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 ∈ UPGraph ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 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 } ∪ { 2 , 3 } ) |
11 |
9 10
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( { 0 , 1 } ∪ { 2 , 3 } ) ) |
12 |
11
|
raleqdv |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ↔ ∀ 𝑘 ∈ ( { 0 , 1 } ∪ { 2 , 3 } ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ) ) |
13 |
|
ralunb |
⊢ ( ∀ 𝑘 ∈ ( { 0 , 1 } ∪ { 2 , 3 } ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ↔ ( ∀ 𝑘 ∈ { 0 , 1 } { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ∧ ∀ 𝑘 ∈ { 2 , 3 } { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ) ) |
14 |
|
c0ex |
⊢ 0 ∈ V |
15 |
|
1ex |
⊢ 1 ∈ 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 ↔ { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 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 ↔ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ) |
27 |
14 15 19 26
|
ralpr |
⊢ ( ∀ 𝑘 ∈ { 0 , 1 } { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ↔ ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ) |
28 |
|
2ex |
⊢ 2 ∈ V |
29 |
|
3ex |
⊢ 3 ∈ 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 ↔ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 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 ↔ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) |
44 |
28 29 36 43
|
ralpr |
⊢ ( ∀ 𝑘 ∈ { 2 , 3 } { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ↔ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) |
45 |
27 44
|
anbi12i |
⊢ ( ( ∀ 𝑘 ∈ { 0 , 1 } { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ∧ ∀ 𝑘 ∈ { 2 , 3 } { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ) ↔ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) ) |
46 |
13 45
|
bitri |
⊢ ( ∀ 𝑘 ∈ ( { 0 , 1 } ∪ { 2 , 3 } ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ↔ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) ) |
47 |
12 46
|
bitrdi |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ↔ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) ) ) |
48 |
8 47
|
anbi12d |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → ( ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ) ↔ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 4 ) ) ∧ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 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 ↔ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) |
51 |
50
|
eqcoms |
⊢ ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 4 ) → ( { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ↔ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) |
52 |
51
|
anbi2d |
⊢ ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 4 ) → ( ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ↔ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) |
53 |
52
|
anbi2d |
⊢ ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 4 ) → ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) ↔ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) ) |
54 |
53
|
adantl |
⊢ ( ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 4 ) ) → ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) ↔ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) ) |
55 |
|
4nn0 |
⊢ 4 ∈ ℕ0 |
56 |
55
|
a1i |
⊢ ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) → 4 ∈ ℕ0 ) |
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 ∈ ℕ0 → 4 ∈ ℕ0 ) |
64 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
65 |
64
|
a1i |
⊢ ( 4 ∈ ℕ0 → 0 ∈ ℕ0 ) |
66 |
|
4pos |
⊢ 0 < 4 |
67 |
66
|
a1i |
⊢ ( 4 ∈ ℕ0 → 0 < 4 ) |
68 |
63 65 67
|
3jca |
⊢ ( 4 ∈ ℕ0 → ( 4 ∈ ℕ0 ∧ 0 ∈ ℕ0 ∧ 0 < 4 ) ) |
69 |
|
fvffz0 |
⊢ ( ( ( 4 ∈ ℕ0 ∧ 0 ∈ ℕ0 ∧ 0 < 4 ) ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 0 ) ∈ 𝑉 ) |
70 |
68 69
|
sylan |
⊢ ( ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 0 ) ∈ 𝑉 ) |
71 |
70
|
ad2antlr |
⊢ ( ( ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) ∧ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) → ( 𝑃 ‘ 0 ) ∈ 𝑉 ) |
72 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
73 |
72
|
a1i |
⊢ ( 4 ∈ ℕ0 → 1 ∈ ℕ0 ) |
74 |
