| 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 29518 |
. . . 4
⊢ (𝜑 → 𝐺 ∈ USPGraph) |
| 6 | | uspgrushgr 29194 |
. . . 4
⊢ (𝐺 ∈ USPGraph → 𝐺 ∈
USHGraph) |
| 7 | 5, 6 | syl 17 |
. . 3
⊢ (𝜑 → 𝐺 ∈ USHGraph) |
| 8 | 3, 1 | eleqtrrd 2844 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (Vtx‘𝐺)) |
| 9 | | eqid 2737 |
. . . 4
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
| 10 | | eqid 2737 |
. . . 4
⊢
(Edg‘𝐺) =
(Edg‘𝐺) |
| 11 | | eqid 2737 |
. . . 4
⊢
(VtxDeg‘𝐺) =
(VtxDeg‘𝐺) |
| 12 | 9, 10, 11 | vtxdushgrfvedg 29508 |
. . 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 584 |
. 2
⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒}) +𝑒
(♯‘{𝑒 ∈
(Edg‘𝐺) ∣ 𝑒 = {𝑁}}))) |
| 14 | | snex 5436 |
. . . . . . . 8
⊢ {𝑁} ∈ V |
| 15 | | sneq 4636 |
. . . . . . . . 9
⊢ (𝑎 = {𝑁} → {𝑎} = {{𝑁}}) |
| 16 | 15 | eqeq2d 2748 |
. . . . . . . 8
⊢ (𝑎 = {𝑁} → ({{𝑁}} = {𝑎} ↔ {{𝑁}} = {{𝑁}})) |
| 17 | | eqid 2737 |
. . . . . . . 8
⊢ {{𝑁}} = {{𝑁}} |
| 18 | 14, 16, 17 | ceqsexv2d 3533 |
. . . . . . 7
⊢
∃𝑎{{𝑁}} = {𝑎} |
| 19 | 18 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ∃𝑎{{𝑁}} = {𝑎}) |
| 20 | | snidg 4660 |
. . . . . . . . . 10
⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝑁}) |
| 21 | 3, 20 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ {𝑁}) |
| 22 | 21 | iftrued 4533 |
. . . . . . . 8
⊢ (𝜑 → if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {{𝑁}}) |
| 23 | 22 | eqeq1d 2739 |
. . . . . . 7
⊢ (𝜑 → (if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎} ↔ {{𝑁}} = {𝑎})) |
| 24 | 23 | exbidv 1921 |
. . . . . 6
⊢ (𝜑 → (∃𝑎if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎} ↔ ∃𝑎{{𝑁}} = {𝑎})) |
| 25 | 19, 24 | mpbird 257 |
. . . . 5
⊢ (𝜑 → ∃𝑎if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎}) |
| 26 | 1, 2, 3, 4 | 1loopgredg 29519 |
. . . . . . . . 9
⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}}) |
| 27 | 26 | rabeqdv 3452 |
. . . . . . . 8
⊢ (𝜑 → {𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑒 ∈ {{𝑁}} ∣ 𝑁 ∈ 𝑒}) |
| 28 | | eleq2 2830 |
. . . . . . . . 9
⊢ (𝑒 = {𝑁} → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ {𝑁})) |
| 29 | 28 | rabsnif 4723 |
. . . . . . . 8
⊢ {𝑒 ∈ {{𝑁}} ∣ 𝑁 ∈ 𝑒} = if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) |
| 30 | 27, 29 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝜑 → {𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅)) |
| 31 | 30 | eqeq1d 2739 |
. . . . . 6
⊢ (𝜑 → ({𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑎} ↔ if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎})) |
| 32 | 31 | exbidv 1921 |
. . . . 5
⊢ (𝜑 → (∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑎} ↔ ∃𝑎if(𝑁 ∈ {𝑁}, {{𝑁}}, ∅) = {𝑎})) |
