Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofscom Unicode version

Theorem ofscom 24702
Description: The outer five segment predicate commutes. (Contributed by Scott Fenton, 26-Sep-2013.)
Assertion
Ref Expression
ofscom  |-  ( ( ( 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 ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. 
<-> 
<. <. E ,  F >. ,  <. G ,  H >. >. 
OuterFiveSeg  <. <. A ,  B >. ,  <. C ,  D >. >. ) )

Proof of Theorem ofscom
StepHypRef Expression
1 ancom 437 . . . 4  |-  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  <-> 
( F  Btwn  <. E ,  G >.  /\  B  Btwn  <. A ,  C >. ) )
21a1i 10 . . 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 ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  <-> 
( F  Btwn  <. E ,  G >.  /\  B  Btwn  <. A ,  C >. ) ) )
3 simp11 985 . . . . 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 ) ) )  ->  N  e.  NN )
4 simp12 986 . . . . 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 ) ) )  ->  A  e.  ( EE `  N
) )
5 simp13 987 . . . . 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 ) ) )  ->  B  e.  ( EE `  N
) )
6 simp23 990 . . . . 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 ) ) )  ->  E  e.  ( EE `  N
) )
7 simp31 991 . . . . 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 ) ) )  ->  F  e.  ( EE `  N
) )
8 cgrcom 24685 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr <. E ,  F >.  <->  <. E ,  F >.Cgr <. A ,  B >. ) )
93, 4, 5, 6, 7, 8syl122anc 1191 . . . 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 ) ) )  ->  ( <. A ,  B >.Cgr <. E ,  F >.  <->  <. E ,  F >.Cgr <. A ,  B >. ) )
10 simp21 988 . . . . 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 ) ) )  ->  C  e.  ( EE `  N
) )
11 simp32 992 . . . . 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 ) ) )  ->  G  e.  ( EE `  N
) )
12 cgrcom 24685 . . . . 5  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N ) ) )  ->  ( <. B ,  C >.Cgr <. F ,  G >.  <->  <. F ,  G >.Cgr <. B ,  C >. ) )
133, 5, 10, 7, 11, 12syl122anc 1191 . . . 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 ) ) )  ->  ( <. B ,  C >.Cgr <. F ,  G >.  <->  <. F ,  G >.Cgr <. B ,  C >. ) )
149, 13anbi12d 691 . . 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 ) ) )  ->  (
( <. A ,  B >.Cgr
<. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  <-> 
( <. E ,  F >.Cgr
<. A ,  B >.  /\ 
<. F ,  G >.Cgr <. B ,  C >. ) ) )
15 simp22 989 . . . . 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 ) ) )  ->  D  e.  ( EE `  N
) )
16 simp33 993 . . . . 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 ) ) )  ->  H  e.  ( EE `  N
) )
17 cgrcom 24685 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  H  e.  ( EE `  N ) ) )  ->  ( <. A ,  D >.Cgr <. E ,  H >.  <->  <. E ,  H >.Cgr <. A ,  D >. ) )
183, 4, 15, 6, 16, 17syl122anc 1191 . . . 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 ) ) )  ->  ( <. A ,  D >.Cgr <. E ,  H >.  <->  <. E ,  H >.Cgr <. A ,  D >. ) )
19 cgrcom 24685 . . . . 5  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  H  e.  ( EE `  N ) ) )  ->  ( <. B ,  D >.Cgr <. F ,  H >.  <->  <. F ,  H >.Cgr <. B ,  D >. ) )
203, 5, 15, 7, 16, 19syl122anc 1191 . . . 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 ) ) )  ->  ( <. B ,  D >.Cgr <. F ,  H >.  <->  <. F ,  H >.Cgr <. B ,  D >. ) )
2118, 20anbi12d 691 . . 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 ) ) )  ->  (
( <. A ,  D >.Cgr
<. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. )  <-> 
( <. E ,  H >.Cgr
<. A ,  D >.  /\ 
<. F ,  H >.Cgr <. B ,  D >. ) ) )
222, 14, 213anbi123d 1252 . 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 ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr
<. F ,  H >. ) )  <->  ( ( F 
Btwn  <. E ,  G >.  /\  B  Btwn  <. A ,  C >. )  /\  ( <. E ,  F >.Cgr <. A ,  B >.  /\ 
<. F ,  G >.Cgr <. B ,  C >. )  /\  ( <. E ,  H >.Cgr <. A ,  D >.  /\  <. F ,  H >.Cgr
<. B ,  D >. ) ) ) )
23 brofs 24700 . 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 ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. 
<->  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr
<. F ,  H >. ) ) ) )
24 brofs 24700 . . 3  |-  ( ( ( N  e.  NN  /\  E  e.  ( EE
`  N )  /\  F  e.  ( EE `  N ) )  /\  ( G  e.  ( EE `  N )  /\  H  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  ( <. <. E ,  F >. ,  <. G ,  H >. >. 
OuterFiveSeg  <. <. A ,  B >. ,  <. C ,  D >. >. 
<->  ( ( F  Btwn  <. E ,  G >.  /\  B  Btwn  <. A ,  C >. )  /\  ( <. E ,  F >.Cgr <. A ,  B >.  /\ 
<. F ,  G >.Cgr <. B ,  C >. )  /\  ( <. E ,  H >.Cgr <. A ,  D >.  /\  <. F ,  H >.Cgr
<. B ,  D >. ) ) ) )
253, 6, 7, 11, 16, 4, 5, 10, 15, 24syl333anc 1214 . 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 ) ) )  ->  ( <. <. E ,  F >. ,  <. G ,  H >. >. 
OuterFiveSeg  <. <. A ,  B >. ,  <. C ,  D >. >. 
<->  ( ( F  Btwn  <. E ,  G >.  /\  B  Btwn  <. A ,  C >. )  /\  ( <. E ,  F >.Cgr <. A ,  B >.  /\ 
<. F ,  G >.Cgr <. B ,  C >. )  /\  ( <. E ,  H >.Cgr <. A ,  D >.  /\  <. F ,  H >.Cgr
<. B ,  D >. ) ) ) )
2622, 23, 253bitr4d 276 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 ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. 
<-> 
<. <. E ,  F >. ,  <. G ,  H >. >. 
OuterFiveSeg  <. <. A ,  B >. ,  <. C ,  D >. >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696   <.cop 3656   class class class wbr 4039   ` cfv 5271   NNcn 9762   EEcee 24588    Btwn cbtwn 24589  Cgrccgr 24590    OuterFiveSeg cofs 24677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-exp 11121  df-sum 12175  df-ee 24591  df-cgr 24593  df-ofs 24678
  Copyright terms: Public domain W3C validator