| 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 ) ) ) ) |