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Theorem cgrsub 24670
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 732 . . . 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 733 . . . 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 958 . . . . . 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 1033 . . . . . 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 1036 . . . . . 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 24627 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  ->  <. A ,  A >.Cgr <. D ,  D >. )
73, 4, 5, 6syl3anc 1182 . . . . 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 740 . . . . . 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 1035 . . . . . . 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 1038 . . . . . . 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 24623 . . . . . . 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 1191 . . . . . 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 201 . . . . 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 518 . . . 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 1034 . . . . 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 1037 . . . . 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 24668 . . . . . 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 24669 . . . . . 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 226 . . . . 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 1214 . . . 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 1280 . . 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 24623 . . . 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 1191 . . 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 201 . 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 423 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1686   <.cop 3645   class class class wbr 4025   ` cfv 5257   NNcn 9748   EEcee 24518    Btwn cbtwn 24519  Cgrccgr 24520    InnerFiveSeg cifs 24660
This theorem is referenced by:  brifs2  24703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-sum 12161  df-ee 24521  df-btwn 24522  df-cgr 24523  df-ofs 24608  df-ifs 24664
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