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Theorem tgfscgr 23682
Description: Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Hypotheses
Ref Expression
tglngval.p  |-  P  =  ( Base `  G
)
tglngval.l  |-  L  =  (LineG `  G )
tglngval.i  |-  I  =  (Itv `  G )
tglngval.g  |-  ( ph  ->  G  e. TarskiG )
tglngval.x  |-  ( ph  ->  X  e.  P )
tglngval.y  |-  ( ph  ->  Y  e.  P )
tgcolg.z  |-  ( ph  ->  Z  e.  P )
lnxfr.r  |-  .~  =  (cgrG `  G )
lnxfr.a  |-  ( ph  ->  A  e.  P )
lnxfr.b  |-  ( ph  ->  B  e.  P )
lnxfr.d  |-  .-  =  ( dist `  G )
tgfscgr.t  |-  ( ph  ->  T  e.  P )
tgfscgr.c  |-  ( ph  ->  C  e.  P )
tgfscgr.d  |-  ( ph  ->  D  e.  P )
tgfscgr.1  |-  ( ph  ->  ( Y  e.  ( X L Z )  \/  X  =  Z ) )
tgfscgr.2  |-  ( ph  ->  <" X Y Z ">  .~  <" A B C "> )
tgfscgr.3  |-  ( ph  ->  ( X  .-  T
)  =  ( A 
.-  D ) )
tgfscgr.4  |-  ( ph  ->  ( Y  .-  T
)  =  ( B 
.-  D ) )
tgfscgr.5  |-  ( ph  ->  X  =/=  Y )
Assertion
Ref Expression
tgfscgr  |-  ( ph  ->  ( Z  .-  T
)  =  ( C 
.-  D ) )

Proof of Theorem tgfscgr
StepHypRef Expression
1 tglngval.p . . 3  |-  P  =  ( Base `  G
)
2 lnxfr.d . . 3  |-  .-  =  ( dist `  G )
3 tglngval.i . . 3  |-  I  =  (Itv `  G )
4 tglngval.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  G  e. TarskiG )
6 tglngval.x . . . 4  |-  ( ph  ->  X  e.  P )
76adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  X  e.  P )
8 tglngval.y . . . 4  |-  ( ph  ->  Y  e.  P )
98adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Y  e.  P )
10 tgcolg.z . . . 4  |-  ( ph  ->  Z  e.  P )
1110adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Z  e.  P )
12 lnxfr.a . . . 4  |-  ( ph  ->  A  e.  P )
1312adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  A  e.  P )
14 lnxfr.b . . . 4  |-  ( ph  ->  B  e.  P )
1514adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  B  e.  P )
16 tgfscgr.c . . . 4  |-  ( ph  ->  C  e.  P )
1716adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  C  e.  P )
18 tgfscgr.t . . . 4  |-  ( ph  ->  T  e.  P )
1918adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  T  e.  P )
20 tgfscgr.d . . . 4  |-  ( ph  ->  D  e.  P )
2120adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  D  e.  P )
22 tgfscgr.5 . . . 4  |-  ( ph  ->  X  =/=  Y )
2322adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  X  =/=  Y )
24 simpr 461 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Y  e.  ( X I Z ) )
25 lnxfr.r . . . 4  |-  .~  =  (cgrG `  G )
26 tgfscgr.2 . . . . 5  |-  ( ph  ->  <" X Y Z ">  .~  <" A B C "> )
2726adantr 465 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  <" X Y Z ">  .~  <" A B C "> )
281, 2, 3, 25, 5, 7, 9, 11, 13, 15, 17, 27, 24tgbtwnxfr 23646 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  B  e.  ( A I C ) )
291, 2, 3, 25, 5, 7, 9, 11, 13, 15, 17, 27cgr3simp1 23639 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  ( X  .-  Y )  =  ( A  .-  B ) )
301, 2, 3, 25, 5, 7, 9, 11, 13, 15, 17, 27cgr3simp2 23640 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  ( Y  .-  Z )  =  ( B  .-  C ) )
31 tgfscgr.3 . . . 4  |-  ( ph  ->  ( X  .-  T
)  =  ( A 
.-  D ) )
3231adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  ( X  .-  T )  =  ( A  .-  D ) )
33 tgfscgr.4 . . . 4  |-  ( ph  ->  ( Y  .-  T
)  =  ( B 
.-  D ) )
3433adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  ( Y  .-  T )  =  ( B  .-  D ) )
351, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 24, 28, 29, 30, 32, 34axtg5seg 23590 . 2  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  ( Z  .-  T )  =  ( C  .-  D ) )
364adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  G  e. TarskiG )
378adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  Y  e.  P )
386adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  X  e.  P )
3910adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  Z  e.  P )
4014adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  B  e.  P )
4112adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  A  e.  P )
4216adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  C  e.  P )
