 691 . . . . . . 7
2221riotabidv 6306 . . . . . 6
23 sneq 3651 . . . . . . 7
2423adantr 451 . . . . . 6
2518, 22, 24ifbieq12d 3587 . . . . 5
26 eqid 2283 . . . . 5
2725, 26ovmpt2ga 5977 . . . 4
2810, 11, 15, 27syl3anc 1182 . . 3
29 lineval2.7 . . . 4
30 iftrue 3571 . . . 4
3129, 30syl 15 . . 3
3228, 31eqtrd 2315 . 2
333, 9, 323eqtrd 2319 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684   wne 2446  cvv 2788  cif 3565  csn 3640  cfv 5255  (class class class)co 5858   cmpt2 5860  crio 6297  PPointscpoints 26056  PLinescplines 26058  Igcig 26060  cline 26076 This theorem is referenced by:  lineval42  26080  lineval12  26081  lineval22  26082  lineval4a  26087 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-li 26077
 Copyright terms: Public domain W3C validator