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Theorem cgr3tr4 30819
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 1349 . . 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 459 . . . . 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 1092 . . . . 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 1093 . . . . 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 1095 . . . . 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 1096 . . . . 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 1098 . . . . 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 1099 . . . . 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 24945 . . . . 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 1291 . . . 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 1094 . . . . 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 1097 . . . . 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 1100 . . . . 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 24945 . . . . 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 1291 . . . 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 24945 . . . . 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 1291 . . . 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 1346 . . 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 222 . 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 30813 . . . 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 1219 . . 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 30813 . . . 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 1218 . . 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 717 . 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 30813 . . 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 1217 . 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 272 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 188    /\ wa 371    /\ w3a 985    e. wcel 1887   <.cop 3974   class class class wbr 4402   ` cfv 5582   NNcn 10609   EEcee 24918  Cgrccgr 24920  Cgr3ccgr3 30803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-seq 12214  df-sum 13753  df-ee 24921  df-cgr 24923  df-cgr3 30808
This theorem is referenced by:  btwnxfr  30823
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