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Theorem linecom 30461
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 997 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  N  e.  NN )
2 simp3 999 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  x  e.  ( EE `  N
) )
3 simp21 1030 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  P  e.  ( EE `  N
) )
4 simp22 1031 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  /\  x  e.  ( EE `  N
) )  ->  Q  e.  ( EE `  N
) )
5 colinearperm1 30373 . . . . 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 1232 . . . 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 1197 . . 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 3049 . 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 30457 . 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 2672 . . . . 5  |-  ( P  =/=  Q  <->  Q  =/=  P )
11103anbi3i 1190 . . . 4  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  <->  ( P  e.  ( EE `  N
)  /\  Q  e.  ( EE `  N )  /\  Q  =/=  P
) )
12 3ancoma 981 . . . 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 30457 . . 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 473 . 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 2453 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2757   <.cop 3977   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   NNcn 10496   EEcee 24489    Colinear ccolin 30348  Linecline2 30445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-ec 7270  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-ico 11506  df-icc 11507  df-fz 11644  df-fzo 11768  df-seq 12062  df-exp 12121  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-sum 13565  df-ee 24492  df-btwn 24493  df-cgr 24494  df-colinear 30350  df-line2 30448
This theorem is referenced by:  linerflx2  30462  linethru  30464
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