Step | Hyp | Ref
| Expression |
1 | | eleq1 2676 |
. . . . . . . . 9
⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝑋𝐻(𝑁 + 2)) ↔ 𝑥 ∈ (𝑋𝐻(𝑁 + 2)))) |
2 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → (𝑅‘𝑦) = (𝑅‘𝑥)) |
3 | | oveq1 6556 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → (𝑦 substr 〈0, (𝑁 + 1)〉) = (𝑥 substr 〈0, (𝑁 + 1)〉)) |
4 | 2, 3 | eqeq12d 2625 |
. . . . . . . . 9
⊢ (𝑦 = 𝑥 → ((𝑅‘𝑦) = (𝑦 substr 〈0, (𝑁 + 1)〉) ↔ (𝑅‘𝑥) = (𝑥 substr 〈0, (𝑁 + 1)〉))) |
5 | 1, 4 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑦 = 𝑥 → ((𝑦 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑦) = (𝑦 substr 〈0, (𝑁 + 1)〉)) ↔ (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 substr 〈0, (𝑁 + 1)〉)))) |
6 | 5 | imbi2d 329 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑦 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑦) = (𝑦 substr 〈0, (𝑁 + 1)〉))) ↔ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 substr 〈0, (𝑁 + 1)〉))))) |
7 | | av-numclwwlk.v |
. . . . . . . 8
⊢ 𝑉 = (Vtx‘𝐺) |
8 | | av-numclwwlk.q |
. . . . . . . 8
⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) |
9 | | av-numclwwlk.f |
. . . . . . . 8
⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ (𝑤‘0) = 𝑣}) |
10 | | av-numclwwlk.h |
. . . . . . . 8
⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) |
11 | | av-numclwwlk.r |
. . . . . . . 8
⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr 〈0, (𝑁 + 1)〉)) |
12 | 7, 8, 9, 10, 11 | av-numclwlk2lem2fv 41534 |
. . . . . . 7
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑦 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑦) = (𝑦 substr 〈0, (𝑁 + 1)〉))) |
13 | 6, 12 | chvarv 2251 |
. . . . . 6
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 substr 〈0, (𝑁 + 1)〉))) |
14 | 13 | 3adant1 1072 |
. . . . 5
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 substr 〈0, (𝑁 + 1)〉))) |
15 | 14 | imp 444 |
. . . 4
⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑥) = (𝑥 substr 〈0, (𝑁 + 1)〉)) |
16 | 7, 8, 9, 10, 11 | av-numclwlk2lem2f 41533 |
. . . . 5
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁)) |
17 | 16 | ffvelrnda 6267 |
. . . 4
⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑥) ∈ (𝑋𝑄𝑁)) |
18 | 15, 17 | eqeltrrd 2689 |
. . 3
⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑥 substr 〈0, (𝑁 + 1)〉) ∈ (𝑋𝑄𝑁)) |
19 | 18 | ralrimiva 2949 |
. 2
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 substr 〈0, (𝑁 + 1)〉) ∈ (𝑋𝑄𝑁)) |
20 | 7, 8, 9, 10 | av-numclwwlk2lem1 41532 |
. . . . 5
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2)))) |
21 | 20 | imp 444 |
. . . 4
⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑢 ∈ (𝑋𝑄𝑁)) → ∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2))) |
22 | 7, 8 | av-numclwwlkovq 41529 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}) |
23 | 22 | eleq2d 2673 |
. . . . . . . 8
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) ↔ 𝑢 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})) |
24 | 23 | 3adant1 1072 |
. . . . . . 7
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) ↔ 𝑢 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})) |
25 | | fveq1 6102 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑢 → (𝑤‘0) = (𝑢‘0)) |
26 | 25 | eqeq1d 2612 |
. . . . . . . . 9
⊢ (𝑤 = 𝑢 → ((𝑤‘0) = 𝑋 ↔ (𝑢‘0) = 𝑋)) |
27 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑢 → ( lastS ‘𝑤) = ( lastS ‘𝑢)) |
28 | 27 | neeq1d 2841 |
. . . . . . . . 9
⊢ (𝑤 = 𝑢 → (( lastS ‘𝑤) ≠ 𝑋 ↔ ( lastS ‘𝑢) ≠ 𝑋)) |
29 | 26, 28 | anbi12d 743 |
. . . . . . . 8
⊢ (𝑤 = 𝑢 → (((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋) ↔ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋))) |
30 | 29 | elrab 3331 |
. . . . . . 7
⊢ (𝑢 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} ↔ (𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋))) |
31 | 24, 30 | syl6bb 275 |
. . . . . 6
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) ↔ (𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)))) |
32 | | wwlknbp2 41063 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 ∈ (𝑁 WWalkSN 𝐺) → (𝑢 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑢) = (𝑁 + 1))) |
33 | 7 | wrdeqi 13183 |
. . . . . . . . . . . . . . . . 17
⊢ Word
𝑉 = Word (Vtx‘𝐺) |
34 | 33 | eleq2i 2680 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 ∈ Word 𝑉 ↔ 𝑢 ∈ Word (Vtx‘𝐺)) |
35 | 34 | anbi1i 727 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ↔ (𝑢 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑢) = (𝑁 + 1))) |
36 | 32, 35 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢ (𝑢 ∈ (𝑁 WWalkSN 𝐺) → (𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1))) |
37 | | simpll 786 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑢 ∈ Word 𝑉) |
38 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℕ) |
39 | | 2nn 11062 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 2 ∈
ℕ |
40 | 39 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℕ → 2 ∈
ℕ) |
41 | 38, 40 | nnaddcld 10944 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈
ℕ) |
42 | 7, 8, 9, 10 | av-numclwwlkovh 41531 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ ℕ) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))}) |
43 | 41, 42 | sylan2 490 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))}) |
44 | 43 | eleq2d 2673 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ 𝑥 ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})) |
45 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0)) |
46 | 45 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑤 = 𝑥 → ((𝑤‘0) = 𝑋 ↔ (𝑥‘0) = 𝑋)) |
47 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑤 = 𝑥 → (𝑤‘((𝑁 + 2) − 2)) = (𝑥‘((𝑁 + 2) − 2))) |
48 | 47, 45 | neeq12d 2843 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑤 = 𝑥 → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))) |
49 | 46, 48 | anbi12d 743 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 = 𝑥 → (((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0)) ↔ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))) |
50 | 49 | elrab 3331 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))} ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))) |
51 | 44, 50 | syl6bb 275 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))))) |
52 | 51 | 3adant1 1072 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))))) |
53 | 52 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))))) |
54 | 7 | clwwlknbp 41193 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 2))) |
55 | | lencl 13179 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑢 ∈ Word 𝑉 → (#‘𝑢) ∈
ℕ0) |
56 | | simprr 792 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((#‘𝑢)
∈ ℕ0 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → 𝑥 ∈ Word 𝑉) |
57 | | df-2 10956 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ 2 = (1 +
1) |
58 | 57 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑁 ∈ ℕ → 2 = (1 +
1)) |
59 | 58 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑁 ∈ ℕ → (𝑁 + 2) = (𝑁 + (1 + 1))) |
60 | | nncn 10905 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℂ) |
61 | | 1cnd 9935 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑁 ∈ ℕ → 1 ∈
ℂ) |
62 | 60, 61, 61 | addassd 9941 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1))) |
63 | 59, 62 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑁 ∈ ℕ → (𝑁 + 2) = ((𝑁 + 1) + 1)) |
64 | 63 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((#‘𝑢)
∈ ℕ0 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 2) = ((𝑁 + 1) + 1)) |
65 | 64 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
((((#‘𝑢)
∈ ℕ0 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((#‘𝑥) = (𝑁 + 2) ↔ (#‘𝑥) = ((𝑁 + 1) + 1))) |
66 | 65 | biimpcd 238 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((#‘𝑥) =
(𝑁 + 2) →
((((#‘𝑢) ∈
ℕ0 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (#‘𝑥) = ((𝑁 + 1) + 1))) |
67 | 66 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(((#‘𝑥) =
(𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → ((((#‘𝑢) ∈ ℕ0 ∧
(#‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (#‘𝑥) = ((𝑁 + 1) + 1))) |
68 | 67 | impcom 445 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((#‘𝑢)
∈ ℕ0 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → (#‘𝑥) = ((𝑁 + 1) + 1)) |
69 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
((#‘𝑢) =
(𝑁 + 1) →
((#‘𝑢) + 1) = ((𝑁 + 1) + 1)) |
70 | 69 | ad3antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((#‘𝑢)
∈ ℕ0 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → ((#‘𝑢) + 1) = ((𝑁 + 1) + 1)) |
71 | 68, 70 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((#‘𝑢)
∈ ℕ0 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → (#‘𝑥) = ((#‘𝑢) + 1)) |
72 | 56, 71 | jca 553 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((((#‘𝑢)
∈ ℕ0 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))) |
73 | 72 | exp31 628 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((#‘𝑢) ∈
ℕ0 ∧ (#‘𝑢) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))))) |
74 | 55, 73 | sylan 487 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))))) |
75 | 74 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈ ℕ → ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) → (((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))))) |
