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Theorem seglelin 24146
Description: Linearity law for segment comparison. Theorem 5.10 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
Assertion
Ref Expression
seglelin  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.  Seg<_  <. C ,  D >.  \/  <. C ,  D >. 
Seg<_ 
<. A ,  B >. ) )
Dummy variable  x is distinct from all other variables.

Proof of Theorem seglelin
StepHypRef Expression
1 segcon2 24135 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( ( B  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  B >. )  /\  <. A ,  x >.Cgr <. C ,  D >. ) )
2 andir 840 . . . . 5  |-  ( ( ( B  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  B >. )  /\  <. A ,  x >.Cgr
<. C ,  D >. )  <-> 
( ( B  Btwn  <. A ,  x >.  /\ 
<. A ,  x >.Cgr <. C ,  D >. )  \/  ( x  Btwn  <. A ,  B >.  /\ 
<. A ,  x >.Cgr <. C ,  D >. ) ) )
3 simpl1 960 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  N  e.  NN )
4 simpl2l 1010 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  A  e.  ( EE `  N ) )
5 simpr 449 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  x  e.  ( EE `  N ) )
6 simpl3 962 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )
7 cgrcom 24020 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. A ,  x >.Cgr <. C ,  D >.  <->  <. C ,  D >.Cgr <. A ,  x >. ) )
83, 4, 5, 6, 7syl121anc 1189 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( <. A ,  x >.Cgr
<. C ,  D >.  <->  <. C ,  D >.Cgr <. A ,  x >. ) )
98anbi2d 686 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( x  Btwn  <. A ,  B >.  /\ 
<. A ,  x >.Cgr <. C ,  D >. )  <-> 
( x  Btwn  <. A ,  B >.  /\  <. C ,  D >.Cgr <. A ,  x >. ) ) )
109orbi2d 684 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( ( B 
Btwn  <. A ,  x >.  /\  <. A ,  x >.Cgr
<. C ,  D >. )  \/  ( x  Btwn  <. A ,  B >.  /\ 
<. A ,  x >.Cgr <. C ,  D >. ) )  <->  ( ( B 
Btwn  <. A ,  x >.  /\  <. A ,  x >.Cgr
<. C ,  D >. )  \/  ( x  Btwn  <. A ,  B >.  /\ 
<. C ,  D >.Cgr <. A ,  x >. ) ) ) )
112, 10syl5bb 250 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( ( B 
Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  B >. )  /\  <. A ,  x >.Cgr <. C ,  D >. )  <->  ( ( B  Btwn  <. A ,  x >.  /\  <. A ,  x >.Cgr
<. C ,  D >. )  \/  ( x  Btwn  <. A ,  B >.  /\ 
<. C ,  D >.Cgr <. A ,  x >. ) ) ) )
1211rexbidva 2561 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( E. x  e.  ( EE `  N
) ( ( B 
Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  B >. )  /\  <. A ,  x >.Cgr <. C ,  D >. )  <->  E. x  e.  ( EE `  N
) ( ( B 
Btwn  <. A ,  x >.  /\  <. A ,  x >.Cgr
<. C ,  D >. )  \/  ( x  Btwn  <. A ,  B >.  /\ 
<. C ,  D >.Cgr <. A ,  x >. ) ) ) )
13 brsegle2 24139 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.  Seg<_  <. C ,  D >.  <->  E. x  e.  ( EE `  N ) ( B  Btwn  <. A ,  x >.  /\  <. A ,  x >.Cgr <. C ,  D >. ) ) )
14 brsegle 24138 . . . . . 6  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( <. C ,  D >.  Seg<_  <. A ,  B >.  <->  E. x  e.  ( EE `  N ) ( x  Btwn  <. A ,  B >.  /\  <. C ,  D >.Cgr <. A ,  x >. ) ) )
15143com23 1159 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. C ,  D >.  Seg<_  <. A ,  B >.  <->  E. x  e.  ( EE `  N ) ( x  Btwn  <. A ,  B >.  /\  <. C ,  D >.Cgr <. A ,  x >. ) ) )
1613, 15orbi12d 692 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.  Seg<_  <. C ,  D >.  \/ 
<. C ,  D >.  Seg<_  <. A ,  B >. )  <-> 
( E. x  e.  ( EE `  N
) ( B  Btwn  <. A ,  x >.  /\ 
<. A ,  x >.Cgr <. C ,  D >. )  \/  E. x  e.  ( EE `  N
) ( x  Btwn  <. A ,  B >.  /\ 
<. C ,  D >.Cgr <. A ,  x >. ) ) ) )
17 r19.43 2696 . . . 4  |-  ( E. x  e.  ( EE
`  N ) ( ( B  Btwn  <. A ,  x >.  /\  <. A ,  x >.Cgr <. C ,  D >. )  \/  ( x 
Btwn  <. A ,  B >.  /\  <. C ,  D >.Cgr
<. A ,  x >. ) )  <->  ( E. x  e.  ( EE `  N
) ( B  Btwn  <. A ,  x >.  /\ 
<. A ,  x >.Cgr <. C ,  D >. )  \/  E. x  e.  ( EE `  N
) ( x  Btwn  <. A ,  B >.  /\ 
<. C ,  D >.Cgr <. A ,  x >. ) ) )
1816, 17syl6bbr 256 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.  Seg<_  <. C ,  D >.  \/ 
<. C ,  D >.  Seg<_  <. A ,  B >. )  <->  E. x  e.  ( EE `  N ) ( ( B  Btwn  <. A ,  x >.  /\  <. A ,  x >.Cgr <. C ,  D >. )  \/  ( x 
Btwn  <. A ,  B >.  /\  <. C ,  D >.Cgr
<. A ,  x >. ) ) ) )
1912, 18bitr4d 249 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( E. x  e.  ( EE `  N
) ( ( B 
Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  B >. )  /\  <. A ,  x >.Cgr <. C ,  D >. )  <->  ( <. A ,  B >.  Seg<_  <. C ,  D >.  \/  <. C ,  D >.  Seg<_  <. A ,  B >. ) ) )
201, 19mpbid 203 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.  Seg<_  <. C ,  D >.  \/  <. C ,  D >. 
Seg<_ 
<. A ,  B >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 936    e. wcel 1685   E.wrex 2545   <.cop 3644   class class class wbr 4024   ` cfv 5221   NNcn 9741   EEcee 23923    Btwn cbtwn 23924  Cgrccgr 23925    Seg<_ csegle 24136
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-oi 7220  df-card 7567  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956  df-sum 12153  df-ee 23926  df-btwn 23927  df-cgr 23928  df-ofs 24013  df-ifs 24069  df-cgr3 24070  df-colinear 24071  df-fs 24072  df-segle 24137
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