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Theorem tglinecom 24141
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 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  G  e. TarskiG )
6 tglineelsb2.2 . . . . 5  |-  ( ph  ->  Q  e.  B )
76adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  Q  e.  B )
8 tglineelsb2.1 . . . . 5  |-  ( ph  ->  P  e.  B )
98adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  P  e.  B )
10 tglineelsb2.4 . . . . . 6  |-  ( ph  ->  P  =/=  Q )
111, 3, 2, 4, 8, 6, 10tglnssp 24065 . . . . 5  |-  ( ph  ->  ( P L Q )  C_  B )
1211sselda 3499 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  B )
1310necomd 2728 . . . . 5  |-  ( ph  ->  Q  =/=  P )
1413adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  Q  =/=  P )
15 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  ( P L Q ) )
161, 2, 3, 5, 7, 9, 12, 14, 15lncom 24128 . . 3  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  ( Q L P ) )
174adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  G  e. TarskiG )
188adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  P  e.  B )
196adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  Q  e.  B )
201, 3, 2, 4, 6, 8, 13tglnssp 24065 . . . . 5  |-  ( ph  ->  ( Q L P )  C_  B )
2120sselda 3499 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  x  e.  B )
2210adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  P  =/=  Q )
23 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  x  e.  ( Q L P ) )
241, 2, 3, 17, 18, 19, 21, 22, 23lncom 24128 . . 3  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  x  e.  ( P L Q ) )
2516, 24impbida 832 . 2  |-  ( ph  ->  ( x  e.  ( P L Q )  <-> 
x  e.  ( Q L P ) ) )
2625eqrdv 2454 1  |-  ( ph  ->  ( P L Q )  =  ( Q L P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   ` cfv 5594  (class class class)co 6296   Basecbs 14644  TarskiGcstrkg 23951  Itvcitv 23958  LineGclng 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976
This theorem is referenced by:  tglinethru  24142  coltr3  24155  footeq  24224  colperpexlem3  24232  mideulem2  24234  opphllem  24235  midex  24237  opphllem3  24247  opphllem5  24249  lmicom  24280  lmiisolem  24287
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