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

Theorem linecom 29374
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 996 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  N  e.  NN )
2 simp3 998 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  x  e.  ( EE `  N
) )
3 simp21 1029 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  P  e.  ( EE `  N
) )
4 simp22 1030 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  Q  e.  ( EE `  N
) )
5 colinearperm1 29286 . . . . 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 1230 . . . 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 1196 . . 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 3104 . 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 29370 . 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 2736 . . . . 5  |-  ( P  =/=  Q  <->  Q  =/=  P )
11103anbi3i 1189 . . . 4  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  <->  ( P  e.  ( EE `  N
)  /\  Q  e.  ( EE `  N )  /\  Q  =/=  P
) )
12 3ancoma 980 . . . 4  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  Q  =/=  P )  <->  ( Q  e.  ( EE `  N
)  /\  P  e.  ( EE `  N )  /\  Q  =/=  P
) )
1311, 12bitri 249 . . 3  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  <->  ( Q  e.  ( EE `  N
)  /\  P  e.  ( EE `  N )  /\  Q  =/=  P
) )
14 fvline2 29370 . . 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 475 . 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 2518 1  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  -> 
( PLine Q )  =  ( QLine P
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818   <.cop 4033   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   NNcn 10532   EEcee 23864    Colinear ccolin 29261  Linecline2 29358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-ec 7310  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-sum 13465  df-ee 23867  df-btwn 23868  df-cgr 23869  df-colinear 29263  df-line2 29361
This theorem is referenced by:  linerflx2  29375  linethru  29377
  Copyright terms: Public domain W3C validator