Step |
Hyp |
Ref |
Expression |
1 |
|
upgr3v3e3cycl.e |
|- E = ( Edg ` G ) |
2 |
|
upgr3v3e3cycl.v |
|- V = ( Vtx ` G ) |
3 |
|
cyclprop |
|- ( F ( Cycles ` G ) P -> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) |
4 |
|
pthiswlk |
|- ( F ( Paths ` G ) P -> F ( Walks ` G ) P ) |
5 |
1
|
upgrwlkvtxedg |
|- ( ( G e. UPGraph /\ F ( Walks ` G ) P ) -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) |
6 |
|
fveq2 |
|- ( ( # ` F ) = 3 -> ( P ` ( # ` F ) ) = ( P ` 3 ) ) |
7 |
6
|
eqeq2d |
|- ( ( # ` F ) = 3 -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) <-> ( P ` 0 ) = ( P ` 3 ) ) ) |
8 |
7
|
anbi2d |
|- ( ( # ` F ) = 3 -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) <-> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) ) ) |
9 |
|
oveq2 |
|- ( ( # ` F ) = 3 -> ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 3 ) ) |
10 |
|
fzo0to3tp |
|- ( 0 ..^ 3 ) = { 0 , 1 , 2 } |
11 |
9 10
|
eqtrdi |
|- ( ( # ` F ) = 3 -> ( 0 ..^ ( # ` F ) ) = { 0 , 1 , 2 } ) |
12 |
11
|
raleqdv |
|- ( ( # ` F ) = 3 -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> A. k e. { 0 , 1 , 2 } { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) ) |
13 |
|
c0ex |
|- 0 e. _V |
14 |
|
1ex |
|- 1 e. _V |
15 |
|
2ex |
|- 2 e. _V |
16 |
|
fveq2 |
|- ( k = 0 -> ( P ` k ) = ( P ` 0 ) ) |
17 |
|
fv0p1e1 |
|- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) ) |
18 |
16 17
|
preq12d |
|- ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } ) |
19 |
18
|
eleq1d |
|- ( k = 0 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> { ( P ` 0 ) , ( P ` 1 ) } e. E ) ) |
20 |
|
fveq2 |
|- ( k = 1 -> ( P ` k ) = ( P ` 1 ) ) |
21 |
|
oveq1 |
|- ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) ) |
22 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
23 |
21 22
|
eqtrdi |
|- ( k = 1 -> ( k + 1 ) = 2 ) |
24 |
23
|
fveq2d |
|- ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) ) |
25 |
20 24
|
preq12d |
|- ( k = 1 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } ) |
26 |
25
|
eleq1d |
|- ( k = 1 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> { ( P ` 1 ) , ( P ` 2 ) } e. E ) ) |
27 |
|
fveq2 |
|- ( k = 2 -> ( P ` k ) = ( P ` 2 ) ) |
28 |
|
oveq1 |
|- ( k = 2 -> ( k + 1 ) = ( 2 + 1 ) ) |
29 |
|
2p1e3 |
|- ( 2 + 1 ) = 3 |
30 |
28 29
|
eqtrdi |
|- ( k = 2 -> ( k + 1 ) = 3 ) |
31 |
30
|
fveq2d |
|- ( k = 2 -> ( P ` ( k + 1 ) ) = ( P ` 3 ) ) |
32 |
27 31
|
preq12d |
|- ( k = 2 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 2 ) , ( P ` 3 ) } ) |
33 |
32
|
eleq1d |
|- ( k = 2 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) |
34 |
13 14 15 19 26 33
|
raltp |
|- ( A. k e. { 0 , 1 , 2 } { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) |
35 |
12 34
|
bitrdi |
|- ( ( # ` F ) = 3 -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) |
36 |
8 35
|
anbi12d |
|- ( ( # ` F ) = 3 -> ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) /\ A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) <-> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) ) |
37 |
2
|
wlkp |