|
1lt4 |
⊢ 1 < 4 |
75 |
74
|
a1i |
⊢ ( 4 ∈ ℕ0 → 1 < 4 ) |
76 |
63 73 75
|
3jca |
⊢ ( 4 ∈ ℕ0 → ( 4 ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ 1 < 4 ) ) |
77 |
|
fvffz0 |
⊢ ( ( ( 4 ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ 1 < 4 ) ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 1 ) ∈ 𝑉 ) |
78 |
76 77
|
sylan |
⊢ ( ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 1 ) ∈ 𝑉 ) |
79 |
78
|
ad2antlr |
⊢ ( ( ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) ∧ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) → ( 𝑃 ‘ 1 ) ∈ 𝑉 ) |
80 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
81 |
80
|
a1i |
⊢ ( 4 ∈ ℕ0 → 2 ∈ ℕ0 ) |
82 |
|
2lt4 |
⊢ 2 < 4 |
83 |
82
|
a1i |
⊢ ( 4 ∈ ℕ0 → 2 < 4 ) |
84 |
63 81 83
|
3jca |
⊢ ( 4 ∈ ℕ0 → ( 4 ∈ ℕ0 ∧ 2 ∈ ℕ0 ∧ 2 < 4 ) ) |
85 |
|
fvffz0 |
⊢ ( ( ( 4 ∈ ℕ0 ∧ 2 ∈ ℕ0 ∧ 2 < 4 ) ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 2 ) ∈ 𝑉 ) |
86 |
84 85
|
sylan |
⊢ ( ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 2 ) ∈ 𝑉 ) |
87 |
86
|
ad2antlr |
⊢ ( ( ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) ∧ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) → ( 𝑃 ‘ 2 ) ∈ 𝑉 ) |
88 |
|
3nn0 |
⊢ 3 ∈ ℕ0 |
89 |
88
|
a1i |
⊢ ( 4 ∈ ℕ0 → 3 ∈ ℕ0 ) |
90 |
|
3lt4 |
⊢ 3 < 4 |
91 |
90
|
a1i |
⊢ ( 4 ∈ ℕ0 → 3 < 4 ) |
92 |
63 89 91
|
3jca |
⊢ ( 4 ∈ ℕ0 → ( 4 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 3 < 4 ) ) |
93 |
|
fvffz0 |
⊢ ( ( ( 4 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 3 < 4 ) ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 3 ) ∈ 𝑉 ) |
94 |
92 93
|
sylan |
⊢ ( ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) → ( 𝑃 ‘ 3 ) ∈ 𝑉 ) |
95 |
94
|
ad2antlr |
⊢ ( ( ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) ∧ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) → ( 𝑃 ‘ 3 ) ∈ 𝑉 ) |
96 |
|
simpr |
⊢ ( ( ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) ∧ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) → ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) |
97 |
|
simplr |
⊢ ( ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( 4 ∈ ℕ0 ∧ 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 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) → 1 < ( ♯ ‘ 𝐹 ) ) |
101 |
|
simpll |
⊢ ( ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) → ( ♯ ‘ 𝐹 ) = 4 ) |
102 |
9
|
ad2antrr |
⊢ ( ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( 4 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) |
103 |
|
4nn |
⊢ 4 ∈ ℕ |
104 |
|
lbfzo0 |
⊢ ( 0 ∈ ( 0 ..^ 4 ) ↔ 4 ∈ ℕ ) |
105 |
103 104
|
mpbir |
⊢ 0 ∈ ( 0 ..^ 4 ) |
106 |
|
eleq2 |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) → ( 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ 0 ∈ ( 0 ..^ 4 ) ) ) |
107 |
105 106
|
mpbiri |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
108 |
107
|
adantl |
⊢ ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
109 |
|
pthdadjvtx |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ∧ 0 ∈ ( 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 ∈ ( 0 ..^ 4 ) ↔ ( 2 ∈ ℕ0 ∧ 4 ∈ ℕ ∧ 2 < 4 ) ) |
117 |
80 103 82 116
|
mpbir3an |
⊢ 2 ∈ ( 0 ..^ 4 ) |
118 |
|
2ne0 |
⊢ 2 ≠ 0 |
119 |
|
fzo1fzo0n0 |
⊢ ( 2 ∈ ( 1 ..^ 4 ) ↔ ( 2 ∈ ( 0 ..^ 4 ) ∧ 2 ≠ 0 ) ) |
120 |
117 118 119
|
mpbir2an |
⊢ 2 ∈ ( 1 ..^ 4 ) |
121 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → ( 1 ..^ ( ♯ ‘ 𝐹 ) ) = ( 1 ..^ 4 ) ) |
122 |
120 121
|
eleqtrrid |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → 2 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) |
123 |
|
0elfz |
⊢ ( 4 ∈ ℕ0 → 0 ∈ ( 0 ... 4 ) ) |
124 |
55 123
|
ax-mp |
⊢ 0 ∈ ( 0 ... 4 ) |
125 |
124 58
|
eleqtrrid |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → 0 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ) |
126 |
118
|
a1i |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → 2 ≠ 0 ) |
127 |
122 125 126
|
3jca |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → ( 2 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 0 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 2 ≠ 0 ) ) |
128 |
127
|
adantr |
