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Theorem linecgr 25919
Description: Congruence rule for lines. Theorem 4.17 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
Assertion
Ref Expression
linecgr  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  -> 
( ( ( A  =/=  B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\  <. B ,  P >.Cgr
<. B ,  Q >. ) )  ->  <. C ,  P >.Cgr <. C ,  Q >. ) )

Proof of Theorem linecgr
StepHypRef Expression
1 simprlr 740 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) ) )  ->  A  Colinear  <. B ,  C >. )
2 cgr3rflx 25892 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >. )
323adant3 977 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >. )
43adantr 452 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >. )
5 simprr 734 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) ) )  ->  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) )
61, 4, 53jca 1134 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) ) )  ->  ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >.  /\  ( <. A ,  P >.Cgr
<. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) ) )
7 simprll 739 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) ) )  ->  A  =/=  B )
86, 7jca 519 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) ) )  ->  (
( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >.  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\  <. B ,  P >.Cgr <. B ,  Q >. ) )  /\  A  =/=  B ) )
98ex 424 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  -> 
( ( ( A  =/=  B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\  <. B ,  P >.Cgr
<. B ,  Q >. ) )  ->  ( ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >.  /\  ( <. A ,  P >.Cgr
<. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) )  /\  A  =/= 
B ) ) )
10 simp1 957 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  ->  N  e.  NN )
11 simp21 990 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
12 simp22 991 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
13 simp23 992 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
14 simp3l 985 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  ->  P  e.  ( EE `  N ) )
15 simp3r 986 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  ->  Q  e.  ( EE `  N ) )
16 brfs 25917 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  Q  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  P >. >.  FiveSeg  <. <. A ,  B >. ,  <. C ,  Q >. >. 
<->  ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >.  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\  <. B ,  P >.Cgr <. B ,  Q >. ) ) ) )
1716anbi1d 686 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  Q  e.  ( EE `  N ) ) )  ->  (
( <. <. A ,  B >. ,  <. C ,  P >. >.  FiveSeg  <. <. A ,  B >. ,  <. C ,  Q >. >.  /\  A  =/=  B )  <->  ( ( A 
Colinear 
<. B ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >.  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) )  /\  A  =/= 
B ) ) )
18 fscgr 25918 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  Q  e.  ( EE `  N ) ) )  ->  (
( <. <. A ,  B >. ,  <. C ,  P >. >.  FiveSeg  <. <. A ,  B >. ,  <. C ,  Q >. >.  /\  A  =/=  B )  ->  <. C ,  P >.Cgr <. C ,  Q >. ) )
1917, 18sylbird 227 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  Q  e.  ( EE `  N ) ) )  ->  (
( ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >.  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) )  /\  A  =/= 
B )  ->  <. C ,  P >.Cgr <. C ,  Q >. ) )
2010, 11, 12, 13, 14, 11, 12, 13, 15, 19syl333anc 1216 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  -> 
( ( ( A 
Colinear 
<. B ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >.  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\ 
<. B ,  P >.Cgr <. B ,  Q >. ) )  /\  A  =/= 
B )  ->  <. C ,  P >.Cgr <. C ,  Q >. ) )
219, 20syld 42 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  -> 
( ( ( A  =/=  B  /\  A  Colinear  <. B ,  C >. )  /\  ( <. A ,  P >.Cgr <. A ,  Q >.  /\  <. B ,  P >.Cgr
<. B ,  Q >. ) )  ->  <. C ,  P >.Cgr <. C ,  Q >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721    =/= wne 2567   <.cop 3777   class class class wbr 4172   ` cfv 5413   NNcn 9956   EEcee 25731  Cgrccgr 25733  Cgr3ccgr3 25874    Colinear ccolin 25875    FiveSeg cfs 25876
This theorem is referenced by:  linecgrand  25920  lineid  25921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-ee 25734  df-btwn 25735  df-cgr 25736  df-ofs 25821  df-ifs 25877  df-cgr3 25878  df-colinear 25879  df-fs 25880
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