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Theorem tglinecom 24680
 Description: Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
Hypotheses
Ref Expression
tglineelsb2.p
tglineelsb2.i Itv
tglineelsb2.l LineG
tglineelsb2.g TarskiG
tglineelsb2.1
tglineelsb2.2
tglineelsb2.4
Assertion
Ref Expression
tglinecom

Proof of Theorem tglinecom
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 tglineelsb2.p . . . 4
2 tglineelsb2.i . . . 4 Itv
3 tglineelsb2.l . . . 4 LineG
4 tglineelsb2.g . . . . 5 TarskiG
54adantr 467 . . . 4 TarskiG
6 tglineelsb2.2 . . . . 5
76adantr 467 . . . 4
8 tglineelsb2.1 . . . . 5
98adantr 467 . . . 4
10 tglineelsb2.4 . . . . . 6
111, 3, 2, 4, 8, 6, 10tglnssp 24597 . . . . 5
1211sselda 3432 . . . 4
1310necomd 2679 . . . . 5
1413adantr 467 . . . 4
15 simpr 463 . . . 4
161, 2, 3, 5, 7, 9, 12, 14, 15lncom 24667 . . 3
174adantr 467 . . . 4 TarskiG
188adantr 467 . . . 4
196adantr 467 . . . 4
201, 3, 2, 4, 6, 8, 13tglnssp 24597 . . . . 5
2120sselda 3432 . . . 4
2210adantr 467 . . . 4
23 simpr 463 . . . 4
241, 2, 3, 17, 18, 19, 21, 22, 23lncom 24667 . . 3
2516, 24impbida 843 . 2
2625eqrdv 2449 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371   wceq 1444   wcel 1887   wne 2622  cfv 5582  (class class class)co 6290  cbs 15121  TarskiGcstrkg 24478  Itvcitv 24484  LineGclng 24485 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-trkgc 24496  df-trkgb 24497  df-trkgcb 24498  df-trkg 24501 This theorem is referenced by:  tglinethru  24681  coltr3  24693  footeq  24766  colperpexlem3  24774  mideulem2  24776  opphllem  24777  midex  24779  opphllem3  24791  opphllem5  24793  lmicom  24830  lmiisolem  24838  lnperpex  24845  trgcopy  24846
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