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Theorem frgrregord013 30192
Description: If a finite friendship graph is 𝐾-regular, then it must have order 0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
Hypothesis
Ref Expression
frgrreggt1.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
frgrregord013 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))

Proof of Theorem frgrregord013
Dummy variables 𝑣 π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hashcl 14339 . . 3 (𝑉 ∈ Fin β†’ (β™―β€˜π‘‰) ∈ β„•0)
2 ax-1 6 . . . . 5 (((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3) β†’ ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
3 3ioran 1104 . . . . . 6 (Β¬ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3) ↔ (Β¬ (β™―β€˜π‘‰) = 0 ∧ Β¬ (β™―β€˜π‘‰) = 1 ∧ Β¬ (β™―β€˜π‘‰) = 3))
4 df-ne 2936 . . . . . . . . . . . . 13 ((β™―β€˜π‘‰) β‰  0 ↔ Β¬ (β™―β€˜π‘‰) = 0)
5 hasheq0 14346 . . . . . . . . . . . . . . . . . 18 (𝑉 ∈ Fin β†’ ((β™―β€˜π‘‰) = 0 ↔ 𝑉 = βˆ…))
65necon3bid 2980 . . . . . . . . . . . . . . . . 17 (𝑉 ∈ Fin β†’ ((β™―β€˜π‘‰) β‰  0 ↔ 𝑉 β‰  βˆ…))
76biimpa 476 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) β‰  0) β†’ 𝑉 β‰  βˆ…)
8 elnnne0 12508 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘‰) ∈ β„• ↔ ((β™―β€˜π‘‰) ∈ β„•0 ∧ (β™―β€˜π‘‰) β‰  0))
9 df-ne 2936 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜π‘‰) β‰  1 ↔ Β¬ (β™―β€˜π‘‰) = 1)
10 eluz2b3 12928 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ↔ ((β™―β€˜π‘‰) ∈ β„• ∧ (β™―β€˜π‘‰) β‰  1))
11 hash2prde 14455 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) = 2) β†’ βˆƒπ‘Žβˆƒπ‘(π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}))
12 vex 3473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 π‘Ž ∈ V
1312a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘Ž β‰  𝑏 β†’ π‘Ž ∈ V)
14 vex 3473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 𝑏 ∈ V
1514a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘Ž β‰  𝑏 β†’ 𝑏 ∈ V)
16 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (π‘Ž β‰  𝑏 β†’ π‘Ž β‰  𝑏)
1713, 15, 163jca 1126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (π‘Ž β‰  𝑏 β†’ (π‘Ž ∈ V ∧ 𝑏 ∈ V ∧ π‘Ž β‰  𝑏))
18 frgrreggt1.v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 𝑉 = (Vtxβ€˜πΊ)
1918eqeq1i 2732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑉 = {π‘Ž, 𝑏} ↔ (Vtxβ€˜πΊ) = {π‘Ž, 𝑏})
2019biimpi 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑉 = {π‘Ž, 𝑏} β†’ (Vtxβ€˜πΊ) = {π‘Ž, 𝑏})
21 nfrgr2v 30069 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((π‘Ž ∈ V ∧ 𝑏 ∈ V ∧ π‘Ž β‰  𝑏) ∧ (Vtxβ€˜πΊ) = {π‘Ž, 𝑏}) β†’ 𝐺 βˆ‰ FriendGraph )
2217, 20, 21syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ 𝐺 βˆ‰ FriendGraph )
23 df-nel 3042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝐺 βˆ‰ FriendGraph ↔ Β¬ 𝐺 ∈ FriendGraph )
2422, 23sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ Β¬ 𝐺 ∈ FriendGraph )
2524pm2.21d 121 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
2625com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
2726exlimivv 1928 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (βˆƒπ‘Žβˆƒπ‘(π‘Ž β‰  𝑏 ∧ 𝑉 = {π‘Ž, 𝑏}) β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
2811, 27syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) = 2) β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
2928ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑉 ∈ Fin β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
3029com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑉 ∈ Fin β†’ (𝑉 β‰  βˆ… β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
3130com14 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
3231a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
33323imp 1109 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 2 β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
3433com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜π‘‰) = 2 β†’ (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
35 eqid 2727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (VtxDegβ€˜πΊ) = (VtxDegβ€˜πΊ)
3618, 35rusgrprop0 29368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝐺 RegUSGraph 𝐾 β†’ (𝐺 ∈ USGraph ∧ 𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾))
37 eluz2gt1 12926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ 1 < (β™―β€˜π‘‰))
3837anim1ci 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ) β†’ (𝐺 ∈ FriendGraph ∧ 1 < (β™―β€˜π‘‰)))
