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Theorem cgrsub 30860
Description: Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
Assertion
Ref Expression
cgrsub  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )  ->  <. A ,  B >.Cgr <. D ,  E >. ) )

Proof of Theorem cgrsub
StepHypRef Expression
1 simprl 769 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. ) )
2 simprr 771 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  ( <. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )
3 simpl1 1017 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  N  e.  NN )
4 simpl21 1092 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  A  e.  ( EE `  N
) )
5 simpl31 1095 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  D  e.  ( EE `  N
) )
6 cgrtriv 30817 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  ->  <. A ,  A >.Cgr <. D ,  D >. )
73, 4, 5, 6syl3anc 1276 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  <. A ,  A >.Cgr <. D ,  D >. )
8 simprrl 779 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  <. A ,  C >.Cgr <. D ,  F >. )
9 simpl23 1094 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  C  e.  ( EE `  N
) )
10 simpl33 1097 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  F  e.  ( EE `  N
) )
11 cgrcomlr 30813 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  C >.Cgr <. D ,  F >.  <->  <. C ,  A >.Cgr <. F ,  D >. ) )
123, 4, 9, 5, 10, 11syl122anc 1285 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  ( <. A ,  C >.Cgr <. D ,  F >.  <->  <. C ,  A >.Cgr <. F ,  D >. ) )
138, 12mpbid 215 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  <. C ,  A >.Cgr <. F ,  D >. )
147, 13jca 539 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  ( <. A ,  A >.Cgr <. D ,  D >.  /\ 
<. C ,  A >.Cgr <. F ,  D >. ) )
15 simpl22 1093 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  B  e.  ( EE `  N
) )
16 simpl32 1096 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  E  e.  ( EE `  N
) )
17 brifs 30858 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  A >. >. 
InnerFiveSeg  <. <. D ,  E >. ,  <. F ,  D >. >. 
<->  ( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  /\  ( <. A ,  A >.Cgr <. D ,  D >.  /\  <. C ,  A >.Cgr
<. F ,  D >. ) ) ) )
18 ifscgr 30859 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  A >. >. 
InnerFiveSeg  <. <. D ,  E >. ,  <. F ,  D >. >.  ->  <. B ,  A >.Cgr <. E ,  D >. ) )
1917, 18sylbird 243 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  /\  ( <. A ,  A >.Cgr <. D ,  D >.  /\  <. C ,  A >.Cgr
<. F ,  D >. ) )  ->  <. B ,  A >.Cgr <. E ,  D >. ) )
203, 4, 15, 9, 4, 5, 16, 10, 5, 19syl333anc 1308 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  /\  ( <. A ,  A >.Cgr <. D ,  D >.  /\  <. C ,  A >.Cgr
<. F ,  D >. ) )  ->  <. B ,  A >.Cgr <. E ,  D >. ) )
211, 2, 14, 20mp3and 1376 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  <. B ,  A >.Cgr <. E ,  D >. )
22 cgrcomlr 30813 . . . 4  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. B ,  A >.Cgr <. E ,  D >.  <->  <. A ,  B >.Cgr <. D ,  E >. ) )
233, 15, 4, 16, 5, 22syl122anc 1285 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  ( <. B ,  A >.Cgr <. E ,  D >.  <->  <. A ,  B >.Cgr <. D ,  E >. ) )
2421, 23mpbid 215 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )  ->  <. A ,  B >.Cgr <. D ,  E >. )
2524ex 440 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )  ->  <. A ,  B >.Cgr <. D ,  E >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    e. wcel 1897   <.cop 3985   class class class wbr 4415   ` cfv 5600   NNcn 10636   EEcee 24966    Btwn cbtwn 24967  Cgrccgr 24968    InnerFiveSeg cifs 30850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-sup 7981  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-rp 11331  df-ico 11669  df-icc 11670  df-fz 11813  df-fzo 11946  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-clim 13600  df-sum 13801  df-ee 24969  df-btwn 24970  df-cgr 24971  df-ofs 30798  df-ifs 30855
This theorem is referenced by:  brifs2  30893
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