Step |
Hyp |
Ref |
Expression |
1 |
|
eupthp1.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
eupthp1.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
eupthp1.f |
⊢ ( 𝜑 → Fun 𝐼 ) |
4 |
|
eupthp1.a |
⊢ ( 𝜑 → 𝐼 ∈ Fin ) |
5 |
|
eupthp1.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
6 |
|
eupthp1.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
7 |
|
eupthp1.d |
⊢ ( 𝜑 → ¬ 𝐵 ∈ dom 𝐼 ) |
8 |
|
eupthp1.p |
⊢ ( 𝜑 → 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 ) |
9 |
|
eupthp1.n |
⊢ 𝑁 = ( ♯ ‘ 𝐹 ) |
10 |
|
eupthp1.e |
⊢ ( 𝜑 → 𝐸 ∈ ( Edg ‘ 𝐺 ) ) |
11 |
|
eupthp1.x |
⊢ ( 𝜑 → { ( 𝑃 ‘ 𝑁 ) , 𝐶 } ⊆ 𝐸 ) |
12 |
|
eupthp1.u |
⊢ ( iEdg ‘ 𝑆 ) = ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) |
13 |
|
eupthp1.h |
⊢ 𝐻 = ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) |
14 |
|
eupthp1.q |
⊢ 𝑄 = ( 𝑃 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) |
15 |
|
eupthp1.s |
⊢ ( Vtx ‘ 𝑆 ) = 𝑉 |
16 |
|
eupthp1.l |
⊢ ( ( 𝜑 ∧ 𝐶 = ( 𝑃 ‘ 𝑁 ) ) → 𝐸 = { 𝐶 } ) |
17 |
|
eupthiswlk |
⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
18 |
8 17
|
syl |
⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
19 |
12
|
a1i |
⊢ ( 𝜑 → ( iEdg ‘ 𝑆 ) = ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) ) |
20 |
15
|
a1i |
⊢ ( 𝜑 → ( Vtx ‘ 𝑆 ) = 𝑉 ) |
21 |
1 2 3 4 5 6 7 18 9 10 11 19 13 14 20 16
|
wlkp1 |
⊢ ( 𝜑 → 𝐻 ( Walks ‘ 𝑆 ) 𝑄 ) |
22 |
2
|
eupthi |
⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1-onto→ dom 𝐼 ) ) |
23 |
9
|
eqcomi |
⊢ ( ♯ ‘ 𝐹 ) = 𝑁 |
24 |
23
|
oveq2i |
⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 𝑁 ) |
25 |
|
f1oeq2 |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 𝑁 ) → ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1-onto→ dom 𝐼 ↔ 𝐹 : ( 0 ..^ 𝑁 ) –1-1-onto→ dom 𝐼 ) ) |
26 |
24 25
|
ax-mp |
⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1-onto→ dom 𝐼 ↔ 𝐹 : ( 0 ..^ 𝑁 ) –1-1-onto→ dom 𝐼 ) |
27 |
26
|
biimpi |
⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1-onto→ dom 𝐼 → 𝐹 : ( 0 ..^ 𝑁 ) –1-1-onto→ dom 𝐼 ) |
28 |
27
|
adantl |
⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1-onto→ dom 𝐼 ) → 𝐹 : ( 0 ..^ 𝑁 ) –1-1-onto→ dom 𝐼 ) |
29 |
8 22 28
|
3syl |
⊢ ( 𝜑 → 𝐹 : ( 0 ..^ 𝑁 ) –1-1-onto→ dom 𝐼 ) |
30 |
9
|
fvexi |
⊢ 𝑁 ∈ V |
31 |
|
f1osng |
⊢ ( ( 𝑁 ∈ V ∧ 𝐵 ∈ 𝑊 ) → { 〈 𝑁 , 𝐵 〉 } : { 𝑁 } –1-1-onto→ { 𝐵 } ) |
32 |
30 5 31
|
sylancr |
⊢ ( 𝜑 → { 〈 𝑁 , 𝐵 〉 } : { 𝑁 } –1-1-onto→ { 𝐵 } ) |
33 |
|
dmsnopg |
⊢ ( 𝐸 ∈ ( Edg ‘ 𝐺 ) → dom { 〈 𝐵 , 𝐸 〉 } = { 𝐵 } ) |
34 |
10 33
|
syl |
⊢ ( 𝜑 → dom { 〈 𝐵 , 𝐸 〉 } = { 𝐵 } ) |
35 |
34
|
f1oeq3d |
⊢ ( 𝜑 → ( { 〈 𝑁 , 𝐵 〉 } : { 𝑁 } –1-1-onto→ dom { 〈 𝐵 , 𝐸 〉 } ↔ { 〈 𝑁 , 𝐵 〉 } : { 𝑁 } –1-1-onto→ { 𝐵 } ) ) |
36 |
32 35
|
mpbird |
⊢ ( 𝜑 → { 〈 𝑁 , 𝐵 〉 } : { 𝑁 } –1-1-onto→ dom { 〈 𝐵 , 𝐸 〉 } ) |
37 |
|
fzodisjsn |
