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Theorem cgr3tr4 28228
Description: Transitivity law for three-place congruence. (Contributed by Scott Fenton, 5-Oct-2013.)
Assertion
Ref Expression
cgr3tr4  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. G ,  <. H ,  I >. >. )  ->  <. D ,  <. E ,  F >. >.Cgr3 <. G ,  <. H ,  I >. >. ) )

Proof of Theorem cgr3tr4
StepHypRef Expression
1 3an6 1300 . . 3  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. A ,  B >.Cgr <. G ,  H >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. A ,  C >.Cgr
<. G ,  I >. )  /\  ( <. B ,  C >.Cgr <. E ,  F >.  /\  <. B ,  C >.Cgr
<. H ,  I >. ) )  <->  ( ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  /\  ( <. A ,  B >.Cgr <. G ,  H >.  /\  <. A ,  C >.Cgr
<. G ,  I >.  /\ 
<. B ,  C >.Cgr <. H ,  I >. ) ) )
2 simpl 457 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  N  e.  NN )
3 simpr11 1072 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  A  e.  ( EE `  N ) )
4 simpr12 1073 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  B  e.  ( EE `  N ) )
5 simpr21 1075 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  D  e.  ( EE `  N ) )
6 simpr22 1076 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  E  e.  ( EE `  N ) )
7 simpr31 1078 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  G  e.  ( EE `  N ) )
8 simpr32 1079 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  H  e.  ( EE `  N ) )
9 axcgrtr 23314 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. A ,  B >.Cgr <. G ,  H >. )  ->  <. D ,  E >.Cgr
<. G ,  H >. ) )
102, 3, 4, 5, 6, 7, 8, 9syl133anc 1242 . . . 4  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  B >.Cgr
<. G ,  H >. )  ->  <. D ,  E >.Cgr
<. G ,  H >. ) )
11 simpr13 1074 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  C  e.  ( EE `  N ) )
12 simpr23 1077 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  F  e.  ( EE `  N ) )
13 simpr33 1080 . . . . 5  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  ->  I  e.  ( EE `  N ) )
14 axcgrtr 23314 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) )  ->  (
( <. A ,  C >.Cgr
<. D ,  F >.  /\ 
<. A ,  C >.Cgr <. G ,  I >. )  ->  <. D ,  F >.Cgr
<. G ,  I >. ) )
152, 3, 11, 5, 12, 7, 13, 14syl133anc 1242 . . . 4  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. A ,  C >.Cgr
<. G ,  I >. )  ->  <. D ,  F >.Cgr
<. G ,  I >. ) )
16 axcgrtr 23314 . . . . 5  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) )  /\  ( F  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) )  ->  (
( <. B ,  C >.Cgr
<. E ,  F >.  /\ 
<. B ,  C >.Cgr <. H ,  I >. )  ->  <. E ,  F >.Cgr
<. H ,  I >. ) )
172, 4, 11, 6, 12, 8, 13, 16syl133anc 1242 . . . 4  |-  ( ( 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( ( <. B ,  C >.Cgr <. E ,  F >.  /\  <. B ,  C >.Cgr
<. H ,  I >. )  ->  <. E ,  F >.Cgr
<. H ,  I >. ) )
1810, 15, 173anim123d 1297 . . 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( ( ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  B >.Cgr <. G ,  H >. )  /\  ( <. A ,  C >.Cgr <. D ,  F >.  /\  <. A ,  C >.Cgr
<. G ,  I >. )  /\  ( <. B ,  C >.Cgr <. E ,  F >.  /\  <. B ,  C >.Cgr
<. H ,  I >. ) )  ->  ( <. D ,  E >.Cgr <. G ,  H >.  /\  <. D ,  F >.Cgr <. G ,  I >.  /\  <. E ,  F >.Cgr
<. H ,  I >. ) ) )
191, 18syl5bir 218 . 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( ( ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  /\  ( <. A ,  B >.Cgr <. G ,  H >.  /\  <. A ,  C >.Cgr
<. G ,  I >.  /\ 
<. B ,  C >.Cgr <. H ,  I >. ) )  ->  ( <. D ,  E >.Cgr <. G ,  H >.  /\  <. D ,  F >.Cgr <. G ,  I >.  /\  <. E ,  F >.Cgr
<. H ,  I >. ) ) )
20 brcgr3 28222 . . . 4  |-  ( ( 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 >. ) ) )
21203adant3r3 1199 . . 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  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 >. ) ) )
22 brcgr3 28222 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. G ,  <. H ,  I >. >.  <->  ( <. A ,  B >.Cgr <. G ,  H >.  /\  <. A ,  C >.Cgr <. G ,  I >.  /\  <. B ,  C >.Cgr
<. H ,  I >. ) ) )
23223adant3r2 1198 . . 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( <. A ,  <. B ,  C >. >.Cgr3 <. G ,  <. H ,  I >. >.  <->  (
<. A ,  B >.Cgr <. G ,  H >.  /\ 
<. A ,  C >.Cgr <. G ,  I >.  /\ 
<. B ,  C >.Cgr <. H ,  I >. ) ) )
2421, 23anbi12d 710 . 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. G ,  <. H ,  I >. >. )  <->  ( ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  /\  ( <. A ,  B >.Cgr <. G ,  H >.  /\  <. A ,  C >.Cgr
<. G ,  I >.  /\ 
<. B ,  C >.Cgr <. H ,  I >. ) ) ) )
25 brcgr3 28222 . . 3  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) )  ->  ( <. D ,  <. E ,  F >. >.Cgr3 <. G ,  <. H ,  I >. >.  <->  ( <. D ,  E >.Cgr <. G ,  H >.  /\  <. D ,  F >.Cgr <. G ,  I >.  /\  <. E ,  F >.Cgr
<. H ,  I >. ) ) )
26253adant3r1 1197 . 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( <. D ,  <. E ,  F >. >.Cgr3 <. G ,  <. H ,  I >. >.  <->  (
<. D ,  E >.Cgr <. G ,  H >.  /\ 
<. D ,  F >.Cgr <. G ,  I >.  /\ 
<. E ,  F >.Cgr <. H ,  I >. ) ) )
2719, 24, 263imtr4d 268 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 ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N
)  /\  I  e.  ( EE `  N ) ) ) )  -> 
( ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. G ,  <. H ,  I >. >. )  ->  <. D ,  <. E ,  F >. >.Cgr3 <. G ,  <. H ,  I >. >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758   <.cop 3992   class class class wbr 4401   ` cfv 5527   NNcn 10434   EEcee 23287  Cgrccgr 23289  Cgr3ccgr3 28212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-seq 11925  df-sum 13283  df-ee 23290  df-cgr 23292  df-cgr3 28217
This theorem is referenced by:  btwnxfr  28232
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