|- ( F ( Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> V ) |
38 |
|
oveq2 |
|- ( ( # ` F ) = 3 -> ( 0 ... ( # ` F ) ) = ( 0 ... 3 ) ) |
39 |
38
|
feq2d |
|- ( ( # ` F ) = 3 -> ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 3 ) --> V ) ) |
40 |
|
id |
|- ( P : ( 0 ... 3 ) --> V -> P : ( 0 ... 3 ) --> V ) |
41 |
|
3nn0 |
|- 3 e. NN0 |
42 |
|
0elfz |
|- ( 3 e. NN0 -> 0 e. ( 0 ... 3 ) ) |
43 |
41 42
|
mp1i |
|- ( P : ( 0 ... 3 ) --> V -> 0 e. ( 0 ... 3 ) ) |
44 |
40 43
|
ffvelrnd |
|- ( P : ( 0 ... 3 ) --> V -> ( P ` 0 ) e. V ) |
45 |
|
1nn0 |
|- 1 e. NN0 |
46 |
|
1lt3 |
|- 1 < 3 |
47 |
|
fvffz0 |
|- ( ( ( 3 e. NN0 /\ 1 e. NN0 /\ 1 < 3 ) /\ P : ( 0 ... 3 ) --> V ) -> ( P ` 1 ) e. V ) |
48 |
47
|
ex |
|- ( ( 3 e. NN0 /\ 1 e. NN0 /\ 1 < 3 ) -> ( P : ( 0 ... 3 ) --> V -> ( P ` 1 ) e. V ) ) |
49 |
41 45 46 48
|
mp3an |
|- ( P : ( 0 ... 3 ) --> V -> ( P ` 1 ) e. V ) |
50 |
|
2nn0 |
|- 2 e. NN0 |
51 |
|
2lt3 |
|- 2 < 3 |
52 |
|
fvffz0 |
|- ( ( ( 3 e. NN0 /\ 2 e. NN0 /\ 2 < 3 ) /\ P : ( 0 ... 3 ) --> V ) -> ( P ` 2 ) e. V ) |
53 |
52
|
ex |
|- ( ( 3 e. NN0 /\ 2 e. NN0 /\ 2 < 3 ) -> ( P : ( 0 ... 3 ) --> V -> ( P ` 2 ) e. V ) ) |
54 |
41 50 51 53
|
mp3an |
|- ( P : ( 0 ... 3 ) --> V -> ( P ` 2 ) e. V ) |
55 |
44 49 54
|
3jca |
|- ( P : ( 0 ... 3 ) --> V -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) |
56 |
39 55
|
syl6bi |
|- ( ( # ` F ) = 3 -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) ) |
57 |
56
|
com12 |
|- ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) ) |
58 |
4 37 57
|
3syl |
|- ( F ( Paths ` G ) P -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) ) |
59 |
58
|
adantr |
|- ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) ) |
60 |
59
|
adantr |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) ) |
61 |
60
|
impcom |
|- ( ( ( # ` F ) = 3 /\ ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) |
62 |
|
preq2 |
|- ( ( P ` 3 ) = ( P ` 0 ) -> { ( P ` 2 ) , ( P ` 3 ) } = { ( P ` 2 ) , ( P ` 0 ) } ) |
63 |
62
|
eqcoms |
|- ( ( P ` 0 ) = ( P ` 3 ) -> { ( P ` 2 ) , ( P ` 3 ) } = { ( P ` 2 ) , ( P ` 0 ) } ) |
64 |
63
|
adantl |
|- ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> { ( P ` 2 ) , ( P ` 3 ) } = { ( P ` 2 ) , ( P ` 0 ) } ) |
65 |
64
|
eleq1d |
|- ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( { ( P ` 2 ) , ( P ` 3 ) } e. E <-> { ( P ` 2 ) , ( P ` 0 ) } e. E ) ) |
66 |
65
|
3anbi3d |
|- ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) ) ) |
67 |
66
|
biimpa |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) -> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) ) |
68 |
67
|
adantl |
|- ( ( ( # ` F ) = 3 /\ ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) -> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) ) |
69 |
|
simpll |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> F ( Paths ` G ) P ) |
70 |
|
breq2 |
|- ( ( # ` F ) = 3 -> ( 1 < ( # ` F ) <-> 1 < 3 ) ) |
71 |
46 70
|
mpbiri |
|- ( ( # ` F ) = 3 -> 1 < ( # ` F ) ) |
72 |
71
|
adantl |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> 1 < ( # ` F ) ) |
73 |
|