76 | 75 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) → (((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))))) |
77 | 76 | impcom 445 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (((#‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1)))) |
78 | 77 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((#‘𝑥) =
(𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1)))) |
79 | 78 | ancoms 468 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 2)) → (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1)))) |
80 | 54, 79 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) → (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1)))) |
81 | 80 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))) → (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1)))) |
82 | 81 | com12 32 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑥 ∈ ((𝑁 + 2) ClWWalkSN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1)))) |
83 | 53, 82 | sylbid 229 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1)))) |
84 | 83 | ralrimiv 2948 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))) |
85 | 37, 84 | jca 553 |
. . . . . . . . . . . . . . 15
⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1)))) |
86 | 85 | ex 449 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))))) |
87 | 36, 86 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑢 ∈ (𝑁 WWalkSN 𝐺) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))))) |
88 | 87 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))))) |
89 | 88 | imp 444 |
. . . . . . . . . . 11
⊢ (((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1)))) |
90 | | reuccats1 13332 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑢) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (#‘𝑢)〉))) |
91 | 89, 90 | syl 17 |
. . . . . . . . . 10
⊢ (((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (#‘𝑢)〉))) |
92 | 91 | imp 444 |
. . . . . . . . 9
⊢ ((((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2))) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (#‘𝑢)〉)) |
93 | | simpr 476 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) → (#‘𝑢) = (𝑁 + 1)) |
94 | 93 | eqcomd 2616 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = (𝑁 + 1)) → (𝑁 + 1) = (#‘𝑢)) |
95 | 36, 94 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑢 ∈ (𝑁 WWalkSN 𝐺) → (𝑁 + 1) = (#‘𝑢)) |
96 | 95 | ad4antr 764 |
. . . . . . . . . . . . 13
⊢
(((((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑁 + 1) = (#‘𝑢)) |
97 | 96 | opeq2d 4347 |
. . . . . . . . . . . 12
⊢
(((((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → 〈0, (𝑁 + 1)〉 = 〈0, (#‘𝑢)〉) |
98 | 97 | oveq2d 6565 |
. . . . . . . . . . 11
⊢
(((((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑥 substr 〈0, (𝑁 + 1)〉) = (𝑥 substr 〈0, (#‘𝑢)〉)) |
99 | 98 | eqeq2d 2620 |
. . . . . . . . . 10
⊢
(((((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉) ↔ 𝑢 = (𝑥 substr 〈0, (#‘𝑢)〉))) |
100 | 99 | reubidva 3102 |
. . . . . . . . 9
⊢ ((((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2))) → (∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉) ↔ ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (#‘𝑢)〉))) |
101 | 92, 100 | mpbird 246 |
. . . . . . . 8
⊢ ((((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2))) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉)) |
102 | 101 | exp31 628 |
. . . . . . 7
⊢ ((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉)))) |
103 | 102 | com12 32 |
. . . . . 6
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑢 ∈ (𝑁 WWalkSN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ ( lastS ‘𝑢) ≠ 𝑋)) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉)))) |
104 | 31, 103 | sylbid 229 |
. . . . 5
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉)))) |
105 | 104 | imp 444 |
. . . 4
⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑢 ∈ (𝑋𝑄𝑁)) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ 〈“𝑣”〉) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉))) |
106 | 21, 105 | mpd 15 |
. . 3
⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑢 ∈ (𝑋𝑄𝑁)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉)) |
107 | 106 | ralrimiva 2949 |
. 2
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∀𝑢 ∈ (𝑋𝑄𝑁)∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉)) |
108 | 11 | f1ompt 6290 |
. 2
⊢ (𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁) ↔ (∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 substr 〈0, (𝑁 + 1)〉) ∈ (𝑋𝑄𝑁) ∧ ∀𝑢 ∈ (𝑋𝑄𝑁)∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 substr 〈0, (𝑁 + 1)〉))) |
109 | 19, 107, 108 | sylanbrc 695 |
1
⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁)) |