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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  |-  B  =  ( Base `  G
)
tglineelsb2.i  |-  I  =  (Itv `  G )
tglineelsb2.l  |-  L  =  (LineG `  G )
tglineelsb2.g  |-  ( ph  ->  G  e. TarskiG )
tglineelsb2.1  |-  ( ph  ->  P  e.  B )
tglineelsb2.2  |-  ( ph  ->  Q  e.  B )
tglineelsb2.4  |-  ( ph  ->  P  =/=  Q )
Assertion
Ref Expression
tglinecom  |-  ( ph  ->  ( P L Q )  =  ( Q L P ) )

Proof of Theorem tglinecom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tglineelsb2.p . . . 4  |-  B  =  ( Base `  G
)
2 tglineelsb2.i . . . 4  |-  I  =  (Itv `  G )
3 tglineelsb2.l . . . 4  |-  L  =  (LineG `  G )
4 tglineelsb2.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
54adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  G  e. TarskiG )
6 tglineelsb2.2 . . . . 5  |-  ( ph  ->  Q  e.  B )
76adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  Q  e.  B )
8 tglineelsb2.1 . . . . 5  |-  ( ph  ->  P  e.  B )
98adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  P  e.  B )
10 tglineelsb2.4 . . . . . 6  |-  ( ph  ->  P  =/=  Q )
111, 3, 2, 4, 8, 6, 10tglnssp 24597 . . . . 5  |-  ( ph  ->  ( P L Q )  C_  B )
1211sselda 3432 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  B )
1310necomd 2679 . . . . 5  |-  ( ph  ->  Q  =/=  P )
1413adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  Q  =/=  P )
15 simpr 463 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  ( P L Q ) )
161, 2, 3, 5, 7, 9, 12, 14, 15lncom 24667 . . 3  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  ( Q L P ) )
174adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  G  e. TarskiG )
188adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  P  e.  B )
196adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  Q  e.  B )
201, 3, 2, 4, 6, 8, 13tglnssp 24597 . . . . 5  |-  ( ph  ->  ( Q L P )  C_  B )
2120sselda 3432 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  x  e.  B )
2210adantr 467 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  P  =/=  Q )
23 simpr 463 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  x  e.  ( Q L P ) )
241, 2, 3, 17, 18, 19, 21, 22, 23lncom 24667 . . 3  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  x  e.  ( P L Q ) )
2516, 24impbida 843 . 2  |-  ( ph  ->  ( x  e.  ( P L Q )  <-> 
x  e.  ( Q L P ) ) )
2625eqrdv 2449 1  |-  ( ph  ->  ( P L Q )  =  ( Q L P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   ` cfv 5582  (class class class)co 6290   Basecbs 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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