3918vdgn0frgrv2 30092 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉) β†’ (1 < (β™―β€˜π‘‰) β†’ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
4039impancom 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ((𝐺 ∈ FriendGraph ∧ 1 < (β™―β€˜π‘‰)) β†’ (𝑣 ∈ 𝑉 β†’ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
4140ralrimiv 3140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ((𝐺 ∈ FriendGraph ∧ 1 < (β™―β€˜π‘‰)) β†’ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0)
42 eqeq2 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 (𝐾 = 0 β†’ (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 ↔ ((VtxDegβ€˜πΊ)β€˜π‘£) = 0))
4342ralbidv 3172 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 ↔ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 0))
44 r19.26 3106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 (βˆ€π‘£ ∈ 𝑉 (((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
45 nne 2939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 (Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ↔ ((VtxDegβ€˜πΊ)β€˜π‘£) = 0)
4645bicomi 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 (((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ↔ Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0)
4746anbi1i 623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ((((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ (Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
48 ancom 460 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ((Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ (((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0))
49 pm3.24 402 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Β¬ (((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0)
5049bifal 1550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ((((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 ∧ Β¬ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ βŠ₯)
5147, 48, 503bitri 297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ((((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ βŠ₯)
5251ralbii 3088 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 (βˆ€π‘£ ∈ 𝑉 (((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) ↔ βˆ€π‘£ ∈ 𝑉 βŠ₯)
53 r19.3rzv 4494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 (𝑉 β‰  βˆ… β†’ (βŠ₯ ↔ βˆ€π‘£ ∈ 𝑉 βŠ₯))
54 falim 1551 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 (βŠ₯ β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))
5553, 54syl6bir 254 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 (𝑉 β‰  βˆ… β†’ (βˆ€π‘£ ∈ 𝑉 βŠ₯ β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
5655adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (βˆ€π‘£ ∈ 𝑉 βŠ₯ β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
5756com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 (βˆ€π‘£ ∈ 𝑉 βŠ₯ β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
5852, 57sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 (βˆ€π‘£ ∈ 𝑉 (((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
5944, 58sylbir 234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 ((βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 0 ∧ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0) β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
6059ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
6143, 60biimtrdi 252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
6261com4t 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) β‰  0 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
6338, 41, 623syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ) β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
6463ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
6564com25 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
6665adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
6766com15 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝐺 ∈ FriendGraph β†’ (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
6867com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (𝐺 ∈ FriendGraph β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
69683ad2ant3 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ β„•0* ∧ βˆ€π‘£ ∈ 𝑉 ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ (𝐺 ∈ FriendGraph β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
7036, 69syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝐺 RegUSGraph 𝐾 β†’ (𝐺 ∈ FriendGraph β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