3nn |
|- 3 e. NN |
74 |
|
lbfzo0 |
|- ( 0 e. ( 0 ..^ 3 ) <-> 3 e. NN ) |
75 |
73 74
|
mpbir |
|- 0 e. ( 0 ..^ 3 ) |
76 |
75 9
|
eleqtrrid |
|- ( ( # ` F ) = 3 -> 0 e. ( 0 ..^ ( # ` F ) ) ) |
77 |
76
|
adantl |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> 0 e. ( 0 ..^ ( # ` F ) ) ) |
78 |
|
pthdadjvtx |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ 0 e. ( 0 ..^ ( # ` F ) ) ) -> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) ) |
79 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
80 |
79
|
fveq2i |
|- ( P ` 1 ) = ( P ` ( 0 + 1 ) ) |
81 |
80
|
neeq2i |
|- ( ( P ` 0 ) =/= ( P ` 1 ) <-> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) ) |
82 |
78 81
|
sylibr |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ 0 e. ( 0 ..^ ( # ` F ) ) ) -> ( P ` 0 ) =/= ( P ` 1 ) ) |
83 |
69 72 77 82
|
syl3anc |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( P ` 0 ) =/= ( P ` 1 ) ) |
84 |
|
elfzo0 |
|- ( 1 e. ( 0 ..^ 3 ) <-> ( 1 e. NN0 /\ 3 e. NN /\ 1 < 3 ) ) |
85 |
45 73 46 84
|
mpbir3an |
|- 1 e. ( 0 ..^ 3 ) |
86 |
85 9
|
eleqtrrid |
|- ( ( # ` F ) = 3 -> 1 e. ( 0 ..^ ( # ` F ) ) ) |
87 |
86
|
adantl |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> 1 e. ( 0 ..^ ( # ` F ) ) ) |
88 |
|
pthdadjvtx |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ 1 e. ( 0 ..^ ( # ` F ) ) ) -> ( P ` 1 ) =/= ( P ` ( 1 + 1 ) ) ) |
89 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
90 |
89
|
fveq2i |
|- ( P ` 2 ) = ( P ` ( 1 + 1 ) ) |
91 |
90
|
neeq2i |
|- ( ( P ` 1 ) =/= ( P ` 2 ) <-> ( P ` 1 ) =/= ( P ` ( 1 + 1 ) ) ) |
92 |
88 91
|
sylibr |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ 1 e. ( 0 ..^ ( # ` F ) ) ) -> ( P ` 1 ) =/= ( P ` 2 ) ) |
93 |
69 72 87 92
|
syl3anc |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( P ` 1 ) =/= ( P ` 2 ) ) |
94 |
|
elfzo0 |
|- ( 2 e. ( 0 ..^ 3 ) <-> ( 2 e. NN0 /\ 3 e. NN /\ 2 < 3 ) ) |
95 |
50 73 51 94
|
mpbir3an |
|- 2 e. ( 0 ..^ 3 ) |
96 |
95 9
|
eleqtrrid |
|- ( ( # ` F ) = 3 -> 2 e. ( 0 ..^ ( # ` F ) ) ) |
97 |
96
|
adantl |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> 2 e. ( 0 ..^ ( # ` F ) ) ) |
98 |
|
pthdadjvtx |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ 2 e. ( 0 ..^ ( # ` F ) ) ) -> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) |
99 |
69 72 97 98
|
syl3anc |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) |
100 |
|
neeq2 |
|- ( ( P ` 0 ) = ( P ` 3 ) -> ( ( P ` 2 ) =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` 3 ) ) ) |
101 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
102 |
101
|
fveq2i |
|- ( P ` 3 ) = ( P ` ( 2 + 1 ) ) |
103 |
102
|
neeq2i |
|- ( ( P ` 2 ) =/= ( P ` 3 ) <-> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) |
104 |
100 103
|
bitrdi |
|- ( ( P ` 0 ) = ( P ` 3 ) -> ( ( P ` 2 ) =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) ) |
105 |
104
|
adantl |
|- ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( ( P ` 2 ) =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) ) |
106 |
105
|
adantr |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( ( P ` 2 ) =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) ) |
