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Theorem tgellng 24458
 Description: Property of lying on the line going through points and . Definition 4.10 of [Schwabhauser] p. 36. We choose the notation LineG instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-2019.)
Hypotheses
Ref Expression
tglngval.p
tglngval.l LineG
tglngval.i Itv
tglngval.g TarskiG
tglngval.x
tglngval.y
tglngval.z
tgellng.z
Assertion
Ref Expression
tgellng

Proof of Theorem tgellng
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 tglngval.p . . . . 5
2 tglngval.l . . . . 5 LineG
3 tglngval.i . . . . 5 Itv
4 tglngval.g . . . . 5 TarskiG
5 tglngval.x . . . . 5
6 tglngval.y . . . . 5
7 tglngval.z . . . . 5
81, 2, 3, 4, 5, 6, 7tglngval 24456 . . . 4
98eleq2d 2499 . . 3
10 eleq1 2501 . . . . 5
11 oveq1 6312 . . . . . 6
1211eleq2d 2499 . . . . 5
13 oveq2 6313 . . . . . 6
1413eleq2d 2499 . . . . 5
1510, 12, 143orbi123d 1334 . . . 4
1615elrab 3235 . . 3
179, 16syl6bb 264 . 2
18 tgellng.z . . 3
1918biantrurd 510 . 2
2017, 19bitr4d 259 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   w3o 981   wceq 1437   wcel 1870   wne 2625  crab 2786  cfv 5601  (class class class)co 6305  cbs 15084  TarskiGcstrkg 24341  Itvcitv 24347  LineGclng 24348 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-trkg 24364 This theorem is referenced by:  tgcolg  24459  hlln  24512  btwnlng1  24523  btwnlng2  24524  btwnlng3  24525  lncom  24526  lnrot1  24527  lnrot2  24528  tglineeltr  24535  colmid  24590  cgracol  24724
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