7170impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) β†’ ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
7271impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (𝐾 = 0 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
7318frrusgrord 30138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜π‘‰) = ((𝐾 Β· (𝐾 βˆ’ 1)) + 1)))
7473imp 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (β™―β€˜π‘‰) = ((𝐾 Β· (𝐾 βˆ’ 1)) + 1))
75 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝐾 = 2 β†’ 𝐾 = 2)
76 oveq1 7421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝐾 = 2 β†’ (𝐾 βˆ’ 1) = (2 βˆ’ 1))
7775, 76oveq12d 7432 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝐾 = 2 β†’ (𝐾 Β· (𝐾 βˆ’ 1)) = (2 Β· (2 βˆ’ 1)))
7877oveq1d 7429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝐾 = 2 β†’ ((𝐾 Β· (𝐾 βˆ’ 1)) + 1) = ((2 Β· (2 βˆ’ 1)) + 1))
79 2m1e1 12360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 (2 βˆ’ 1) = 1
8079oveq2i 7425 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (2 Β· (2 βˆ’ 1)) = (2 Β· 1)
81 2t1e2 12397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 (2 Β· 1) = 2
8280, 81eqtri 2755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (2 Β· (2 βˆ’ 1)) = 2
8382oveq1i 7424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((2 Β· (2 βˆ’ 1)) + 1) = (2 + 1)
84 2p1e3 12376 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (2 + 1) = 3
8583, 84eqtri 2755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((2 Β· (2 βˆ’ 1)) + 1) = 3
8678, 85eqtrdi 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝐾 = 2 β†’ ((𝐾 Β· (𝐾 βˆ’ 1)) + 1) = 3)
8786eqeq2d 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝐾 = 2 β†’ ((β™―β€˜π‘‰) = ((𝐾 Β· (𝐾 βˆ’ 1)) + 1) ↔ (β™―β€˜π‘‰) = 3))
88 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (Β¬ (β™―β€˜π‘‰) = 3 β†’ ((β™―β€˜π‘‰) = 3 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
8988ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 3 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
9089com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((β™―β€˜π‘‰) = 3 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
9187, 90biimtrdi 252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝐾 = 2 β†’ ((β™―β€˜π‘‰) = ((𝐾 Β· (𝐾 βˆ’ 1)) + 1) β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
9274, 91syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (𝐾 = 2 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
9318frgrreg 30191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝐾 = 0 ∨ 𝐾 = 2)))
9493imp 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (𝐾 = 0 ∨ 𝐾 = 2))
9572, 92, 94mpjaod 859 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ (𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾)) β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
9695exp32 420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
9796com34 91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐺 ∈ FriendGraph β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
9897com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (((Β¬ (β™―β€˜π‘‰) = 3 ∧ Β¬ (β™―β€˜π‘‰) = 2) ∧ (β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2)) β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
9998exp4c 432 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
10099com34 91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
101100com25 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
102101ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑉 ∈ Fin β†’ (𝑉 β‰  βˆ… β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
103102com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
104103com14 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1051043imp 1109 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 2 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
106105com3r 87 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Β¬ (β™―β€˜π‘‰) = 2 β†’ (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
10734, 106pm2.61i 182 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ…) β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
1081073exp 1117 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜π‘‰) ∈ (β„€β‰₯β€˜2) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
10910, 108sylbir 234 . . . . . . . . . . . . . . . . . . . . . . 23 (((β™―β€˜π‘‰) ∈ β„• ∧ (β™―β€˜π‘‰) β‰  1) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