4318adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  T  e.  P )
4420adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  D  e.  P )
4522necomd 2738 . . . 4  |-  ( ph  ->  Y  =/=  X )
4645adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  Y  =/=  X )
47 simpr 461 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  X  e.  ( Y I Z ) )
4826adantr 465 . . . . 5  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  <" X Y Z ">  .~  <" A B C "> )
491, 2, 3, 25, 36, 38, 37, 39, 41, 40, 42, 48cgr3swap12 23642 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  <" Y X Z ">  .~  <" B A C "> )
501, 2, 3, 25, 36, 37, 38, 39, 40, 41, 42, 49, 47tgbtwnxfr 23646 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  A  e.  ( B I C ) )
511, 2, 3, 25, 36, 37, 38, 39, 40, 41, 42, 49cgr3simp1 23639 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  ( Y  .-  X )  =  ( B  .-  A ) )
521, 2, 3, 25, 36, 37, 38, 39, 40, 41, 42, 49cgr3simp2 23640 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  ( X  .-  Z )  =  ( A  .-  C ) )
5333adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  ( Y  .-  T )  =  ( B  .-  D ) )
5431adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  ( X  .-  T )  =  ( A  .-  D ) )
551, 2, 3, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 50, 51, 52, 53, 54axtg5seg 23590 . 2  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  ( Z  .-  T )  =  ( C  .-  D ) )
564adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  G  e. TarskiG )
576adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  X  e.  P )
5810adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  Z  e.  P )
598adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  Y  e.  P )
6018adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  T  e.  P )
6112adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  A  e.  P )
6216adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  C  e.  P )
6314adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  B  e.  P )
6420adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  D  e.  P )
65 simpr 461 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  Z  e.  ( X I Y ) )
6626adantr 465 . . . . 5  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  <" X Y Z ">  .~  <" A B C "> )
671, 2, 3, 25, 56, 57, 59, 58, 61, 63, 62, 66cgr3swap23 23643 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  <" X Z Y ">  .~  <" A C B "> )
681, 2, 3, 25, 56, 57, 58, 59, 61, 62, 63, 67, 65tgbtwnxfr 23646 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  C  e.  ( A I B ) )
691, 2, 3, 25, 56, 57, 59, 58, 61, 63, 62, 66cgr3simp1 23639 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  ( X  .-  Y )  =  ( A  .-  B ) )
701, 2, 3, 25, 56, 57, 58, 59, 61, 62, 63, 67cgr3simp2 23640 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  ( Z  .-  Y )  =  ( C  .-  B ) )
7131adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  ( X  .-  T )  =  ( A  .-  D ) )
7233adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  ( Y  .-  T )  =  ( B  .-  D ) )
731, 2, 3, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72tgifscgr 23628 . 2  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  ( Z  .-  T )  =  ( C  .-  D ) )
74 tgfscgr.1 . . 3  |-  ( ph  ->  ( Y  e.  ( X L Z )  \/  X  =  Z ) )
75 tglngval.l . . . 4  |-  L  =  (LineG `  G )
761, 75, 3, 4, 6, 10, 8tgcolg 23669 . . 3  |-  ( ph  ->  ( ( Y  e.  ( X L Z )  \/  X  =  Z )  <->  ( Y  e.  ( X I Z )  \/  X  e.  ( Y I Z )  \/  Z  e.  ( X I Y ) ) ) )
7774, 76mpbid 210 . 2  |-  ( ph  ->  ( Y  e.  ( X I Z )  \/  X  e.  ( Y I Z )  \/  Z  e.  ( X I Y ) ) )
7835, 55, 73, 77mpjao3dan 1295 1  |-  ( ph  ->  ( Z  .-  T
)  =  ( C 
.-  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   <"cs3 12766   Basecbs 14486   distcds 14560  TarskiGcstrkg 23553  Itvcitv 23560  LineGclng 23561  cgrGccgrg 23630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-s2 12772  df-s3 12773  df-trkgc 23572  df-trkgb 23573  df-trkgcb 23574  df-trkg 23578  df-cgrg 23631
This theorem is referenced by:  lncgr  23683  symquadlem  23774
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