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Theorem linecgr 29884
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 29857 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. B ,  C >. >. )
323adant3 1014 . . . . . 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 463 . . . . 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 755 . . . . 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 1174 . . . 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 530 . . 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 432 . 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 994 . . 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 1027 . . 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 1028 . . 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 1029 . . 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 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
) ) )  ->  P  e.  ( EE `  N ) )
15 simp3r 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
) ) )  ->  Q  e.  ( EE `  N ) )
16 brfs 29882 . . . . 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 702 . . . 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 29883 . . . 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 1258 . 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 367    /\ w3a 971    e. wcel 1826    =/= wne 2577   <.cop 3950   class class class wbr 4367   ` cfv 5496   NNcn 10452   EEcee 24312  Cgrccgr 24314  Cgr3ccgr3 29839    Colinear ccolin 29840    FiveSeg cfs 29841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-sum 13511  df-ee 24315  df-btwn 24316  df-cgr 24317  df-ofs 29786  df-colinear 29842  df-ifs 29843  df-cgr3 29844  df-fs 29845
This theorem is referenced by:  linecgrand  29885  lineid  29886
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