⊢ ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) → ( 2 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 0 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 2 ≠ 0 ) ) |
129 |
128
|
3ad2ant3 |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ∧ ( ( ♯ ‘ 𝐹 ) = 4 ∧ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) ) → ( 2 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 0 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 2 ≠ 0 ) ) |
130 |
|
pthdivtx |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 2 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 0 ∈ ( 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 ∈ ( 0 ..^ 4 ) ↔ ( 3 ∈ ℕ0 ∧ 4 ∈ ℕ ∧ 3 < 4 ) ) |
134 |
88 103 90 133
|
mpbir3an |
⊢ 3 ∈ ( 0 ..^ 4 ) |
135 |
|
3ne0 |
⊢ 3 ≠ 0 |
136 |
|
fzo1fzo0n0 |
⊢ ( 3 ∈ ( 1 ..^ 4 ) ↔ ( 3 ∈ ( 0 ..^ 4 ) ∧ 3 ≠ 0 ) ) |
137 |
134 135 136
|
mpbir2an |
⊢ 3 ∈ ( 1 ..^ 4 ) |
138 |
137 121
|
eleqtrrid |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → 3 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) |
139 |
135
|
a1i |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → 3 ≠ 0 ) |
140 |
138 125 139
|
3jca |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → ( 3 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 0 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 3 ≠ 0 ) ) |
141 |
140
|
adantr |
⊢ ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) → ( 3 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 0 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 3 ≠ 0 ) ) |
142 |
141
|
3ad2ant3 |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ∧ ( ( ♯ ‘ 𝐹 ) = 4 ∧ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) ) → ( 3 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 0 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 3 ≠ 0 ) ) |
143 |
|
pthdivtx |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 3 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 0 ∈ ( 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 ∈ ( 0 ..^ 4 ) ↔ ( 1 ∈ ℕ0 ∧ 4 ∈ ℕ ∧ 1 < 4 ) ) |
148 |
72 103 74 147
|
mpbir3an |
⊢ 1 ∈ ( 0 ..^ 4 ) |
149 |
|
eleq2 |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) → ( 1 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ 1 ∈ ( 0 ..^ 4 ) ) ) |
150 |
148 149
|
mpbiri |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) → 1 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
151 |
150
|
adantl |
⊢ ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) → 1 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
152 |
|
pthdadjvtx |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ∧ 1 ∈ ( 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 ∈ ( 1 ..^ 4 ) ↔ ( 1 ∈ ( 0 ..^ 4 ) ∧ 1 ≠ 0 ) ) |
160 |
148 158 159
|
mpbir2an |
⊢ 1 ∈ ( 1 ..^ 4 ) |
161 |
160 121
|
eleqtrrid |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → 1 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) |
162 |
|
3re |
⊢ 3 ∈ ℝ |
163 |
|
4re |
⊢ 4 ∈ ℝ |
164 |
162 163 90
|
ltleii |
⊢ 3 ≤ 4 |
165 |
|
elfz2nn0 |
⊢ ( 3 ∈ ( 0 ... 4 ) ↔ ( 3 ∈ ℕ0 ∧ 4 ∈ ℕ0 ∧ 3 ≤ 4 ) ) |
166 |
88 55 164 165
|
mpbir3an |
⊢ 3 ∈ ( 0 ... 4 ) |
167 |
166 58
|
eleqtrrid |
⊢ ( ( ♯ ‘ 𝐹 ) = 4 → 3 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ) |
168 |
|
1re |
⊢ 1 ∈ ℝ |
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 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 3 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 1 ≠ 3 ) ) |
173 |
172
|
adantr |
⊢ ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) → ( 1 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 3 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 1 ≠ 3 ) ) |
174 |
173
|
3ad2ant3 |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ∧ ( ( ♯ ‘ 𝐹 ) = 4 ∧ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) ) → ( 1 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 3 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 1 ≠ 3 ) ) |
175 |
|
pthdivtx |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 1 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 3 ∈ ( 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 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ 2 ∈ ( 0 ..^ 4 ) ) ) |
178 |
117 177
|
