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Theorem usg2wlkonot 30399
Description: A walk of length 2 between two vertices as ordered triple in an undirected simple graph. This theorem would also hold for undirected multigraphs, but to proof this the cases  A  =  B and/or  B  =  C must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
Assertion
Ref Expression
usg2wlkonot  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <-> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )

Proof of Theorem usg2wlkonot
Dummy variables  f  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 23268 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2wlkonotot 30389 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
32expcom 435 . . . . 5  |-  ( ( A  e.  V  /\  C  e.  V )  ->  ( ( V  e. 
_V  /\  E  e.  _V )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
433adant2 1007 . . . 4  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( V  e. 
_V  /\  E  e.  _V )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
54impcom 430 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
61, 5sylan 471 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
7 vex 2973 . . . . . . . . . . . 12  |-  f  e. 
_V
8 vex 2973 . . . . . . . . . . . 12  |-  p  e. 
_V
97, 8pm3.2i 455 . . . . . . . . . . 11  |-  ( f  e.  _V  /\  p  e.  _V )
10 is2wlk 23462 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( f  e.  _V  /\  p  e.  _V )
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) ) ) )
111, 9, 10sylancl 662 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) ) ) )
12 preq12 3954 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 ) )  ->  { A ,  B }  =  {
( p `  0
) ,  ( p `
 1 ) } )
13123adant3 1008 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  { A ,  B }  =  {
( p `  0
) ,  ( p `
 1 ) } )
1413eqeq2d 2452 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( E `
 ( f ` 
0 ) )  =  { A ,  B } 
<->  ( E `  (
f `  0 )
)  =  { ( p `  0 ) ,  ( p ` 
1 ) } ) )
15 preq12 3954 . . . . . . . . . . . . . . . . 17  |-  ( ( B  =  ( p `
 1 )  /\  C  =  ( p `  2 ) )  ->  { B ,  C }  =  {
( p `  1
) ,  ( p `
 2 ) } )
16153adant1 1006 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  { B ,  C }  =  {
( p `  1
) ,  ( p `
 2 ) } )
1716eqeq2d 2452 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( E `
 ( f ` 
1 ) )  =  { B ,  C } 
<->  ( E `  (
f `  1 )
)  =  { ( p `  1 ) ,  ( p ` 
2 ) } ) )
1814, 17anbi12d 710 . . . . . . . . . . . . . 14  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( ( E `  ( f `
 0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1 ) )  =  { B ,  C } )  <->  ( ( E `  ( f `  0 ) )  =  { ( p `
 0 ) ,  ( p `  1
) }  /\  ( E `  ( f `  1 ) )  =  { ( p `
 1 ) ,  ( p `  2
) } ) ) )
1918bicomd 201 . . . . . . . . . . . . 13  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( ( E `  ( f `
 0 ) )  =  { ( p `
 0 ) ,  ( p `  1
) }  /\  ( E `  ( f `  1 ) )  =  { ( p `
 1 ) ,  ( p `  2
) } )  <->  ( ( E `  ( f `  0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1 ) )  =  { B ,  C } ) ) )
20193anbi3d 1295 . . . . . . . . . . . 12  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  { A ,  B }  /\  ( E `  (
f `  1 )
)  =  { B ,  C } ) ) ) )
21 usgrafun 23275 . . . . . . . . . . . . . . . . . 18  |-  ( V USGrph  E  ->  Fun  E )
22 c0ex 9378 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  0  e.  _V
2322prid1 3981 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  { 0 ,  1 }
24 fzo0to2pr 11612 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0..^ 2 )  =  {
0 ,  1 }
2523, 24eleqtrri 2514 . . . . . . . . . . . . . . . . . . . . . 22  |-  0  e.  ( 0..^ 2 )
26 ffvelrn 5839 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  0  e.  (
0..^ 2 ) )  ->  ( f ` 
0 )  e.  dom  E )
2725, 26mpan2 671 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  0 )  e.  dom  E )
28 fvelrn 5837 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Fun  E  /\  (
f `  0 )  e.  dom  E )  -> 
( E `  (
f `  0 )
)  e.  ran  E
)
2927, 28sylan2 474 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  0 )
)  e.  ran  E
)
30 eleq1 2501 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( f `
 0 ) )  =  { A ,  B }  ->  ( ( E `  ( f `
 0 ) )  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
3129, 30syl5ibcom 220 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  0
) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
32 1ex 9379 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  1  e.  _V
3332prid2 3982 . . . . . . . . . . . . . . . . . . . . . . 23  |-  1  e.  { 0 ,  1 }
3433, 24eleqtrri 2514 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  ( 0..^ 2 )
35 ffvelrn 5839 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  1  e.  (
0..^ 2 ) )  ->  ( f ` 
1 )  e.  dom  E )
3634, 35mpan2 671 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  1 )  e.  dom  E )
37 fvelrn 5837 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Fun  E  /\  (
f `  1 )  e.  dom  E )  -> 
( E `  (
f `  1 )
)  e.  ran  E
)
3836, 37sylan2 474 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  1 )
)  e.  ran  E
)
39 eleq1 2501 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E `  ( f `
 1 ) )  =  { B ,  C }  ->  ( ( E `  ( f `
 1 ) )  e.  ran  E  <->  { B ,  C }  e.  ran  E ) )
4038, 39syl5ibcom 220 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  1
) )  =  { B ,  C }  ->  { B ,  C }  e.  ran  E ) )
4131, 40anim12d 563 . . . . . . . . . . . . . . . . . 18  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
4221, 41sylan 471 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
4342a1d 25 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  (
( ( E `  ( f `  0
) )  =  { A ,  B }  /\  ( E `  (
f `  1 )
)  =  { B ,  C } )  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
4443expcom 435 . . . . . . . . . . . . . . 15  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
)  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4544com24 87 . . . . . . . . . . . . . 14  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( ( ( E `  ( f `  0
) )  =  { A ,  B }  /\  ( E `  (
f `  1 )
)  =  { B ,  C } )  -> 
( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4645a1d 25 . . . . . . . . . . . . 13  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( p : ( 0 ... 2 ) --> V  ->  ( ( ( E `  ( f `
 0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1 ) )  =  { B ,  C } )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) ) )
47463imp 1181 . . . . . . . . . . . 12  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( f ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( f `  1
) )  =  { B ,  C }
) )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
4820, 47syl6bi 228 . . . . . . . . . . 11  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
4948com14 88 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  ->  (
( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
5011, 49sylbid 215 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
5150com14 88 . . . . . . . 8  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
5251expdcom 439 . . . . . . 7  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) ) )
53523imp 1181 . . . . . 6  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
5453com13 80 . . . . 5  |-  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  (
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
5554imp 429 . . . 4  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
5655exlimdvv 1691 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
57 usg2wlk 30306 . . . . 5  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
58573expib 1190 . . . 4  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
5958adantr 465 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
6056, 59impbid 191 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )  <->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
616, 60bitrd 253 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <-> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2970   {cpr 3877   <.cotp 3883   class class class wbr 4290   dom cdm 4838   ran crn 4839   Fun wfun 5410   -->wf 5412   ` cfv 5416  (class class class)co 6089   0cc0 9280   1c1 9281   2c2 10369   ...cfz 11435  ..^cfzo 11546   #chash 12101   USGrph cusg 23262   Walks cwalk 23403   2WalksOnOt c2wlkonot 30371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-ot 3884  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-usgra 23264  df-wlk 23413  df-wlkon 23419  df-2wlkonot 30374
This theorem is referenced by:  usg2spthonot  30404  usg2spthonot0  30405  frg2woteu  30645  frg2woteqm  30649
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