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Theorem cgr3tr4 24675
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 24543 . . . . 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 24543 . . . . 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 24543 . . . . 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 24669 . . . 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 24669 . . . 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 24669 . . 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 1684   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 24516  Cgrccgr 24518  Cgr3ccgr3 24659
This theorem is referenced by:  btwnxfr  24679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-sum 12159  df-ee 24519  df-cgr 24521  df-cgr3 24663
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