mpbiri |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) → 2 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
179 |
178
|
adantl |
⊢ ( ( ( ♯ ‘ 𝐹 ) = 4 ∧ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 4 ) ) → 2 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
180 |
|
pthdadjvtx |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ∧ 2 ∈ ( 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 ∈ ℕ0 ∧ 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 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) ∧ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 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 ↔ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ) |
192 |
191
|
anbi2d |
⊢ ( 𝑐 = ( 𝑃 ‘ 2 ) → ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ↔ ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ) ) |
193 |
|
preq1 |
⊢ ( 𝑐 = ( 𝑃 ‘ 2 ) → { 𝑐 , 𝑑 } = { ( 𝑃 ‘ 2 ) , 𝑑 } ) |
194 |
193
|
eleq1d |
⊢ ( 𝑐 = ( 𝑃 ‘ 2 ) → ( { 𝑐 , 𝑑 } ∈ 𝐸 ↔ { ( 𝑃 ‘ 2 ) , 𝑑 } ∈ 𝐸 ) ) |
195 |
194
|
anbi1d |
⊢ ( 𝑐 = ( 𝑃 ‘ 2 ) → ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ↔ ( { ( 𝑃 ‘ 2 ) , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) |
196 |
192 195
|
anbi12d |
⊢ ( 𝑐 = ( 𝑃 ‘ 2 ) → ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ↔ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ∧ ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ 𝑐 ∧ ( 𝑃 ‘ 0 ) ≠ 𝑑 ) ∧ ( ( 𝑃 ‘ 1 ) ≠ 𝑐 ∧ ( 𝑃 ‘ 1 ) ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ↔ ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ↔ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ) ) |
206 |
|
preq1 |
⊢ ( 𝑑 = ( 𝑃 ‘ 3 ) → { 𝑑 , ( 𝑃 ‘ 0 ) } = { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ) |
207 |
206
|
eleq1d |
⊢ ( 𝑑 = ( 𝑃 ‘ 3 ) → ( { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ↔ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) |
208 |
205 207
|
anbi12d |
⊢ ( 𝑑 = ( 𝑃 ‘ 3 ) → ( ( { ( 𝑃 ‘ 2 ) , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ↔ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) |
209 |
208
|
anbi2d |
⊢ ( 𝑑 = ( 𝑃 ‘ 3 ) → ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ↔ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 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 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 3 ) ∈ 𝑉 ∧ ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 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 ) ) ) ) ) → ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) ∧ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) → ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) ) ∧ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ∧ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∈ ℕ0 ∧ 𝑃 : ( 0 ... 4 ) ⟶ 𝑉 ) → ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ∧ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ∧ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ∧ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ∧ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ∧ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) ) → ( ( 𝑃 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ∧ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ↔ { ( 𝑃 ‘ 0 ) , 𝑏 } ∈ 𝐸 ) ) |
228 |
227
|
anbi1d |
⊢ ( 𝑎 = ( 𝑃 ‘ 0 ) → ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ↔ ( { ( 𝑃 ‘ 0 ) , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ) ) |
229 |
|
preq2 |
⊢ ( 𝑎 = ( 𝑃 ‘ 0 ) → { 𝑑 , 𝑎 } = { 𝑑 , ( 𝑃 ‘ 0 ) } ) |
230 |
229
|
eleq1d |
⊢ ( 𝑎 = ( 𝑃 ‘ 0 ) → ( { 𝑑 , 𝑎 } ∈ 𝐸 ↔ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) |
231 |
230
|
anbi2d |
⊢ ( 𝑎 = ( 𝑃 ‘ 0 ) → ( ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ↔ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ) |
232 |
228 231
|
anbi12d |
⊢ ( 𝑎 = ( 𝑃 ‘ 0 ) → ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ) ↔ ( ( { ( 𝑃 ‘ 0 ) , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ) ∧ ( ( 𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ↔ ( ( ( { ( 𝑃 ‘ 0 ) , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ∧ ( ( ( 𝑃 ‘ 0 ) ≠ 𝑏 ∧ ( 𝑃 ‘ 0 ) ≠ 𝑐 ∧ ( 𝑃 ‘ 0 ) ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ) ) |
