Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cgr3com Unicode version

Theorem cgr3com 25702
Description: Commutativity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Assertion
Ref Expression
cgr3com  |-  ( ( 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 ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. D ,  <. E ,  F >. >.Cgr3 <. A ,  <. B ,  C >. >. ) )

Proof of Theorem cgr3com
StepHypRef Expression
1 id 20 . . . 4  |-  ( N  e.  NN  ->  N  e.  NN )
2 3simpa 954 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )
3 3simpa 954 . . . 4  |-  ( ( D  e.  ( EE
`  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  ->  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )
4 cgrcom 25639 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr <. D ,  E >.  <->  <. D ,  E >.Cgr <. A ,  B >. ) )
51, 2, 3, 4syl3an 1226 . . 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 ) ) )  ->  ( <. A ,  B >.Cgr <. D ,  E >.  <->  <. D ,  E >.Cgr <. A ,  B >. ) )
6 3simpb 955 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )
7 3simpb 955 . . . 4  |-  ( ( D  e.  ( EE
`  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  ->  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )
8 cgrcom 25639 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  C >.Cgr <. D ,  F >.  <->  <. D ,  F >.Cgr <. A ,  C >. ) )
91, 6, 7, 8syl3an 1226 . . 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 ) ) )  ->  ( <. A ,  C >.Cgr <. D ,  F >.  <->  <. D ,  F >.Cgr <. A ,  C >. ) )
10 3simpc 956 . . . 4  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )
11 3simpc 956 . . . 4  |-  ( ( D  e.  ( EE
`  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  ->  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )
12 cgrcom 25639 . . . 4  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  ->  ( <. B ,  C >.Cgr <. E ,  F >.  <->  <. E ,  F >.Cgr <. B ,  C >. ) )
131, 10, 11, 12syl3an 1226 . . 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 ) ) )  ->  ( <. B ,  C >.Cgr <. E ,  F >.  <->  <. E ,  F >.Cgr <. B ,  C >. ) )
145, 9, 133anbi123d 1254 . 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 ) ) )  ->  (
( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  <-> 
( <. D ,  E >.Cgr
<. A ,  B >.  /\ 
<. D ,  F >.Cgr <. A ,  C >.  /\ 
<. E ,  F >.Cgr <. B ,  C >. ) ) )
15 brcgr3 25695 . 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 ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )
16 brcgr3 25695 . . 3  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  ->  ( <. D ,  <. E ,  F >. >.Cgr3 <. A ,  <. B ,  C >. >.  <->  ( <. D ,  E >.Cgr <. A ,  B >.  /\  <. D ,  F >.Cgr <. A ,  C >.  /\  <. E ,  F >.Cgr
<. B ,  C >. ) ) )
17163com23 1159 . 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 ) ) )  ->  ( <. D ,  <. E ,  F >. >.Cgr3 <. A ,  <. B ,  C >. >.  <->  ( <. D ,  E >.Cgr <. A ,  B >.  /\  <. D ,  F >.Cgr <. A ,  C >.  /\  <. E ,  F >.Cgr
<. B ,  C >. ) ) )
1814, 15, 173bitr4d 277 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 ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  <->  <. D ,  <. E ,  F >. >.Cgr3 <. A ,  <. B ,  C >. >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717   <.cop 3761   class class class wbr 4154   ` cfv 5395   NNcn 9933   EEcee 25542  Cgrccgr 25544  Cgr3ccgr3 25685
This theorem is referenced by:  btwnxfr  25705
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252  df-exp 11311  df-sum 12408  df-ee 25545  df-cgr 25547  df-cgr3 25689
  Copyright terms: Public domain W3C validator