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Theorem tglinethru 24220
Description: If  A is a line containing two distinct points  P and  Q, then  A is the line through  P and  Q. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 25-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 )
tglinethru.0  |-  ( ph  ->  P  =/=  Q )
tglinethru.1  |-  ( ph  ->  A  e.  ran  L
)
tglinethru.2  |-  ( ph  ->  P  e.  A )
tglinethru.3  |-  ( ph  ->  Q  e.  A )
Assertion
Ref Expression
tglinethru  |-  ( ph  ->  A  =  ( P L Q ) )

Proof of Theorem tglinethru
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglineelsb2.p . . . . 5  |-  B  =  ( Base `  G
)
2 tglineelsb2.i . . . . 5  |-  I  =  (Itv `  G )
3 tglineelsb2.l . . . . 5  |-  L  =  (LineG `  G )
4 tglineelsb2.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
54ad4antr 729 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  G  e. TarskiG )
6 simp-4r 766 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  x  e.  B )
7 simpllr 758 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  y  e.  B )
8 simplrr 760 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  x  =/=  y )
9 tglineelsb2.2 . . . . . 6  |-  ( ph  ->  Q  e.  B )
109ad4antr 729 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  Q  e.  B )
11 tglinethru.0 . . . . . . . 8  |-  ( ph  ->  P  =/=  Q )
1211ad4antr 729 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  P  =/=  Q )
1312necomd 2725 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  Q  =/=  P )
14 simpr 459 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  P  =  x )
1513, 14neeqtrd 2749 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  Q  =/=  x )
16 tglinethru.3 . . . . . . 7  |-  ( ph  ->  Q  e.  A )
1716ad4antr 729 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  Q  e.  A )
18 simplrl 759 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  A  =  ( x L y ) )
1917, 18eleqtrd 2544 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  Q  e.  ( x L y ) )
201, 2, 3, 5, 6, 7, 8, 10, 15, 19tglineelsb2 24216 . . . 4  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  ( x L y )  =  ( x L Q ) )
2114oveq1d 6285 . . . 4  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  ( P L Q )  =  ( x L Q ) )
2220, 18, 213eqtr4d 2505 . . 3  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =  x )  ->  A  =  ( P L Q ) )
23 simplrl 759 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  A  =  ( x L y ) )
244ad4antr 729 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  G  e. TarskiG )
25 simp-4r 766 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  x  e.  B )
26 simpllr 758 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  -> 
y  e.  B )
27 simplrr 760 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  x  =/=  y )
28 tglineelsb2.1 . . . . . . 7  |-  ( ph  ->  P  e.  B )
2928ad4antr 729 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  P  e.  B )
30 simpr 459 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  P  =/=  x )
31 tglinethru.2 . . . . . . . 8  |-  ( ph  ->  P  e.  A )
3231ad4antr 729 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  P  e.  A )
3332, 23eleqtrd 2544 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  P  e.  ( x L y ) )
341, 2, 3, 24, 25, 26, 27, 29, 30, 33tglineelsb2 24216 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  -> 
( x L y )  =  ( x L P ) )
3530necomd 2725 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  x  =/=  P )
361, 2, 3, 24, 25, 29, 35tglinecom 24219 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  -> 
( x L P )  =  ( P L x ) )
3723, 34, 363eqtrd 2499 . . . 4  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  A  =  ( P L x ) )
389ad4antr 729 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  Q  e.  B )
3911ad4antr 729 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  P  =/=  Q )
4039necomd 2725 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  Q  =/=  P )
4116ad4antr 729 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  Q  e.  A )
4241, 37eleqtrd 2544 . . . . 5  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  Q  e.  ( P L x ) )
431, 2, 3, 24, 29, 25, 30, 38, 40, 42tglineelsb2 24216 . . . 4  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  -> 
( P L x )  =  ( P L Q ) )
4437, 43eqtrd 2495 . . 3  |-  ( ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  /\  P  =/=  x )  ->  A  =  ( P L Q ) )
4522, 44pm2.61dane 2772 . 2  |-  ( ( ( ( ph  /\  x  e.  B )  /\  y  e.  B
)  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  ->  A  =  ( P L Q ) )
46 tglinethru.1 . . 3  |-  ( ph  ->  A  e.  ran  L
)
471, 2, 3, 4, 46tgisline 24211 . 2  |-  ( ph  ->  E. x  e.  B  E. y  e.  B  ( A  =  (
x L y )  /\  x  =/=  y
) )
4845, 47r19.29vva 2998 1  |-  ( ph  ->  A  =  ( P L Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   ran crn 4989   ` cfv 5570  (class class class)co 6270   Basecbs 14719  TarskiGcstrkg 24026  Itvcitv 24033  LineGclng 24034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-concat 12531  df-s1 12532  df-s2 12807  df-s3 12808  df-trkgc 24045  df-trkgb 24046  df-trkgcb 24047  df-trkg 24051  df-cgrg 24107
This theorem is referenced by:  tghilberti2  24222  tglineintmo  24226  colline  24234  tglowdim2ln  24236  mirln  24260  mirln2  24261  perpneq  24295  ragperp  24298  footex  24299  perpdragALT  24305  perpdrag  24306  colperp  24307  opphllem1  24323  opphllem2  24324  opphllem3  24325  opphllem4  24326  opphllem5  24327  opphllem6  24328  opphl  24329  hpgerlem  24338  lmiisolem  24365
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