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Theorem linecgr 28251
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 762 . . . . 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 28224 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >. )
323adant3 1008 . . . . . 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 465 . . . . 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 756 . . . . 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 1168 . . . 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 761 . . . 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 532 . . 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 434 . 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 988 . . 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 1021 . . 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 1022 . . 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 1023 . . 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 1016 . . 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 1017 . . 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 28249 . . . . 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 704 . . . 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 28250 . . . 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 235 . . 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 1251 . 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 44 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2645   <.cop 3986   class class class wbr 4395   ` cfv 5521   NNcn 10428   EEcee 23281  Cgrccgr 23283  Cgr3ccgr3 28206    Colinear ccolin 28207    FiveSeg cfs 28208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-ico 11412  df-icc 11413  df-fz 11550  df-fzo 11661  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-sum 13277  df-ee 23284  df-btwn 23285  df-cgr 23286  df-ofs 28153  df-colinear 28209  df-ifs 28210  df-cgr3 28211  df-fs 28212
This theorem is referenced by:  linecgrand  28252  lineid  28253
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