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Theorem linevala2 26078
 Description: Definition of the line xy. It also defines a degenerate line. (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
Assertion
Ref Expression
linevala2
Distinct variable groups:   ,,,   ,   ,,
Allowed substitution hints:   (,,)   ()   (,)   (,,)

Proof of Theorem linevala2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 lineval2.3 . 2
2 lineval2.4 . . 3 Ig
3 lineval2.1 . . . . . 6 PPoints
4 fvex 5539 . . . . . 6 PPoints
53, 4eqeltri 2353 . . . . 5
65, 5pm3.2i 441 . . . 4
7 mpt2exga 6197 . . . 4
86, 7mp1i 11 . . 3
9 fveq2 5525 . . . . . 6 PPoints PPoints
109, 3syl6eqr 2333 . . . . 5 PPoints
11 fveq2 5525 . . . . . . . 8 PLines PLines
12 lineval2.2 . . . . . . . 8 PLines
1311, 12syl6eqr 2333 . . . . . . 7 PLines
1413riotaeqdv 6305 . . . . . 6 PLines
1514ifeq1d 3579 . . . . 5 PLines
1610, 10, 15mpt2eq123dv 5910 . . . 4 PPoints PPoints PLines
17 df-li 26077 . . . 4 Ig PPoints PPoints PLines
1816, 17fvmptg 5600 . . 3 Ig
192, 8, 18syl2anc 642 . 2
201, 19syl5eq 2327 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   cmpt2 5860  crio 6297  PPointscpoints 26056  PLinescplines 26058  Igcig 26060  cline 26076 This theorem is referenced by:  lineval222  26079  lineval3a  26083 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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-li 26077
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