Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  linecom Unicode version

Theorem linecom 25988
Description: Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
linecom  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  -> 
( PLine Q )  =  ( QLine P
) )

Proof of Theorem linecom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp1 957 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  N  e.  NN )
2 simp3 959 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  x  e.  ( EE `  N
) )
3 simp21 990 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  P  e.  ( EE `  N
) )
4 simp22 991 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  Q  e.  ( EE `  N
) )
5 colinearperm1 25900 . . . . 5  |-  ( ( N  e.  NN  /\  ( x  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N
) ) )  -> 
( x  Colinear  <. P ,  Q >. 
<->  x  Colinear  <. Q ,  P >. ) )
61, 2, 3, 4, 5syl13anc 1186 . . . 4  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  (
x  Colinear  <. P ,  Q >.  <-> 
x  Colinear  <. Q ,  P >. ) )
763expa 1153 . . 3  |-  ( ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q
) )  /\  x  e.  ( EE `  N
) )  ->  (
x  Colinear  <. P ,  Q >.  <-> 
x  Colinear  <. Q ,  P >. ) )
87rabbidva 2907 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  { x  e.  ( EE `  N )  |  x  Colinear  <. P ,  Q >. }  =  { x  e.  ( EE `  N
)  |  x  Colinear  <. Q ,  P >. } )
9 fvline2 25984 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  -> 
( PLine Q )  =  { x  e.  ( EE `  N
)  |  x  Colinear  <. P ,  Q >. } )
10 necom 2648 . . . . 5  |-  ( P  =/=  Q  <->  Q  =/=  P )
11103anbi3i 1146 . . . 4  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  <->  ( P  e.  ( EE `  N
)  /\  Q  e.  ( EE `  N )  /\  Q  =/=  P
) )
12 3ancoma 943 . . . 4  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  Q  =/=  P )  <->  ( Q  e.  ( EE `  N
)  /\  P  e.  ( EE `  N )  /\  Q  =/=  P
) )
1311, 12bitri 241 . . 3  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  <->  ( Q  e.  ( EE `  N
)  /\  P  e.  ( EE `  N )  /\  Q  =/=  P
) )
14 fvline2 25984 . . 3  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  P  e.  ( EE `  N )  /\  Q  =/=  P ) )  -> 
( QLine P )  =  { x  e.  ( EE `  N
)  |  x  Colinear  <. Q ,  P >. } )
1513, 14sylan2b 462 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  -> 
( QLine P )  =  { x  e.  ( EE `  N
)  |  x  Colinear  <. Q ,  P >. } )
168, 9, 153eqtr4d 2446 1  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  -> 
( PLine Q )  =  ( QLine P
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   <.cop 3777   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   NNcn 9956   EEcee 25731    Colinear ccolin 25875  Linecline2 25972
This theorem is referenced by:  linerflx2  25989  linethru  25991
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-ec 6866  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-ee 25734  df-btwn 25735  df-cgr 25736  df-colinear 25879  df-line2 25975
  Copyright terms: Public domain W3C validator