107 |
99 106
|
mpbird |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( P ` 2 ) =/= ( P ` 0 ) ) |
108 |
83 93 107
|
3jca |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( # ` F ) = 3 ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) |
109 |
108
|
ex |
|- ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) ) |
110 |
109
|
adantr |
|- ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) -> ( ( # ` F ) = 3 -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) ) |
111 |
110
|
impcom |
|- ( ( ( # ` F ) = 3 /\ ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) |
112 |
|
preq1 |
|- ( a = ( P ` 0 ) -> { a , b } = { ( P ` 0 ) , b } ) |
113 |
112
|
eleq1d |
|- ( a = ( P ` 0 ) -> ( { a , b } e. E <-> { ( P ` 0 ) , b } e. E ) ) |
114 |
|
preq2 |
|- ( a = ( P ` 0 ) -> { c , a } = { c , ( P ` 0 ) } ) |
115 |
114
|
eleq1d |
|- ( a = ( P ` 0 ) -> ( { c , a } e. E <-> { c , ( P ` 0 ) } e. E ) ) |
116 |
113 115
|
3anbi13d |
|- ( a = ( P ` 0 ) -> ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) <-> ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E /\ { c , ( P ` 0 ) } e. E ) ) ) |
117 |
|
neeq1 |
|- ( a = ( P ` 0 ) -> ( a =/= b <-> ( P ` 0 ) =/= b ) ) |
118 |
|
neeq2 |
|- ( a = ( P ` 0 ) -> ( c =/= a <-> c =/= ( P ` 0 ) ) ) |
119 |
117 118
|
3anbi13d |
|- ( a = ( P ` 0 ) -> ( ( a =/= b /\ b =/= c /\ c =/= a ) <-> ( ( P ` 0 ) =/= b /\ b =/= c /\ c =/= ( P ` 0 ) ) ) ) |
120 |
116 119
|
anbi12d |
|- ( a = ( P ` 0 ) -> ( ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) <-> ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E /\ { c , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= b /\ b =/= c /\ c =/= ( P ` 0 ) ) ) ) ) |
121 |
|
preq2 |
|- ( b = ( P ` 1 ) -> { ( P ` 0 ) , b } = { ( P ` 0 ) , ( P ` 1 ) } ) |
122 |
121
|
eleq1d |
|- ( b = ( P ` 1 ) -> ( { ( P ` 0 ) , b } e. E <-> { ( P ` 0 ) , ( P ` 1 ) } e. E ) ) |
123 |
|
preq1 |
|- ( b = ( P ` 1 ) -> { b , c } = { ( P ` 1 ) , c } ) |
124 |
123
|
eleq1d |
|- ( b = ( P ` 1 ) -> ( { b , c } e. E <-> { ( P ` 1 ) , c } e. E ) ) |
125 |
122 124
|
3anbi12d |
|- ( b = ( P ` 1 ) -> ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E /\ { c , ( P ` 0 ) } e. E ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E /\ { c , ( P ` 0 ) } e. E ) ) ) |
126 |
|
neeq2 |
|- ( b = ( P ` 1 ) -> ( ( P ` 0 ) =/= b <-> ( P ` 0 ) =/= ( P ` 1 ) ) ) |
127 |
|
neeq1 |
|- ( b = ( P ` 1 ) -> ( b =/= c <-> ( P ` 1 ) =/= c ) ) |
128 |
126 127
|
3anbi12d |
|- ( b = ( P ` 1 ) -> ( ( ( P ` 0 ) =/= b /\ b =/= c /\ c =/= ( P ` 0 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= c /\ c =/= ( P ` 0 ) ) ) ) |
129 |
125 128
|
anbi12d |
|- ( b = ( P ` 1 ) -> ( ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E /\ { c , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= b /\ b =/= c /\ c =/= ( P ` 0 ) ) ) <-> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E /\ { c , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= c /\ c =/= ( P ` 0 ) ) ) ) ) |
130 |
|
preq2 |
|- ( c = ( P ` 2 ) -> { ( P ` 1 ) , c } = { ( P ` 1 ) , ( P ` 2 ) } ) |
131 |
130
|
eleq1d |
|- ( c = ( P ` 2 ) -> ( { ( P ` 1 ) , c } e. E <-> { ( P ` 1 ) , ( P ` 2 ) } e. E ) ) |
132 |
|
preq1 |
|- ( c = ( P ` 2 ) -> { c , ( P ` 0 ) } = { ( P ` 2 ) , ( P ` 0 ) } ) |
133 |
132
|
eleq1d |
|- ( c = ( P ` 2 ) -> ( { c , ( P ` 0 ) } e. E <-> { ( P ` 2 ) , ( P ` 0 ) } e. E ) ) |
134 |
131 133
|
3anbi23d |
|- ( c = ( P ` 2 ) -> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E /\ { c , ( P ` 0 ) } e. E ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) ) ) |
135 |
|
neeq2 |
|- ( c = ( P ` 2 ) -> ( ( P ` 1 ) =/= c <-> ( P ` 1 ) =/= ( P ` 2 ) ) ) |
136 |
|
neeq1 |
|- ( c = ( P ` 2 ) -> ( c =/= ( P ` 0 ) <-> ( P ` 2 ) =/= ( P ` 0 ) ) ) |
137 |
135 136
|
3anbi23d |
|- ( c = ( P ` 2 ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= c /\ c =/= ( P ` 0 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) ) |
138 |
134 137
|
anbi12d |
|- ( c = ( P ` 2 ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E /\ { c , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= c /\ c =/= ( P ` 0 ) ) ) <-> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) ) ) |
139 |
120 129 138
|
rspc3ev |
|- ( ( ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 0 ) } e. E ) /\ ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 0 ) ) ) ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) |
140 |
61 68 111 139
|
syl12anc |
|- ( ( ( # ` F ) = 3 /\ ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) |
141 |
140
|
ex |
|- ( ( # ` F ) = 3 -> ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 3 ) ) /\ ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E /\ { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) |
142 |
36 141
|
sylbid |
|- ( ( # ` F ) = 3 -> ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) /\ A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) |
143 |
142
|
expd |
|- ( ( # ` F ) = 3 -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) |
144 |
143
|
com13 |
|- ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) |
145 |
5 144
|
syl |
|- ( ( G e. UPGraph /\ F ( Walks ` G ) P ) -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) |
146 |
145
|
expcom |
|- ( F ( Walks ` G ) P -> ( G e. UPGraph -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) ) |
147 |
146
|
com23 |
|- ( F ( Walks ` G ) P -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) ) |
148 |
147
|
expd |
|- ( F ( Walks ` G ) P -> ( F ( Paths ` G ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) ) ) |
149 |
4 148
|
mpcom |
|- ( F ( Paths ` G ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) ) |
150 |
149
|
imp |
|- ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) |
151 |
3 150
|
syl |
|- ( F ( Cycles ` G ) P -> ( G e. UPGraph -> ( ( # ` F ) = 3 -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) ) ) |
152 |
151
|
3imp21 |
|- ( ( G e. UPGraph /\ F ( Cycles ` G ) P /\ ( # ` F ) = 3 ) -> E. a e. V E. b e. V E. c e. V ( ( { a , b } e. E /\ { b , c } e. E /\ { c , a } e. E ) /\ ( a =/= b /\ b =/= c /\ c =/= a ) ) ) |