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Theorem cgr3tr4 25986
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 1264 . . 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 444 . . . . 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 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 ) ) ) )  ->  A  e.  ( EE `  N ) )
4 simpr12 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 ) ) ) )  ->  B  e.  ( EE `  N ) )
5 simpr21 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 ) ) ) )  ->  D  e.  ( EE `  N ) )
6 simpr22 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 ) ) ) )  ->  E  e.  ( EE `  N ) )
7 simpr31 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 ) ) ) )  ->  G  e.  ( EE `  N ) )
8 simpr32 1048 . . . . 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 25854 . . . . 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 1207 . . . 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 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 ) ) ) )  ->  C  e.  ( EE `  N ) )
12 simpr23 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 ) ) ) )  ->  F  e.  ( EE `  N ) )
13 simpr33 1049 . . . . 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 25854 . . . . 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 1207 . . . 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 25854 . . . . 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 1207 . . . 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 1261 . . 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 210 . 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 25980 . . . 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 1164 . . 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 25980 . . . 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 1163 . . 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 692 . 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 25980 . . 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 1162 . 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 260 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 177    /\ wa 359    /\ w3a 936    e. wcel 1725   <.cop 3817   class class class wbr 4212   ` cfv 5454   NNcn 10000   EEcee 25827  Cgrccgr 25829  Cgr3ccgr3 25970
This theorem is referenced by:  btwnxfr  25990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324  df-sum 12480  df-ee 25830  df-cgr 25832  df-cgr3 25974
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