239 |
238
|
2rexbidv |
⊢ ( 𝑎 = ( 𝑃 ‘ 0 ) → ( ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ) ∧ ( ( 𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ↔ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ↔ { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ) ) |
242 |
|
preq1 |
⊢ ( 𝑏 = ( 𝑃 ‘ 1 ) → { 𝑏 , 𝑐 } = { ( 𝑃 ‘ 1 ) , 𝑐 } ) |
243 |
242
|
eleq1d |
⊢ ( 𝑏 = ( 𝑃 ‘ 1 ) → ( { 𝑏 , 𝑐 } ∈ 𝐸 ↔ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ) |
244 |
241 243
|
anbi12d |
⊢ ( 𝑏 = ( 𝑃 ‘ 1 ) → ( ( { ( 𝑃 ‘ 0 ) , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ↔ ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ) ) |
245 |
244
|
anbi1d |
⊢ ( 𝑏 = ( 𝑃 ‘ 1 ) → ( ( ( { ( 𝑃 ‘ 0 ) , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ↔ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ∧ ( ( ( 𝑃 ‘ 0 ) ≠ 𝑏 ∧ ( 𝑃 ‘ 0 ) ≠ 𝑐 ∧ ( 𝑃 ‘ 0 ) ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ↔ ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ) → ( ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ∧ ( ( ( 𝑃 ‘ 0 ) ≠ 𝑏 ∧ ( 𝑃 ‘ 0 ) ≠ 𝑐 ∧ ( 𝑃 ‘ 0 ) ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ↔ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 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 ) ∈ 𝑉 ∧ ( 𝑃 ‘ 1 ) ∈ 𝑉 ∧ ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 1 ) , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , ( 𝑃 ‘ 0 ) } ∈ 𝐸 ) ) ∧ ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ 𝑐 ∧ ( 𝑃 ‘ 0 ) ≠ 𝑑 ) ∧ ( ( 𝑃 ‘ 1 ) ≠ 𝑐 ∧ ( 𝑃 ‘ 1 ) ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 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 ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ∈ 𝐸 ) ∧ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ∈ 𝐸 ∧ { ( 𝑃 ‘ 3 ) , ( 𝑃 ‘ 4 ) } ∈ 𝐸 ) ) ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 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 ) ) ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 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 ) ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ) ∧ ( ( 𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ) ) ) |
258 |
257
|
com13 |
⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ∈ 𝐸 → ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ( ♯ ‘ 𝐹 ) = 4 → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ) ∧ ( ( 𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ) ) ) |
259 |
5 258
|
syl |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) → ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ( ♯ ‘ 𝐹 ) = 4 → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ) ∧ ( ( 𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ) ) ) |
260 |
259
|
expcom |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ UPGraph → ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ( ♯ ‘ 𝐹 ) = 4 → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 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 ∈ UPGraph → ( ( ♯ ‘ 𝐹 ) = 4 → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 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 ∈ UPGraph → ( ( ♯ ‘ 𝐹 ) = 4 → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 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 ∈ UPGraph → ( ( ♯ ‘ 𝐹 ) = 4 → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 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 ∈ UPGraph → ( ( ♯ ‘ 𝐹 ) = 4 → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ) ∧ ( ( 𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ) ) ) |
265 |
3 264
|
syl |
⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ UPGraph → ( ( ♯ ‘ 𝐹 ) = 4 → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ) ∧ ( ( 𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ) ) ) |
266 |
265
|
3imp21 |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝐹 ) = 4 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ∃ 𝑐 ∈ 𝑉 ∃ 𝑑 ∈ 𝑉 ( ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ { 𝑏 , 𝑐 } ∈ 𝐸 ) ∧ ( { 𝑐 , 𝑑 } ∈ 𝐸 ∧ { 𝑑 , 𝑎 } ∈ 𝐸 ) ) ∧ ( ( 𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑 ) ∧ ( 𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑 ) ) ) ) |