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Theorem cgr3tr4 25061
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 1262 . . 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 443 . . . . 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 1039 . . . . 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 1040 . . . . 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 1042 . . . . 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 1043 . . . . 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 1045 . . . . 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 1046 . . . . 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 24929 . . . . 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 1205 . . . 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 1041 . . . . 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 1044 . . . . 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 1047 . . . . 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 24929 . . . . 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 1205 . . . 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 24929 . . . . 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 1205 . . . 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 1259 . . 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 209 . 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 25055 . . . 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 1162 . . 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 25055 . . . 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 1161 . . 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 691 . 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 25055 . . 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 1160 . 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 259 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1701   <.cop 3677   class class class wbr 4060   ` cfv 5292   NNcn 9791   EEcee 24902  Cgrccgr 24904  Cgr3ccgr3 25045
This theorem is referenced by:  btwnxfr  25065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-seq 11094  df-sum 12206  df-ee 24905  df-cgr 24907  df-cgr3 25049
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