⊢ ( ( 0 ..^ 𝑁 ) ∩ { 𝑁 } ) = ∅ |
38 |
37
|
a1i |
⊢ ( 𝜑 → ( ( 0 ..^ 𝑁 ) ∩ { 𝑁 } ) = ∅ ) |
39 |
34
|
ineq2d |
⊢ ( 𝜑 → ( dom 𝐼 ∩ dom { 〈 𝐵 , 𝐸 〉 } ) = ( dom 𝐼 ∩ { 𝐵 } ) ) |
40 |
|
disjsn |
⊢ ( ( dom 𝐼 ∩ { 𝐵 } ) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐼 ) |
41 |
7 40
|
sylibr |
⊢ ( 𝜑 → ( dom 𝐼 ∩ { 𝐵 } ) = ∅ ) |
42 |
39 41
|
eqtrd |
⊢ ( 𝜑 → ( dom 𝐼 ∩ dom { 〈 𝐵 , 𝐸 〉 } ) = ∅ ) |
43 |
|
f1oun |
⊢ ( ( ( 𝐹 : ( 0 ..^ 𝑁 ) –1-1-onto→ dom 𝐼 ∧ { 〈 𝑁 , 𝐵 〉 } : { 𝑁 } –1-1-onto→ dom { 〈 𝐵 , 𝐸 〉 } ) ∧ ( ( ( 0 ..^ 𝑁 ) ∩ { 𝑁 } ) = ∅ ∧ ( dom 𝐼 ∩ dom { 〈 𝐵 , 𝐸 〉 } ) = ∅ ) ) → ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) : ( ( 0 ..^ 𝑁 ) ∪ { 𝑁 } ) –1-1-onto→ ( dom 𝐼 ∪ dom { 〈 𝐵 , 𝐸 〉 } ) ) |
44 |
29 36 38 42 43
|
syl22anc |
⊢ ( 𝜑 → ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) : ( ( 0 ..^ 𝑁 ) ∪ { 𝑁 } ) –1-1-onto→ ( dom 𝐼 ∪ dom { 〈 𝐵 , 𝐸 〉 } ) ) |
45 |
13
|
a1i |
⊢ ( 𝜑 → 𝐻 = ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) ) |
46 |
1 2 3 4 5 6 7 18 9 10 11 19 13
|
wlkp1lem2 |
⊢ ( 𝜑 → ( ♯ ‘ 𝐻 ) = ( 𝑁 + 1 ) ) |
47 |
46
|
oveq2d |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝐻 ) ) = ( 0 ..^ ( 𝑁 + 1 ) ) ) |
48 |
|
wlkcl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
49 |
9
|
eleq1i |
⊢ ( 𝑁 ∈ ℕ0 ↔ ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
50 |
|
elnn0uz |
⊢ ( 𝑁 ∈ ℕ0 ↔ 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
51 |
49 50
|
sylbb1 |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
52 |
48 51
|
syl |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
53 |
8 17 52
|
3syl |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
54 |
|
fzosplitsn |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( 0 ..^ ( 𝑁 + 1 ) ) = ( ( 0 ..^ 𝑁 ) ∪ { 𝑁 } ) ) |
55 |
53 54
|
syl |
⊢ ( 𝜑 → ( 0 ..^ ( 𝑁 + 1 ) ) = ( ( 0 ..^ 𝑁 ) ∪ { 𝑁 } ) ) |
56 |
47 55
|
eqtrd |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝐻 ) ) = ( ( 0 ..^ 𝑁 ) ∪ { 𝑁 } ) ) |
57 |
|
dmun |
⊢ dom ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) = ( dom 𝐼 ∪ dom { 〈 𝐵 , 𝐸 〉 } ) |
58 |
57
|
a1i |
⊢ ( 𝜑 → dom ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) = ( dom 𝐼 ∪ dom { 〈 𝐵 , 𝐸 〉 } ) ) |
59 |
45 56 58
|
f1oeq123d |
⊢ ( 𝜑 → ( 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) ↔ ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) : ( ( 0 ..^ 𝑁 ) ∪ { 𝑁 } ) –1-1-onto→ ( dom 𝐼 ∪ dom { 〈 𝐵 , 𝐸 〉 } ) ) ) |
60 |
44 59
|
mpbird |
⊢ ( 𝜑 → 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) ) |
61 |
12
|
eqcomi |
⊢ ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) = ( iEdg ‘ 𝑆 ) |
62 |
61
|
iseupthf1o |
⊢ ( 𝐻 ( EulerPaths ‘ 𝑆 ) 𝑄 ↔ ( 𝐻 ( Walks ‘ 𝑆 ) 𝑄 ∧ 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) ) ) |
63 |
21 60 62
|
sylanbrc |
⊢ ( 𝜑 → 𝐻 ( EulerPaths ‘ 𝑆 ) 𝑄 ) |