Proof of Theorem rusgrnumwwlkb0
Step | Hyp | Ref
| Expression |
1 | | simpr 488 |
. . 3
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → 𝑃 ∈ 𝑉) |
2 | | 0nn0 11993 |
. . 3
⊢ 0 ∈
ℕ0 |
3 | | rusgrnumwwlk.v |
. . . 4
⊢ 𝑉 = (Vtx‘𝐺) |
4 | | rusgrnumwwlk.l |
. . . 4
⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦
(♯‘{𝑤 ∈
(𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) |
5 | 3, 4 | rusgrnumwwlklem 27910 |
. . 3
⊢ ((𝑃 ∈ 𝑉 ∧ 0 ∈ ℕ0) →
(𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})) |
6 | 1, 2, 5 | sylancl 589 |
. 2
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})) |
7 | | df-rab 3062 |
. . . . 5
⊢ {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)} |
8 | 7 | a1i 11 |
. . . 4
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)}) |
9 | | wwlksn0s 27801 |
. . . . . . . . 9
⊢ (0
WWalksN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1} |
10 | 9 | a1i 11 |
. . . . . . . 8
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (0 WWalksN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1}) |
11 | 10 | eleq2d 2818 |
. . . . . . 7
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ 𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1})) |
12 | | rabid 3281 |
. . . . . . 7
⊢ (𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1} ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1)) |
13 | 11, 12 | bitrdi 290 |
. . . . . 6
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1))) |
14 | 13 | anbi1d 633 |
. . . . 5
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → ((𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃))) |
15 | 14 | abbidv 2802 |
. . . 4
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)} = {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)}) |
16 | | wrdl1s1 14059 |
. . . . . . . . 9
⊢ (𝑃 ∈ (Vtx‘𝐺) → (𝑣 = 〈“𝑃”〉 ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃))) |
17 | | df-3an 1090 |
. . . . . . . . 9
⊢ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃)) |
18 | 16, 17 | bitr2di 291 |
. . . . . . . 8
⊢ (𝑃 ∈ (Vtx‘𝐺) → (((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃) ↔ 𝑣 = 〈“𝑃”〉)) |
19 | | vex 3402 |
. . . . . . . . 9
⊢ 𝑣 ∈ V |
20 | | eleq1w 2815 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑣 → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑣 ∈ Word (Vtx‘𝐺))) |
21 | | fveqeq2 6685 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑣 → ((♯‘𝑤) = 1 ↔ (♯‘𝑣) = 1)) |
22 | 20, 21 | anbi12d 634 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑣 → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1))) |
23 | | fveq1 6675 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑣 → (𝑤‘0) = (𝑣‘0)) |
24 | 23 | eqeq1d 2740 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑣 → ((𝑤‘0) = 𝑃 ↔ (𝑣‘0) = 𝑃)) |
25 | 22, 24 | anbi12d 634 |
. . . . . . . . 9
⊢ (𝑤 = 𝑣 → (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))) |
26 | 19, 25 | elab 3573 |
. . . . . . . 8
⊢ (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃)) |
27 | | velsn 4532 |
. . . . . . . 8
⊢ (𝑣 ∈ {〈“𝑃”〉} ↔ 𝑣 = 〈“𝑃”〉) |
28 | 18, 26, 27 | 3bitr4g 317 |
. . . . . . 7
⊢ (𝑃 ∈ (Vtx‘𝐺) → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {〈“𝑃”〉})) |
29 | 28, 3 | eleq2s 2851 |
. . . . . 6
⊢ (𝑃 ∈ 𝑉 → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {〈“𝑃”〉})) |
30 | 29 | eqrdv 2736 |
. . . . 5
⊢ (𝑃 ∈ 𝑉 → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {〈“𝑃”〉}) |
31 | 30 | adantl 485 |
. . . 4
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {〈“𝑃”〉}) |
32 | 8, 15, 31 | 3eqtrd 2777 |
. . 3
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {〈“𝑃”〉}) |
33 | 32 | fveq2d 6680 |
. 2
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (♯‘{〈“𝑃”〉})) |
34 | | s1cl 14047 |
. . . 4
⊢ (𝑃 ∈ 𝑉 → 〈“𝑃”〉 ∈ Word 𝑉) |
35 | | hashsng 13824 |
. . . 4
⊢
(〈“𝑃”〉 ∈ Word 𝑉 → (♯‘{〈“𝑃”〉}) =
1) |
36 | 34, 35 | syl 17 |
. . 3
⊢ (𝑃 ∈ 𝑉 → (♯‘{〈“𝑃”〉}) =
1) |
37 | 36 | adantl 485 |
. 2
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (♯‘{〈“𝑃”〉}) =
1) |
38 | 6, 33, 37 | 3eqtrd 2777 |
1
⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1) |