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

Theorem ofscom 29885
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 448 . . . 4  |-  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  <-> 
( F  Btwn  <. E ,  G >.  /\  B  Btwn  <. A ,  C >. ) )
21a1i 11 . . 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 1024 . . . . 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 1025 . . . . 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 1026 . . . . 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 1029 . . . . 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 1030 . . . . 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 29868 . . . . 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 1235 . . . 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 1027 . . . . 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 1031 . . . . 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 29868 . . . . 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 1235 . . . 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 708 . . 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 1028 . . . . 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 1032 . . . . 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 29868 . . . . 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 1235 . . . 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 29868 . . . . 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 1235 . . . 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 708 . . 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 1297 . 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 29883 . 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 29883 . . 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 1258 . 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 285 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    e. wcel 1823   <.cop 4022   class class class wbr 4439   ` cfv 5570   NNcn 10531   EEcee 24393    Btwn cbtwn 24394  Cgrccgr 24395    OuterFiveSeg cofs 29860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-seq 12090  df-exp 12149  df-sum 13591  df-ee 24396  df-cgr 24398  df-ofs 29861
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator