Step |
Hyp |
Ref |
Expression |
1 |
|
upgr4cycl4dv4e.v |
|- V = ( Vtx ` G ) |
2 |
|
upgr4cycl4dv4e.e |
|- E = ( Edg ` 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 |
2
|
upgrwlkvtxedg |
|- ( ( G e. UPGraph /\ F ( Walks ` G ) P ) -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) |
6 |
|
fveq2 |
|- ( ( # ` F ) = 4 -> ( P ` ( # ` F ) ) = ( P ` 4 ) ) |
7 |
6
|
eqeq2d |
|- ( ( # ` F ) = 4 -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) <-> ( P ` 0 ) = ( P ` 4 ) ) ) |
8 |
7
|
anbi2d |
|- ( ( # ` F ) = 4 -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) <-> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 4 ) ) ) ) |
9 |
|
oveq2 |
|- ( ( # ` F ) = 4 -> ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) |
10 |
|
fzo0to42pr |
|- ( 0 ..^ 4 ) = ( { 0 , 1 } u. { 2 , 3 } ) |
11 |
9 10
|
eqtrdi |
|- ( ( # ` F ) = 4 -> ( 0 ..^ ( # ` F ) ) = ( { 0 , 1 } u. { 2 , 3 } ) ) |
12 |
11
|
raleqdv |
|- ( ( # ` F ) = 4 -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> A. k e. ( { 0 , 1 } u. { 2 , 3 } ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) ) |
13 |
|
ralunb |
|- ( A. k e. ( { 0 , 1 } u. { 2 , 3 } ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> ( A. k e. { 0 , 1 } { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E /\ A. k e. { 2 , 3 } { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E ) ) |
14 |
|
c0ex |
|- 0 e. _V |
15 |
|
1ex |
|- 1 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 |
14 15 19 26
|
ralpr |
|- ( A. k e. { 0 , 1 } { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) ) |
28 |
|
2ex |
|- 2 e. _V |
29 |
|
3ex |
|- 3 e. _V |
30 |
|
fveq2 |
|- ( k = 2 -> ( P ` k ) = ( P ` 2 ) ) |
31 |
|
oveq1 |
|- ( k = 2 -> ( k + 1 ) = ( 2 + 1 ) ) |
32 |
|
2p1e3 |
|- ( 2 + 1 ) = 3 |
33 |
31 32
|
eqtrdi |
|- ( k = 2 -> ( k + 1 ) = 3 ) |
34 |
33
|
fveq2d |
|- ( k = 2 -> ( P ` ( k + 1 ) ) = ( P ` 3 ) ) |
35 |
30 34
|
preq12d |
|- ( k = 2 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 2 ) , ( P ` 3 ) } ) |
36 |
35
|
eleq1d |
|- ( k = 2 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) |
37 |
|
fveq2 |
|- ( k = 3 -> ( P ` k ) = ( P ` 3 ) ) |
38 |
|
oveq1 |
|- ( k = 3 -> ( k + 1 ) = ( 3 + 1 ) ) |
39 |
|
3p1e4 |
|- ( 3 + 1 ) = 4 |
40 |
38 39
|
eqtrdi |
|- ( k = 3 -> ( k + 1 ) = 4 ) |
41 |
40
|
fveq2d |
|- ( k = 3 -> ( P ` ( k + 1 ) ) = ( P ` 4 ) ) |
42 |
37 41
|
preq12d |
|- ( k = 3 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 3 ) , ( P ` 4 ) } ) |
43 |
42
|
eleq1d |
|- ( k = 3 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) |
44 |
28 29 36 43
|
ralpr |
|- ( A. k e. { 2 , 3 } { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E <-> ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) |
45 |
27 44
|
anbi12i |
|- ( ( A. k e. { 0 , 1 } { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E /\ A. k e. { 2 , 3 } { ( 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 /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) ) |
46 |
13 45
|
bitri |
|- ( A. k e. ( { 0 , 1 } u. { 2 , 3 } ) { ( 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 /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) ) |
47 |
12 46
|
bitrdi |
|- ( ( # ` F ) = 4 -> ( 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 /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) ) ) |
48 |
8 47
|
anbi12d |
|- ( ( # ` F ) = 4 -> ( ( ( 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 ` 4 ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) ) ) ) |
49 |
|
preq2 |
|- ( ( P ` 4 ) = ( P ` 0 ) -> { ( P ` 3 ) , ( P ` 4 ) } = { ( P ` 3 ) , ( P ` 0 ) } ) |
50 |
49
|
eleq1d |
|- ( ( P ` 4 ) = ( P ` 0 ) -> ( { ( P ` 3 ) , ( P ` 4 ) } e. E <-> { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) |
51 |
50
|
eqcoms |
|- ( ( P ` 0 ) = ( P ` 4 ) -> ( { ( P ` 3 ) , ( P ` 4 ) } e. E <-> { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) |
52 |
51
|
anbi2d |
|- ( ( P ` 0 ) = ( P ` 4 ) -> ( ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) <-> ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) |
53 |
52
|
anbi2d |
|- ( ( P ` 0 ) = ( P ` 4 ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) <-> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) ) |
54 |
53
|
adantl |
|- ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( P ` 0 ) = ( P ` 4 ) ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) <-> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) ) |
55 |
|
4nn0 |
|- 4 e. NN0 |
56 |
55
|
a1i |
|- ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) -> 4 e. NN0 ) |
57 |
1
|
wlkp |
|- ( F ( Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> V ) |
58 |
|
oveq2 |
|- ( ( # ` F ) = 4 -> ( 0 ... ( # ` F ) ) = ( 0 ... 4 ) ) |
59 |
58
|
feq2d |
|- ( ( # ` F ) = 4 -> ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 4 ) --> V ) ) |
60 |
59
|
biimpcd |
|- ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( # ` F ) = 4 -> P : ( 0 ... 4 ) --> V ) ) |
61 |
4 57 60
|
3syl |
|- ( F ( Paths ` G ) P -> ( ( # ` F ) = 4 -> P : ( 0 ... 4 ) --> V ) ) |
62 |
61
|
impcom |
|- ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) -> P : ( 0 ... 4 ) --> V ) |
63 |
|
id |
|- ( 4 e. NN0 -> 4 e. NN0 ) |
64 |
|
0nn0 |
|- 0 e. NN0 |
65 |
64
|
a1i |
|- ( 4 e. NN0 -> 0 e. NN0 ) |
66 |
|
4pos |
|- 0 < 4 |
67 |
66
|
a1i |
|- ( 4 e. NN0 -> 0 < 4 ) |
68 |
63 65 67
|
3jca |
|- ( 4 e. NN0 -> ( 4 e. NN0 /\ 0 e. NN0 /\ 0 < 4 ) ) |
69 |
|
fvffz0 |
|- ( ( ( 4 e. NN0 /\ 0 e. NN0 /\ 0 < 4 ) /\ P : ( 0 ... 4 ) --> V ) -> ( P ` 0 ) e. V ) |
70 |
68 69
|
sylan |
|- ( ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) -> ( P ` 0 ) e. V ) |
71 |
70
|
ad2antlr |
|- ( ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) -> ( P ` 0 ) e. V ) |
72 |
|
1nn0 |
|- 1 e. NN0 |
73 |
72
|
a1i |
|- ( 4 e. NN0 -> 1 e. NN0 ) |
74 |
|
1lt4 |
|- 1 < 4 |
75 |
74
|
a1i |
|- ( 4 e. NN0 -> 1 < 4 ) |
76 |
63 73 75
|
3jca |
|- ( 4 e. NN0 -> ( 4 e. NN0 /\ 1 e. NN0 /\ 1 < 4 ) ) |
77 |
|
fvffz0 |
|- ( ( ( 4 e. NN0 /\ 1 e. NN0 /\ 1 < 4 ) /\ P : ( 0 ... 4 ) --> V ) -> ( P ` 1 ) e. V ) |
78 |
76 77
|
sylan |
|- ( ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) -> ( P ` 1 ) e. V ) |
79 |
78
|
ad2antlr |
|- ( ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) -> ( P ` 1 ) e. V ) |
80 |
|
2nn0 |
|- 2 e. NN0 |
81 |
80
|
a1i |
|- ( 4 e. NN0 -> 2 e. NN0 ) |
82 |
|
2lt4 |
|- 2 < 4 |
83 |
82
|
a1i |
|- ( 4 e. NN0 -> 2 < 4 ) |
84 |
63 81 83
|
3jca |
|- ( 4 e. NN0 -> ( 4 e. NN0 /\ 2 e. NN0 /\ 2 < 4 ) ) |
85 |
|
fvffz0 |
|- ( ( ( 4 e. NN0 /\ 2 e. NN0 /\ 2 < 4 ) /\ P : ( 0 ... 4 ) --> V ) -> ( P ` 2 ) e. V ) |
86 |
84 85
|
sylan |
|- ( ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) -> ( P ` 2 ) e. V ) |
87 |
86
|
ad2antlr |
|- ( ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) -> ( P ` 2 ) e. V ) |
88 |
|
3nn0 |
|- 3 e. NN0 |
89 |
88
|
a1i |
|- ( 4 e. NN0 -> 3 e. NN0 ) |
90 |
|
3lt4 |
|- 3 < 4 |
91 |
90
|
a1i |
|- ( 4 e. NN0 -> 3 < 4 ) |
92 |
63 89 91
|
3jca |
|- ( 4 e. NN0 -> ( 4 e. NN0 /\ 3 e. NN0 /\ 3 < 4 ) ) |
93 |
|
fvffz0 |
|- ( ( ( 4 e. NN0 /\ 3 e. NN0 /\ 3 < 4 ) /\ P : ( 0 ... 4 ) --> V ) -> ( P ` 3 ) e. V ) |
94 |
92 93
|
sylan |
|- ( ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) -> ( P ` 3 ) e. V ) |
95 |
94
|
ad2antlr |
|- ( ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) -> ( P ` 3 ) e. V ) |
96 |
|
simpr |
|- ( ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) -> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) |
97 |
|
simplr |
|- ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) -> F ( Paths ` G ) P ) |
98 |
|
breq2 |
|- ( ( # ` F ) = 4 -> ( 1 < ( # ` F ) <-> 1 < 4 ) ) |
99 |
74 98
|
mpbiri |
|- ( ( # ` F ) = 4 -> 1 < ( # ` F ) ) |
100 |
99
|
ad2antrr |
|- ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) -> 1 < ( # ` F ) ) |
101 |
|
simpll |
|- ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) -> ( # ` F ) = 4 ) |
102 |
9
|
ad2antrr |
|- ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) -> ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) |
103 |
|
4nn |
|- 4 e. NN |
104 |
|
lbfzo0 |
|- ( 0 e. ( 0 ..^ 4 ) <-> 4 e. NN ) |
105 |
103 104
|
mpbir |
|- 0 e. ( 0 ..^ 4 ) |
106 |
|
eleq2 |
|- ( ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) -> ( 0 e. ( 0 ..^ ( # ` F ) ) <-> 0 e. ( 0 ..^ 4 ) ) ) |
107 |
105 106
|
mpbiri |
|- ( ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) -> 0 e. ( 0 ..^ ( # ` F ) ) ) |
108 |
107
|
adantl |
|- ( ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) -> 0 e. ( 0 ..^ ( # ` F ) ) ) |
109 |
|
pthdadjvtx |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ 0 e. ( 0 ..^ ( # ` F ) ) ) -> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) ) |
110 |
108 109
|
syl3an3 |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) ) |
111 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
112 |
111
|
fveq2i |
|- ( P ` 1 ) = ( P ` ( 0 + 1 ) ) |
113 |
112
|
neeq2i |
|- ( ( P ` 0 ) =/= ( P ` 1 ) <-> ( P ` 0 ) =/= ( P ` ( 0 + 1 ) ) ) |
114 |
110 113
|
sylibr |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 0 ) =/= ( P ` 1 ) ) |
115 |
|
simp1 |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> F ( Paths ` G ) P ) |
116 |
|
elfzo0 |
|- ( 2 e. ( 0 ..^ 4 ) <-> ( 2 e. NN0 /\ 4 e. NN /\ 2 < 4 ) ) |
117 |
80 103 82 116
|
mpbir3an |
|- 2 e. ( 0 ..^ 4 ) |
118 |
|
2ne0 |
|- 2 =/= 0 |
119 |
|
fzo1fzo0n0 |
|- ( 2 e. ( 1 ..^ 4 ) <-> ( 2 e. ( 0 ..^ 4 ) /\ 2 =/= 0 ) ) |
120 |
117 118 119
|
mpbir2an |
|- 2 e. ( 1 ..^ 4 ) |
121 |
|
oveq2 |
|- ( ( # ` F ) = 4 -> ( 1 ..^ ( # ` F ) ) = ( 1 ..^ 4 ) ) |
122 |
120 121
|
eleqtrrid |
|- ( ( # ` F ) = 4 -> 2 e. ( 1 ..^ ( # ` F ) ) ) |
123 |
|
0elfz |
|- ( 4 e. NN0 -> 0 e. ( 0 ... 4 ) ) |
124 |
55 123
|
ax-mp |
|- 0 e. ( 0 ... 4 ) |
125 |
124 58
|
eleqtrrid |
|- ( ( # ` F ) = 4 -> 0 e. ( 0 ... ( # ` F ) ) ) |
126 |
118
|
a1i |
|- ( ( # ` F ) = 4 -> 2 =/= 0 ) |
127 |
122 125 126
|
3jca |
|- ( ( # ` F ) = 4 -> ( 2 e. ( 1 ..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 2 =/= 0 ) ) |
128 |
127
|
adantr |
|- ( ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) -> ( 2 e. ( 1 ..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 2 =/= 0 ) ) |
129 |
128
|
3ad2ant3 |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( 2 e. ( 1 ..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 2 =/= 0 ) ) |
130 |
|
pthdivtx |
|- ( ( F ( Paths ` G ) P /\ ( 2 e. ( 1 ..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 2 =/= 0 ) ) -> ( P ` 2 ) =/= ( P ` 0 ) ) |
131 |
115 129 130
|
syl2anc |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 2 ) =/= ( P ` 0 ) ) |
132 |
131
|
necomd |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 0 ) =/= ( P ` 2 ) ) |
133 |
|
elfzo0 |
|- ( 3 e. ( 0 ..^ 4 ) <-> ( 3 e. NN0 /\ 4 e. NN /\ 3 < 4 ) ) |
134 |
88 103 90 133
|
mpbir3an |
|- 3 e. ( 0 ..^ 4 ) |
135 |
|
3ne0 |
|- 3 =/= 0 |
136 |
|
fzo1fzo0n0 |
|- ( 3 e. ( 1 ..^ 4 ) <-> ( 3 e. ( 0 ..^ 4 ) /\ 3 =/= 0 ) ) |
137 |
134 135 136
|
mpbir2an |
|- 3 e. ( 1 ..^ 4 ) |
138 |
137 121
|
eleqtrrid |
|- ( ( # ` F ) = 4 -> 3 e. ( 1 ..^ ( # ` F ) ) ) |
139 |
135
|
a1i |
|- ( ( # ` F ) = 4 -> 3 =/= 0 ) |
140 |
138 125 139
|
3jca |
|- ( ( # ` F ) = 4 -> ( 3 e. ( 1 ..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 3 =/= 0 ) ) |
141 |
140
|
adantr |
|- ( ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) -> ( 3 e. ( 1 ..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 3 =/= 0 ) ) |
142 |
141
|
3ad2ant3 |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( 3 e. ( 1 ..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 3 =/= 0 ) ) |
143 |
|
pthdivtx |
|- ( ( F ( Paths ` G ) P /\ ( 3 e. ( 1 ..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 3 =/= 0 ) ) -> ( P ` 3 ) =/= ( P ` 0 ) ) |
144 |
115 142 143
|
syl2anc |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 3 ) =/= ( P ` 0 ) ) |
145 |
144
|
necomd |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 0 ) =/= ( P ` 3 ) ) |
146 |
114 132 145
|
3jca |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= ( P ` 3 ) ) ) |
147 |
|
elfzo0 |
|- ( 1 e. ( 0 ..^ 4 ) <-> ( 1 e. NN0 /\ 4 e. NN /\ 1 < 4 ) ) |
148 |
72 103 74 147
|
mpbir3an |
|- 1 e. ( 0 ..^ 4 ) |
149 |
|
eleq2 |
|- ( ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) -> ( 1 e. ( 0 ..^ ( # ` F ) ) <-> 1 e. ( 0 ..^ 4 ) ) ) |
150 |
148 149
|
mpbiri |
|- ( ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) -> 1 e. ( 0 ..^ ( # ` F ) ) ) |
151 |
150
|
adantl |
|- ( ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) -> 1 e. ( 0 ..^ ( # ` F ) ) ) |
152 |
|
pthdadjvtx |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ 1 e. ( 0 ..^ ( # ` F ) ) ) -> ( P ` 1 ) =/= ( P ` ( 1 + 1 ) ) ) |
153 |
151 152
|
syl3an3 |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 1 ) =/= ( P ` ( 1 + 1 ) ) ) |
154 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
155 |
154
|
fveq2i |
|- ( P ` 2 ) = ( P ` ( 1 + 1 ) ) |
156 |
155
|
neeq2i |
|- ( ( P ` 1 ) =/= ( P ` 2 ) <-> ( P ` 1 ) =/= ( P ` ( 1 + 1 ) ) ) |
157 |
153 156
|
sylibr |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 1 ) =/= ( P ` 2 ) ) |
158 |
|
ax-1ne0 |
|- 1 =/= 0 |
159 |
|
fzo1fzo0n0 |
|- ( 1 e. ( 1 ..^ 4 ) <-> ( 1 e. ( 0 ..^ 4 ) /\ 1 =/= 0 ) ) |
160 |
148 158 159
|
mpbir2an |
|- 1 e. ( 1 ..^ 4 ) |
161 |
160 121
|
eleqtrrid |
|- ( ( # ` F ) = 4 -> 1 e. ( 1 ..^ ( # ` F ) ) ) |
162 |
|
3re |
|- 3 e. RR |
163 |
|
4re |
|- 4 e. RR |
164 |
162 163 90
|
ltleii |
|- 3 <_ 4 |
165 |
|
elfz2nn0 |
|- ( 3 e. ( 0 ... 4 ) <-> ( 3 e. NN0 /\ 4 e. NN0 /\ 3 <_ 4 ) ) |
166 |
88 55 164 165
|
mpbir3an |
|- 3 e. ( 0 ... 4 ) |
167 |
166 58
|
eleqtrrid |
|- ( ( # ` F ) = 4 -> 3 e. ( 0 ... ( # ` F ) ) ) |
168 |
|
1re |
|- 1 e. RR |
169 |
|
1lt3 |
|- 1 < 3 |
170 |
168 169
|
ltneii |
|- 1 =/= 3 |
171 |
170
|
a1i |
|- ( ( # ` F ) = 4 -> 1 =/= 3 ) |
172 |
161 167 171
|
3jca |
|- ( ( # ` F ) = 4 -> ( 1 e. ( 1 ..^ ( # ` F ) ) /\ 3 e. ( 0 ... ( # ` F ) ) /\ 1 =/= 3 ) ) |
173 |
172
|
adantr |
|- ( ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) -> ( 1 e. ( 1 ..^ ( # ` F ) ) /\ 3 e. ( 0 ... ( # ` F ) ) /\ 1 =/= 3 ) ) |
174 |
173
|
3ad2ant3 |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( 1 e. ( 1 ..^ ( # ` F ) ) /\ 3 e. ( 0 ... ( # ` F ) ) /\ 1 =/= 3 ) ) |
175 |
|
pthdivtx |
|- ( ( F ( Paths ` G ) P /\ ( 1 e. ( 1 ..^ ( # ` F ) ) /\ 3 e. ( 0 ... ( # ` F ) ) /\ 1 =/= 3 ) ) -> ( P ` 1 ) =/= ( P ` 3 ) ) |
176 |
115 174 175
|
syl2anc |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 1 ) =/= ( P ` 3 ) ) |
177 |
|
eleq2 |
|- ( ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) -> ( 2 e. ( 0 ..^ ( # ` F ) ) <-> 2 e. ( 0 ..^ 4 ) ) ) |
178 |
117 177
|
mpbiri |
|- ( ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) -> 2 e. ( 0 ..^ ( # ` F ) ) ) |
179 |
178
|
adantl |
|- ( ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) -> 2 e. ( 0 ..^ ( # ` F ) ) ) |
180 |
|
pthdadjvtx |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ 2 e. ( 0 ..^ ( # ` F ) ) ) -> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) |
181 |
179 180
|
syl3an3 |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) |
182 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
183 |
182
|
fveq2i |
|- ( P ` 3 ) = ( P ` ( 2 + 1 ) ) |
184 |
183
|
neeq2i |
|- ( ( P ` 2 ) =/= ( P ` 3 ) <-> ( P ` 2 ) =/= ( P ` ( 2 + 1 ) ) ) |
185 |
181 184
|
sylibr |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( P ` 2 ) =/= ( P ` 3 ) ) |
186 |
157 176 185
|
3jca |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 3 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) ) |
187 |
146 186
|
jca |
|- ( ( F ( Paths ` G ) P /\ 1 < ( # ` F ) /\ ( ( # ` F ) = 4 /\ ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 4 ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= ( P ` 3 ) ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 3 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) ) ) |
188 |
97 100 101 102 187
|
syl112anc |
|- ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= ( P ` 3 ) ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 3 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) ) ) |
189 |
188
|
adantr |
|- ( ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= ( P ` 3 ) ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 3 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) ) ) |
190 |
|
preq2 |
|- ( c = ( P ` 2 ) -> { ( P ` 1 ) , c } = { ( P ` 1 ) , ( P ` 2 ) } ) |
191 |
190
|
eleq1d |
|- ( c = ( P ` 2 ) -> ( { ( P ` 1 ) , c } e. E <-> { ( P ` 1 ) , ( P ` 2 ) } e. E ) ) |
192 |
191
|
anbi2d |
|- ( c = ( P ` 2 ) -> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) ) ) |
193 |
|
preq1 |
|- ( c = ( P ` 2 ) -> { c , d } = { ( P ` 2 ) , d } ) |
194 |
193
|
eleq1d |
|- ( c = ( P ` 2 ) -> ( { c , d } e. E <-> { ( P ` 2 ) , d } e. E ) ) |
195 |
194
|
anbi1d |
|- ( c = ( P ` 2 ) -> ( ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) <-> ( { ( P ` 2 ) , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) ) |
196 |
192 195
|
anbi12d |
|- ( c = ( P ` 2 ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) <-> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) ) ) |
197 |
|
neeq2 |
|- ( c = ( P ` 2 ) -> ( ( P ` 0 ) =/= c <-> ( P ` 0 ) =/= ( P ` 2 ) ) ) |
198 |
197
|
3anbi2d |
|- ( c = ( P ` 2 ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= d ) ) ) |
199 |
|
neeq2 |
|- ( c = ( P ` 2 ) -> ( ( P ` 1 ) =/= c <-> ( P ` 1 ) =/= ( P ` 2 ) ) ) |
200 |
|
neeq1 |
|- ( c = ( P ` 2 ) -> ( c =/= d <-> ( P ` 2 ) =/= d ) ) |
201 |
199 200
|
3anbi13d |
|- ( c = ( P ` 2 ) -> ( ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) <-> ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= d /\ ( P ` 2 ) =/= d ) ) ) |
202 |
198 201
|
anbi12d |
|- ( c = ( P ` 2 ) -> ( ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) <-> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= d /\ ( P ` 2 ) =/= d ) ) ) ) |
203 |
196 202
|
anbi12d |
|- ( c = ( P ` 2 ) -> ( ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) <-> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= d /\ ( P ` 2 ) =/= d ) ) ) ) ) |
204 |
|
preq2 |
|- ( d = ( P ` 3 ) -> { ( P ` 2 ) , d } = { ( P ` 2 ) , ( P ` 3 ) } ) |
205 |
204
|
eleq1d |
|- ( d = ( P ` 3 ) -> ( { ( P ` 2 ) , d } e. E <-> { ( P ` 2 ) , ( P ` 3 ) } e. E ) ) |
206 |
|
preq1 |
|- ( d = ( P ` 3 ) -> { d , ( P ` 0 ) } = { ( P ` 3 ) , ( P ` 0 ) } ) |
207 |
206
|
eleq1d |
|- ( d = ( P ` 3 ) -> ( { d , ( P ` 0 ) } e. E <-> { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) |
208 |
205 207
|
anbi12d |
|- ( d = ( P ` 3 ) -> ( ( { ( P ` 2 ) , d } e. E /\ { d , ( P ` 0 ) } e. E ) <-> ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) |
209 |
208
|
anbi2d |
|- ( d = ( P ` 3 ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) <-> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) ) |
210 |
|
neeq2 |
|- ( d = ( P ` 3 ) -> ( ( P ` 0 ) =/= d <-> ( P ` 0 ) =/= ( P ` 3 ) ) ) |
211 |
210
|
3anbi3d |
|- ( d = ( P ` 3 ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= d ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= ( P ` 3 ) ) ) ) |
212 |
|
neeq2 |
|- ( d = ( P ` 3 ) -> ( ( P ` 1 ) =/= d <-> ( P ` 1 ) =/= ( P ` 3 ) ) ) |
213 |
|
neeq2 |
|- ( d = ( P ` 3 ) -> ( ( P ` 2 ) =/= d <-> ( P ` 2 ) =/= ( P ` 3 ) ) ) |
214 |
212 213
|
3anbi23d |
|- ( d = ( P ` 3 ) -> ( ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= d /\ ( P ` 2 ) =/= d ) <-> ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 3 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) ) ) |
215 |
211 214
|
anbi12d |
|- ( d = ( P ` 3 ) -> ( ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= d /\ ( P ` 2 ) =/= d ) ) <-> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= ( P ` 3 ) ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 3 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) ) ) ) |
216 |
209 215
|
anbi12d |
|- ( d = ( P ` 3 ) -> ( ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= d /\ ( P ` 2 ) =/= d ) ) ) <-> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= ( P ` 3 ) ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 3 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) ) ) ) ) |
217 |
203 216
|
rspc2ev |
|- ( ( ( P ` 2 ) e. V /\ ( P ` 3 ) e. V /\ ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= ( P ` 2 ) /\ ( P ` 0 ) =/= ( P ` 3 ) ) /\ ( ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 1 ) =/= ( P ` 3 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) ) ) ) -> E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) |
218 |
87 95 96 189 217
|
syl112anc |
|- ( ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) -> E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) |
219 |
71 79 218
|
3jca |
|- ( ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) ) |
220 |
219
|
exp31 |
|- ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) -> ( ( 4 e. NN0 /\ P : ( 0 ... 4 ) --> V ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) ) ) ) |
221 |
56 62 220
|
mp2and |
|- ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) ) ) |
222 |
221
|
adantr |
|- ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( P ` 0 ) = ( P ` 4 ) ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 0 ) } e. E ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) ) ) |
223 |
54 222
|
sylbid |
|- ( ( ( ( # ` F ) = 4 /\ F ( Paths ` G ) P ) /\ ( P ` 0 ) = ( P ` 4 ) ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) ) ) |
224 |
223
|
exp31 |
|- ( ( # ` F ) = 4 -> ( F ( Paths ` G ) P -> ( ( P ` 0 ) = ( P ` 4 ) -> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) ) ) ) ) |
225 |
224
|
imp4c |
|- ( ( # ` F ) = 4 -> ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 4 ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) ) ) |
226 |
|
preq1 |
|- ( a = ( P ` 0 ) -> { a , b } = { ( P ` 0 ) , b } ) |
227 |
226
|
eleq1d |
|- ( a = ( P ` 0 ) -> ( { a , b } e. E <-> { ( P ` 0 ) , b } e. E ) ) |
228 |
227
|
anbi1d |
|- ( a = ( P ` 0 ) -> ( ( { a , b } e. E /\ { b , c } e. E ) <-> ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E ) ) ) |
229 |
|
preq2 |
|- ( a = ( P ` 0 ) -> { d , a } = { d , ( P ` 0 ) } ) |
230 |
229
|
eleq1d |
|- ( a = ( P ` 0 ) -> ( { d , a } e. E <-> { d , ( P ` 0 ) } e. E ) ) |
231 |
230
|
anbi2d |
|- ( a = ( P ` 0 ) -> ( ( { c , d } e. E /\ { d , a } e. E ) <-> ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) ) |
232 |
228 231
|
anbi12d |
|- ( a = ( P ` 0 ) -> ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) <-> ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) ) ) |
233 |
|
neeq1 |
|- ( a = ( P ` 0 ) -> ( a =/= b <-> ( P ` 0 ) =/= b ) ) |
234 |
|
neeq1 |
|- ( a = ( P ` 0 ) -> ( a =/= c <-> ( P ` 0 ) =/= c ) ) |
235 |
|
neeq1 |
|- ( a = ( P ` 0 ) -> ( a =/= d <-> ( P ` 0 ) =/= d ) ) |
236 |
233 234 235
|
3anbi123d |
|- ( a = ( P ` 0 ) -> ( ( a =/= b /\ a =/= c /\ a =/= d ) <-> ( ( P ` 0 ) =/= b /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) ) ) |
237 |
236
|
anbi1d |
|- ( a = ( P ` 0 ) -> ( ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) <-> ( ( ( P ` 0 ) =/= b /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) |
238 |
232 237
|
anbi12d |
|- ( a = ( P ` 0 ) -> ( ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) <-> ( ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= b /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) |
239 |
238
|
2rexbidv |
|- ( a = ( P ` 0 ) -> ( E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) <-> E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= b /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) |
240 |
|
preq2 |
|- ( b = ( P ` 1 ) -> { ( P ` 0 ) , b } = { ( P ` 0 ) , ( P ` 1 ) } ) |
241 |
240
|
eleq1d |
|- ( b = ( P ` 1 ) -> ( { ( P ` 0 ) , b } e. E <-> { ( P ` 0 ) , ( P ` 1 ) } e. E ) ) |
242 |
|
preq1 |
|- ( b = ( P ` 1 ) -> { b , c } = { ( P ` 1 ) , c } ) |
243 |
242
|
eleq1d |
|- ( b = ( P ` 1 ) -> ( { b , c } e. E <-> { ( P ` 1 ) , c } e. E ) ) |
244 |
241 243
|
anbi12d |
|- ( b = ( P ` 1 ) -> ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) ) ) |
245 |
244
|
anbi1d |
|- ( b = ( P ` 1 ) -> ( ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) <-> ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) ) ) |
246 |
|
neeq2 |
|- ( b = ( P ` 1 ) -> ( ( P ` 0 ) =/= b <-> ( P ` 0 ) =/= ( P ` 1 ) ) ) |
247 |
246
|
3anbi1d |
|- ( b = ( P ` 1 ) -> ( ( ( P ` 0 ) =/= b /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) ) ) |
248 |
|
neeq1 |
|- ( b = ( P ` 1 ) -> ( b =/= c <-> ( P ` 1 ) =/= c ) ) |
249 |
|
neeq1 |
|- ( b = ( P ` 1 ) -> ( b =/= d <-> ( P ` 1 ) =/= d ) ) |
250 |
248 249
|
3anbi12d |
|- ( b = ( P ` 1 ) -> ( ( b =/= c /\ b =/= d /\ c =/= d ) <-> ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) |
251 |
247 250
|
anbi12d |
|- ( b = ( P ` 1 ) -> ( ( ( ( P ` 0 ) =/= b /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) <-> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) |
252 |
245 251
|
anbi12d |
|- ( b = ( P ` 1 ) -> ( ( ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= b /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) <-> ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) ) |
253 |
252
|
2rexbidv |
|- ( b = ( P ` 1 ) -> ( E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= b /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) <-> E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) ) |
254 |
239 253
|
rspc2ev |
|- ( ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ E. c e. V E. d e. V ( ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , c } e. E ) /\ ( { c , d } e. E /\ { d , ( P ` 0 ) } e. E ) ) /\ ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 0 ) =/= c /\ ( P ` 0 ) =/= d ) /\ ( ( P ` 1 ) =/= c /\ ( P ` 1 ) =/= d /\ c =/= d ) ) ) ) -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) |
255 |
225 254
|
syl6 |
|- ( ( # ` F ) = 4 -> ( ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` 4 ) ) /\ ( ( { ( P ` 0 ) , ( P ` 1 ) } e. E /\ { ( P ` 1 ) , ( P ` 2 ) } e. E ) /\ ( { ( P ` 2 ) , ( P ` 3 ) } e. E /\ { ( P ` 3 ) , ( P ` 4 ) } e. E ) ) ) -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) |
256 |
48 255
|
sylbid |
|- ( ( # ` F ) = 4 -> ( ( ( 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 E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) |
257 |
256
|
expd |
|- ( ( # ` F ) = 4 -> ( ( 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 E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) ) |
258 |
257
|
com13 |
|- ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 4 -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) ) |
259 |
5 258
|
syl |
|- ( ( G e. UPGraph /\ F ( Walks ` G ) P ) -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 4 -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) ) |
260 |
259
|
expcom |
|- ( F ( Walks ` G ) P -> ( G e. UPGraph -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( ( # ` F ) = 4 -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) ) ) |
261 |
260
|
com23 |
|- ( F ( Walks ` G ) P -> ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 4 -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) ) ) |
262 |
261
|
expd |
|- ( F ( Walks ` G ) P -> ( F ( Paths ` G ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 4 -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) ) ) ) |
263 |
4 262
|
mpcom |
|- ( F ( Paths ` G ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 4 -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) ) ) |
264 |
263
|
imp |
|- ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( G e. UPGraph -> ( ( # ` F ) = 4 -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) ) |
265 |
3 264
|
syl |
|- ( F ( Cycles ` G ) P -> ( G e. UPGraph -> ( ( # ` F ) = 4 -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) ) ) |
266 |
265
|
3imp21 |
|- ( ( G e. UPGraph /\ F ( Cycles ` G ) P /\ ( # ` F ) = 4 ) -> E. a e. V E. b e. V E. c e. V E. d e. V ( ( ( { a , b } e. E /\ { b , c } e. E ) /\ ( { c , d } e. E /\ { d , a } e. E ) ) /\ ( ( a =/= b /\ a =/= c /\ a =/= d ) /\ ( b =/= c /\ b =/= d /\ c =/= d ) ) ) ) |