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Theorem seglecgr12 25759
Description: Substitution law for segment comparison under congruence. Biconditional version. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
seglecgr12  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. A ,  B >.Cgr
<. E ,  F >.  /\ 
<. C ,  D >.Cgr <. G ,  H >. )  ->  ( <. A ,  B >.  Seg<_  <. C ,  D >.  <->  <. E ,  F >.  Seg<_  <. G ,  H >. ) ) )

Proof of Theorem seglecgr12
StepHypRef Expression
1 df-3an 938 . . . 4  |-  ( (
<. A ,  B >.Cgr <. E ,  F >.  /\ 
<. C ,  D >.Cgr <. G ,  H >.  /\ 
<. A ,  B >.  Seg<_  <. C ,  D >. )  <-> 
( ( <. A ,  B >.Cgr <. E ,  F >.  /\  <. C ,  D >.Cgr
<. G ,  H >. )  /\  <. A ,  B >. 
Seg<_ 
<. C ,  D >. ) )
2 seglecgr12im 25758 . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. A ,  B >.Cgr
<. E ,  F >.  /\ 
<. C ,  D >.Cgr <. G ,  H >.  /\ 
<. A ,  B >.  Seg<_  <. C ,  D >. )  ->  <. E ,  F >. 
Seg<_ 
<. G ,  H >. ) )
31, 2syl5bir 210 . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( ( <. A ,  B >.Cgr <. E ,  F >.  /\  <. C ,  D >.Cgr
<. G ,  H >. )  /\  <. A ,  B >. 
Seg<_ 
<. C ,  D >. )  ->  <. E ,  F >. 
Seg<_ 
<. G ,  H >. ) )
43exp3a 426 . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. A ,  B >.Cgr
<. E ,  F >.  /\ 
<. C ,  D >.Cgr <. G ,  H >. )  ->  ( <. A ,  B >.  Seg<_  <. C ,  D >.  ->  <. E ,  F >. 
Seg<_ 
<. G ,  H >. ) ) )
5 simp11 987 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  N  e.  NN )
6 simp12 988 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
7 simp13 989 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
8 simp23 992 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  E  e.  ( EE `  N
) )
9 simp31 993 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N
) )
10 cgrcom 25638 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr <. E ,  F >.  <->  <. E ,  F >.Cgr <. A ,  B >. ) )
115, 6, 7, 8, 9, 10syl122anc 1193 . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr <. E ,  F >.  <->  <. E ,  F >.Cgr <. A ,  B >. ) )
12 simp21 990 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
13 simp22 991 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
14 simp32 994 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  G  e.  ( EE `  N
) )
15 simp33 995 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  H  e.  ( EE `  N
) )
16 cgrcom 25638 . . . . 5  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N ) ) )  ->  ( <. C ,  D >.Cgr <. G ,  H >.  <->  <. G ,  H >.Cgr <. C ,  D >. ) )
175, 12, 13, 14, 15, 16syl122anc 1193 . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. C ,  D >.Cgr <. G ,  H >.  <->  <. G ,  H >.Cgr <. C ,  D >. ) )
1811, 17anbi12d 692 . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. A ,  B >.Cgr
<. E ,  F >.  /\ 
<. C ,  D >.Cgr <. G ,  H >. )  <-> 
( <. E ,  F >.Cgr
<. A ,  B >.  /\ 
<. G ,  H >.Cgr <. C ,  D >. ) ) )
19 df-3an 938 . . . . 5  |-  ( (
<. E ,  F >.Cgr <. A ,  B >.  /\ 
<. G ,  H >.Cgr <. C ,  D >.  /\ 
<. E ,  F >.  Seg<_  <. G ,  H >. )  <-> 
( ( <. E ,  F >.Cgr <. A ,  B >.  /\  <. G ,  H >.Cgr
<. C ,  D >. )  /\  <. E ,  F >. 
Seg<_ 
<. G ,  H >. ) )
20 seglecgr12im 25758 . . . . . 6  |-  ( ( ( N  e.  NN  /\  E  e.  ( EE
`  N )  /\  F  e.  ( EE `  N ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( <. E ,  F >.Cgr
<. A ,  B >.  /\ 
<. G ,  H >.Cgr <. C ,  D >.  /\ 
<. E ,  F >.  Seg<_  <. G ,  H >. )  ->  <. A ,  B >. 
Seg<_ 
<. C ,  D >. ) )
215, 8, 9, 14, 15, 6, 7, 12, 13, 20syl333anc 1216 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. E ,  F >.Cgr
<. A ,  B >.  /\ 
<. G ,  H >.Cgr <. C ,  D >.  /\ 
<. E ,  F >.  Seg<_  <. G ,  H >. )  ->  <. A ,  B >. 
Seg<_ 
<. C ,  D >. ) )
2219, 21syl5bir 210 . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( ( <. E ,  F >.Cgr <. A ,  B >.  /\  <. G ,  H >.Cgr
<. C ,  D >. )  /\  <. E ,  F >. 
Seg<_ 
<. G ,  H >. )  ->  <. A ,  B >. 
Seg<_ 
<. C ,  D >. ) )
2322exp3a 426 . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. E ,  F >.Cgr
<. A ,  B >.  /\ 
<. G ,  H >.Cgr <. C ,  D >. )  ->  ( <. E ,  F >.  Seg<_  <. G ,  H >.  ->  <. A ,  B >. 
Seg<_ 
<. C ,  D >. ) ) )
2418, 23sylbid 207 . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. A ,  B >.Cgr
<. E ,  F >.  /\ 
<. C ,  D >.Cgr <. G ,  H >. )  ->  ( <. E ,  F >.  Seg<_  <. G ,  H >.  ->  <. A ,  B >. 
Seg<_ 
<. C ,  D >. ) ) )
254, 24impbidd 182 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. A ,  B >.Cgr
<. E ,  F >.  /\ 
<. C ,  D >.Cgr <. G ,  H >. )  ->  ( <. A ,  B >.  Seg<_  <. C ,  D >.  <->  <. E ,  F >.  Seg<_  <. G ,  H >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717   <.cop 3760   class class class wbr 4153   ` cfv 5394   NNcn 9932   EEcee 25541  Cgrccgr 25543    Seg<_ csegle 25754
This theorem is referenced by:  colinbtwnle  25766
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407  df-ee 25544  df-btwn 25545  df-cgr 25546  df-ofs 25631  df-cgr3 25688  df-segle 25755
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