Step | Hyp | Ref
| Expression |
1 | | 1loopgruspgr.v |
. . . . 5
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
2 | | 1loopgruspgr.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ 𝑋) |
3 | | 1loopgruspgr.n |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ 𝑉) |
4 | | 1loopgruspgr.i |
. . . . 5
⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝑁}〉}) |
5 | 1, 2, 3, 4 | 1loopgruspgr 40715 |
. . . 4
⊢ (𝜑 → 𝐺 ∈ USPGraph ) |
6 | | uspgrushgr 40405 |
. . . 4
⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph
) |
7 | 5, 6 | syl 17 |
. . 3
⊢ (𝜑 → 𝐺 ∈ USHGraph ) |
8 | 3, 1 | eleqtrrd 2691 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐺)) |
9 | | eqid 2610 |
. . . 4
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
10 | | eqid 2610 |
. . . 4
⊢
(Edg‘𝐺) =
(Edg‘𝐺) |
11 | | eqid 2610 |
. . . 4
⊢
(VtxDeg‘𝐺) =
(VtxDeg‘𝐺) |
12 | 9, 10, 11 | vtxdushgrfvedg 40705 |
. . 3
⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ (Vtx‘𝐺)) → ((VtxDeg‘𝐺)‘𝑁) = ((#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒}) +𝑒 (#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}}))) |
13 | 7, 8, 12 | syl2anc 691 |
. 2
⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = ((#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒}) +𝑒 (#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}}))) |
14 | | snex 4835 |
. . . . . . . 8
⊢ {𝑁} ∈ V |
15 | | sneq 4135 |
. . . . . . . . 9
⊢ (𝑎 = {𝑁} → {𝑎} = {{𝑁}}) |
16 | 15 | eqeq2d 2620 |
. . . . . . . 8
⊢ (𝑎 = {𝑁} → ({{𝑁}} = {𝑎} ↔ {{𝑁}} = {{𝑁}})) |
17 | | eqid 2610 |
. . . . . . . 8
⊢ {{𝑁}} = {{𝑁}} |
18 | 14, 16, 17 | ceqsexv2d 3216 |
. . . . . . 7
⊢
∃𝑎{{𝑁}} = {𝑎} |
19 | 18 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ∃𝑎{{𝑁}} = {𝑎}) |
20 | | snidg 4153 |
. . . . . . . . . 10
⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝑁}) |
21 | 3, 20 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ {𝑁}) |
22 | 21 | iftrued 4044 |
. . . . . . . 8
⊢ (𝜑 → if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {{𝑁}}) |
23 | 22 | eqeq1d 2612 |
. . . . . . 7
⊢ (𝜑 → (if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎} ↔ {{𝑁}} = {𝑎})) |
24 | 23 | exbidv 1837 |
. . . . . 6
⊢ (𝜑 → (∃𝑎if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎} ↔ ∃𝑎{{𝑁}} = {𝑎})) |
25 | 19, 24 | mpbird 246 |
. . . . 5
⊢ (𝜑 → ∃𝑎if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎}) |
26 | 1, 2, 3, 4 | 1loopgredg 40716 |
. . . . . . . . 9
⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}}) |
27 | 26 | rabeqdv 3167 |
. . . . . . . 8
⊢ (𝜑 → {𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑒 ∈ {{𝑁}} ∣ 𝑁 ∈ 𝑒}) |
28 | | eleq2 2677 |
. . . . . . . . 9
⊢ (𝑒 = {𝑁} → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ {𝑁})) |
29 | 28 | rabsnif 4202 |
. . . . . . . 8
⊢ {𝑒 ∈ {{𝑁}} ∣ 𝑁 ∈ 𝑒} = if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) |
30 | 27, 29 | syl6eq 2660 |
. . . . . . 7
⊢ (𝜑 → {𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅)) |
31 | 30 | eqeq1d 2612 |
. . . . . 6
⊢ (𝜑 → ({𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑎} ↔ if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎})) |
32 | 31 | exbidv 1837 |
. . . . 5
⊢ (𝜑 → (∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑎} ↔ ∃𝑎if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎})) |
