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Theorem tglineelsb2 24137
Description: If  S lies on PQ , then PQ = PS . Theorem 6.16 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 )
tglineelsb2.3  |-  ( ph  ->  S  e.  B )
tglineelsb2.5  |-  ( ph  ->  S  =/=  P )
tglineelsb2.6  |-  ( ph  ->  S  e.  ( P L Q ) )
Assertion
Ref Expression
tglineelsb2  |-  ( ph  ->  ( P L Q )  =  ( P L S ) )

Proof of Theorem tglineelsb2
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 tglineelsb2.1 . . . . 5  |-  ( ph  ->  P  e.  B )
76adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  P  e.  B )
8 tglineelsb2.3 . . . . 5  |-  ( ph  ->  S  e.  B )
98adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  S  e.  B )
10 tglineelsb2.5 . . . . . 6  |-  ( ph  ->  S  =/=  P )
1110necomd 2728 . . . . 5  |-  ( ph  ->  P  =/=  S )
1211adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  P  =/=  S )
13 tglineelsb2.2 . . . . 5  |-  ( ph  ->  Q  e.  B )
1413adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  Q  e.  B )
15 tglineelsb2.4 . . . . . 6  |-  ( ph  ->  P  =/=  Q )
1615necomd 2728 . . . . 5  |-  ( ph  ->  Q  =/=  P )
1716adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  Q  =/=  P )
18 tglineelsb2.6 . . . . . . 7  |-  ( ph  ->  S  e.  ( P L Q ) )
1918adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  S  e.  ( P L Q ) )
201, 2, 3, 5, 14, 7, 9, 17, 19lncom 24127 . . . . 5  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  S  e.  ( Q L P ) )
211, 2, 3, 5, 7, 9, 14, 12, 20, 17lnrot1 24128 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  Q  e.  ( P L S ) )
221, 3, 2, 4, 6, 13, 15tglnssp 24064 . . . . 5  |-  ( ph  ->  ( P L Q )  C_  B )
2322sselda 3499 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  B )
24 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  ( P L Q ) )
251, 2, 3, 5, 7, 9, 12, 14, 17, 21, 23, 24tglineeltr 24136 . . 3  |-  ( (
ph  /\  x  e.  ( P L Q ) )  ->  x  e.  ( P L S ) )
264adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  G  e. TarskiG )
276adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  P  e.  B )
2813adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  Q  e.  B )
2915adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  P  =/=  Q )
308adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  S  e.  B )
3110adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  S  =/=  P )
3218adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  S  e.  ( P L Q ) )
331, 3, 2, 4, 6, 8, 11tglnssp 24064 . . . . 5  |-  ( ph  ->  ( P L S )  C_  B )
3433sselda 3499 . . . 4  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  x  e.  B )
35 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  x  e.  ( P L S ) )
361, 2, 3, 26, 27, 28, 29, 30, 31, 32, 34, 35tglineeltr 24136 . . 3  |-  ( (
ph  /\  x  e.  ( P L S ) )  ->  x  e.  ( P L Q ) )
3725, 36impbida 832 . 2  |-  ( ph  ->  ( x  e.  ( P L Q )  <-> 
x  e.  ( P L S ) ) )
3837eqrdv 2454 1  |-  ( ph  ->  ( P L Q )  =  ( P L S ) )
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 14643  TarskiGcstrkg 23950  Itvcitv 23957  LineGclng 23958
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-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
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-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-hash 12408  df-word 12545  df-concat 12547  df-s1 12548  df-s2 12824  df-s3 12825  df-trkgc 23969  df-trkgb 23970  df-trkgcb 23971  df-trkg 23975  df-cgrg 24028
This theorem is referenced by:  tglinethru  24141  ncolncol  24151  coltr3  24154  hlperpnel  24224  colperpexlem3  24231  mideulem2  24233  lmieu  24275  lmiisolem  24286
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