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Theorem axcgrtr 23953
Description: Congruence is transitive. Axiom A2 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
Assertion
Ref Expression
axcgrtr  |-  ( ( 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 >.Cgr
<. C ,  D >.  /\ 
<. A ,  B >.Cgr <. E ,  F >. )  ->  <. C ,  D >.Cgr
<. E ,  F >. ) )

Proof of Theorem axcgrtr
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 eqtr2 2302 . . 3  |-  ( (
sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( E `
 i )  -  ( F `  i ) ) ^ 2 ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( E `  i )  -  ( F `  i )
) ^ 2 ) )
2 simpl1 958 . . . . . 6  |-  ( ( ( 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  e.  ( EE `  N
) )
3 simpl2 959 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
4 simpl3 960 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
5 simpr1 961 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
6 brcgr 23938 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. C ,  D >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) ) )
72, 3, 4, 5, 6syl22anc 1183 . . . . 5  |-  ( ( ( 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 >.Cgr <. C ,  D >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) ) )
8 simpr2 962 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  E  e.  ( EE `  N
) )
9 simpr3 963 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N
) )
10 brcgr 23938 . . . . . 6  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. E ,  F >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( E `
 i )  -  ( F `  i ) ) ^ 2 ) ) )
112, 3, 8, 9, 10syl22anc 1183 . . . . 5  |-  ( ( ( 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 >.Cgr <. E ,  F >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( E `
 i )  -  ( F `  i ) ) ^ 2 ) ) )
127, 11anbi12d 691 . . . 4  |-  ( ( ( 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 >.Cgr
<. C ,  D >.  /\ 
<. A ,  B >.Cgr <. E ,  F >. )  <-> 
( sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( E `
 i )  -  ( F `  i ) ) ^ 2 ) ) ) )
13 brcgr 23938 . . . . 5  |-  ( ( ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( <. C ,  D >.Cgr
<. E ,  F >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( E `
 i )  -  ( F `  i ) ) ^ 2 ) ) )
144, 5, 8, 9, 13syl22anc 1183 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. C ,  D >.Cgr <. E ,  F >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( E `
 i )  -  ( F `  i ) ) ^ 2 ) ) )
1512, 14imbi12d 311 . . 3  |-  ( ( ( 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 >.Cgr <. C ,  D >.  /\  <. A ,  B >.Cgr
<. E ,  F >. )  ->  <. C ,  D >.Cgr
<. E ,  F >. )  <-> 
( ( sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `  i )  -  ( D `  i )
) ^ 2 )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( E `
 i )  -  ( F `  i ) ) ^ 2 ) )  ->  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( E `  i )  -  ( F `  i )
) ^ 2 ) ) ) )
161, 15mpbiri 224 . 2  |-  ( ( ( 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 >.Cgr
<. C ,  D >.  /\ 
<. A ,  B >.Cgr <. E ,  F >. )  ->  <. C ,  D >.Cgr
<. E ,  F >. ) )
17163adant1 973 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 ) ) )  ->  (
( <. A ,  B >.Cgr
<. C ,  D >.  /\ 
<. A ,  B >.Cgr <. E ,  F >. )  ->  <. C ,  D >.Cgr
<. E ,  F >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685   <.cop 3644   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   1c1 8734    - cmin 9033   NNcn 9742   2c2 9791   ...cfz 10778   ^cexp 11100   sum_csu 12154   EEcee 23926  Cgrccgr 23928
This theorem is referenced by:  cgrtr4d  24018  cgrcoml  24029  cgr3tr4  24085
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-n0 9962  df-z 10021  df-uz 10227  df-fz 10779  df-seq 11043  df-sum 12155  df-ee 23929  df-cgr 23931
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