33 | 25, 32 | mpbird 246 |
. . . 4
⊢ (𝜑 → ∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑎}) |
34 | | fvex 6113 |
. . . . . 6
⊢
(Edg‘𝐺) ∈
V |
35 | 34 | rabex 4740 |
. . . . 5
⊢ {𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} ∈ V |
36 | | hash1snb 13068 |
. . . . 5
⊢ ({𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} ∈ V → ((#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒}) = 1 ↔ ∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑎})) |
37 | 35, 36 | ax-mp 5 |
. . . 4
⊢
((#‘{𝑒 ∈
(Edg‘𝐺) ∣ 𝑁 ∈ 𝑒}) = 1 ↔ ∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑎}) |
38 | 33, 37 | sylibr 223 |
. . 3
⊢ (𝜑 → (#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒}) = 1) |
39 | | eqid 2610 |
. . . . . . . . 9
⊢ {𝑁} = {𝑁} |
40 | 39 | iftruei 4043 |
. . . . . . . 8
⊢ if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {{𝑁}} |
41 | 40 | eqeq1i 2615 |
. . . . . . 7
⊢
(if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎} ↔ {{𝑁}} = {𝑎}) |
42 | 41 | exbii 1764 |
. . . . . 6
⊢
(∃𝑎if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎} ↔ ∃𝑎{{𝑁}} = {𝑎}) |
43 | 19, 42 | sylibr 223 |
. . . . 5
⊢ (𝜑 → ∃𝑎if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎}) |
44 | 26 | rabeqdv 3167 |
. . . . . . . 8
⊢ (𝜑 → {𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑒 ∈ {{𝑁}} ∣ 𝑒 = {𝑁}}) |
45 | | eqeq1 2614 |
. . . . . . . . 9
⊢ (𝑒 = {𝑁} → (𝑒 = {𝑁} ↔ {𝑁} = {𝑁})) |
46 | 45 | rabsnif 4202 |
. . . . . . . 8
⊢ {𝑒 ∈ {{𝑁}} ∣ 𝑒 = {𝑁}} = if({𝑁} = {𝑁}, {{𝑁}}, ∅) |
47 | 44, 46 | syl6eq 2660 |
. . . . . . 7
⊢ (𝜑 → {𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = if({𝑁} = {𝑁}, {{𝑁}}, ∅)) |
48 | 47 | eqeq1d 2612 |
. . . . . 6
⊢ (𝜑 → ({𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑎} ↔ if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎})) |
49 | 48 | exbidv 1837 |
. . . . 5
⊢ (𝜑 → (∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑎} ↔ ∃𝑎if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎})) |
50 | 43, 49 | mpbird 246 |
. . . 4
⊢ (𝜑 → ∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑎}) |
51 | 34 | rabex 4740 |
. . . . 5
⊢ {𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} ∈ V |
52 | | hash1snb 13068 |
. . . . 5
⊢ ({𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} ∈ V → ((#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}}) = 1 ↔ ∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑎})) |
53 | 51, 52 | ax-mp 5 |
. . . 4
⊢
((#‘{𝑒 ∈
(Edg‘𝐺) ∣ 𝑒 = {𝑁}}) = 1 ↔ ∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑎}) |
54 | 50, 53 | sylibr 223 |
. . 3
⊢ (𝜑 → (#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}}) = 1) |
55 | 38, 54 | oveq12d 6567 |
. 2
⊢ (𝜑 → ((#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒}) +𝑒 (#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}})) = (1 +𝑒
1)) |
56 | | 1re 9918 |
. . . . 5
⊢ 1 ∈
ℝ |
57 | | rexadd 11937 |
. . . . 5
⊢ ((1
∈ ℝ ∧ 1 ∈ ℝ) → (1 +𝑒 1) = (1
+ 1)) |
58 | 56, 56, 57 | mp2an 704 |
. . . 4
⊢ (1
+𝑒 1) = (1 + 1) |
59 | | 1p1e2 11011 |
. . . 4
⊢ (1 + 1) =
2 |
60 | 58, 59 | eqtri 2632 |
. . 3
⊢ (1
+𝑒 1) = 2 |
61 | 60 | a1i 11 |
. 2
⊢ (𝜑 → (1 +𝑒
1) = 2) |
62 | 13, 55, 61 | 3eqtrd 2648 |
1
⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = 2) |