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Theorem tglinecom 23053
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 22998 . . . . 5  |-  ( ph  ->  ( P L Q )  C_  B )
1211sselda 3368 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  B )
1310necomd 2707 . . . . 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 23041 . . 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 22998 . . . . 5  |-  ( ph  ->  ( Q L P )  C_  B )
2120sselda 3368 . . . 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 23041 . . 3  |-  ( (
ph  /\  x  e.  ( Q L P ) )  ->  x  e.  ( P L Q ) )
2516, 24impbida 828 . 2  |-  ( ph  ->  ( x  e.  ( P L Q )  <-> 
x  e.  ( Q L P ) ) )
2625eqrdv 2441 1  |-  ( ph  ->  ( P L Q )  =  ( Q L P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618   ` cfv 5430  (class class class)co 6103   Basecbs 14186  TarskiGcstrkg 22901  Itvcitv 22909  LineGclng 22910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-trkgc 22921  df-trkgb 22922  df-trkgcb 22923  df-trkg 22928
This theorem is referenced by:  tglinethru  23054
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