| 33 | 25, 32 | mpbird 257 |
. . . 4
⊢ (𝜑 → ∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} = {𝑎}) |
| 34 | | fvex 6919 |
. . . . . 6
⊢
(Edg‘𝐺) ∈
V |
| 35 | 34 | rabex 5339 |
. . . . 5
⊢ {𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒} ∈ V |
| 36 | | hash1snb 14458 |
. . . . 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 234 |
. . 3
⊢ (𝜑 → (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒}) = 1) |
| 39 | | eqid 2737 |
. . . . . . . . 9
⊢ {𝑁} = {𝑁} |
| 40 | 39 | iftruei 4532 |
. . . . . . . 8
⊢ if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {{𝑁}} |
| 41 | 40 | eqeq1i 2742 |
. . . . . . 7
⊢
(if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎} ↔ {{𝑁}} = {𝑎}) |
| 42 | 41 | exbii 1848 |
. . . . . 6
⊢
(∃𝑎if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎} ↔ ∃𝑎{{𝑁}} = {𝑎}) |
| 43 | 19, 42 | sylibr 234 |
. . . . 5
⊢ (𝜑 → ∃𝑎if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎}) |
| 44 | 26 | rabeqdv 3452 |
. . . . . . . 8
⊢ (𝜑 → {𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑒 ∈ {{𝑁}} ∣ 𝑒 = {𝑁}}) |
| 45 | | eqeq1 2741 |
. . . . . . . . 9
⊢ (𝑒 = {𝑁} → (𝑒 = {𝑁} ↔ {𝑁} = {𝑁})) |
| 46 | 45 | rabsnif 4723 |
. . . . . . . 8
⊢ {𝑒 ∈ {{𝑁}} ∣ 𝑒 = {𝑁}} = if({𝑁} = {𝑁}, {{𝑁}}, ∅) |
| 47 | 44, 46 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝜑 → {𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = if({𝑁} = {𝑁}, {{𝑁}}, ∅)) |
| 48 | 47 | eqeq1d 2739 |
. . . . . 6
⊢ (𝜑 → ({𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑎} ↔ if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎})) |
| 49 | 48 | exbidv 1921 |
. . . . 5
⊢ (𝜑 → (∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑎} ↔ ∃𝑎if({𝑁} = {𝑁}, {{𝑁}}, ∅) = {𝑎})) |
| 50 | 43, 49 | mpbird 257 |
. . . 4
⊢ (𝜑 → ∃𝑎{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} = {𝑎}) |
| 51 | 34 | rabex 5339 |
. . . . 5
⊢ {𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}} ∈ V |
| 52 | | hash1snb 14458 |
. . . . 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 234 |
. . 3
⊢ (𝜑 → (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑒 = {𝑁}}) = 1) |
| 55 | 38, 54 | oveq12d 7449 |
. 2
⊢ (𝜑 → ((♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑁 ∈ 𝑒}) +𝑒
(♯‘{𝑒 ∈
(Edg‘𝐺) ∣ 𝑒 = {𝑁}})) = (1 +𝑒
1)) |
| 56 | | 1re 11261 |
. . . . 5
⊢ 1 ∈
ℝ |
| 57 | | rexadd 13274 |
. . . . 5
⊢ ((1
∈ ℝ ∧ 1 ∈ ℝ) → (1 +𝑒 1) = (1
+ 1)) |
| 58 | 56, 56, 57 | mp2an 692 |
. . . 4
⊢ (1
+𝑒 1) = (1 + 1) |
| 59 | | 1p1e2 12391 |
. . . 4
⊢ (1 + 1) =
2 |
| 60 | 58, 59 | eqtri 2765 |
. . 3
⊢ (1
+𝑒 1) = 2 |
| 61 | 60 | a1i 11 |
. 2
⊢ (𝜑 → (1 +𝑒
1) = 2) |
| 62 | 13, 55, 61 | 3eqtrd 2781 |
1
⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = 2) |