110109ex 412 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜π‘‰) ∈ β„• β†’ ((β™―β€˜π‘‰) β‰  1 β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1119, 110biimtrrid 242 . . . . . . . . . . . . . . . . . . . . 21 ((β™―β€˜π‘‰) ∈ β„• β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
112111com25 99 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘‰) ∈ β„• β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1138, 112sylbir 234 . . . . . . . . . . . . . . . . . . 19 (((β™―β€˜π‘‰) ∈ β„•0 ∧ (β™―β€˜π‘‰) β‰  0) β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
114113ex 412 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜π‘‰) ∈ β„•0 β†’ ((β™―β€˜π‘‰) β‰  0 β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))))
115114impcomd 411 . . . . . . . . . . . . . . . . 17 ((β™―β€˜π‘‰) ∈ β„•0 β†’ ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) β‰  0) β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 β‰  βˆ… β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
116115com14 96 . . . . . . . . . . . . . . . 16 (𝑉 β‰  βˆ… β†’ ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) β‰  0) β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ β„•0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1177, 116mpcom 38 . . . . . . . . . . . . . . 15 ((𝑉 ∈ Fin ∧ (β™―β€˜π‘‰) β‰  0) β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ β„•0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))))
118117ex 412 . . . . . . . . . . . . . 14 (𝑉 ∈ Fin β†’ ((β™―β€˜π‘‰) β‰  0 β†’ (𝐺 ∈ FriendGraph β†’ ((β™―β€˜π‘‰) ∈ β„•0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
119118com14 96 . . . . . . . . . . . . 13 ((β™―β€˜π‘‰) ∈ β„•0 β†’ ((β™―β€˜π‘‰) β‰  0 β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1204, 119biimtrrid 242 . . . . . . . . . . . 12 ((β™―β€˜π‘‰) ∈ β„•0 β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ (𝐺 ∈ FriendGraph β†’ (𝑉 ∈ Fin β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
121120com24 95 . . . . . . . . . . 11 ((β™―β€˜π‘‰) ∈ β„•0 β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))))
1221213imp 1109 . . . . . . . . . 10 (((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
123122com25 99 . . . . . . . . 9 (((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) β†’ (𝐺 RegUSGraph 𝐾 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))))
124123imp 406 . . . . . . . 8 ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ (Β¬ (β™―β€˜π‘‰) = 0 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
125124com14 96 . . . . . . 7 (Β¬ (β™―β€˜π‘‰) = 0 β†’ (Β¬ (β™―β€˜π‘‰) = 1 β†’ (Β¬ (β™―β€˜π‘‰) = 3 β†’ ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
1261253imp 1109 . . . . . 6 ((Β¬ (β™―β€˜π‘‰) = 0 ∧ Β¬ (β™―β€˜π‘‰) = 1 ∧ Β¬ (β™―β€˜π‘‰) = 3) β†’ ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
1273, 126sylbi 216 . . . . 5 (Β¬ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3) β†’ ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))
1282, 127pm2.61i 182 . . . 4 ((((β™―β€˜π‘‰) ∈ β„•0 ∧ 𝑉 ∈ Fin ∧ 𝐺 ∈ FriendGraph ) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))
1291283exp1 1350 . . 3 ((β™―β€˜π‘‰) ∈ β„•0 β†’ (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3)))))
1301, 129mpcom 38 . 2 (𝑉 ∈ Fin β†’ (𝐺 ∈ FriendGraph β†’ (𝐺 RegUSGraph 𝐾 β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))))
1311303imp21 1112 1 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∨ wo 846   ∨ w3o 1084   ∧ w3a 1085   = wceq 1534  βŠ₯wfal 1546  βˆƒwex 1774   ∈ wcel 2099   β‰  wne 2935   βˆ‰ wnel 3041  βˆ€wral 3056  Vcvv 3469  βˆ…c0 4318  {cpr 4626   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  Fincfn 8955  0cc0 11130  1c1 11131   + caddc 11133   Β· cmul 11135   < clt 11270   βˆ’ cmin 11466  β„•cn 12234  2c2 12289  3c3 12290  β„•0cn0 12494  β„•0*cxnn0 12566  β„€β‰₯cuz 12844  β™―chash 14313  Vtxcvtx 28796  USGraphcusgr 28949  VtxDegcvtxdg 29266   RegUSGraph crusgr 29357   FriendGraph cfrgr 30055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-ac2 10478  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-ec 8720  df-qs 8724  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-dju 9916  df-card 9954  df-ac 10131  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-xadd 13117  df-ico 13354  df-fz 13509  df-fzo 13652  df-fl 13781  df-mod 13859  df-seq 13991  df-exp 14051  df-hash 14314  df-word 14489  df-lsw 14537  df-concat 14545  df-s1 14570  df-substr 14615  df-pfx 14645  df-reps 14743  df-csh 14763  df-s2 14823  df-s3 14824  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-dvds 16223  df-gcd 16461  df-prm 16634  df-phi 16726  df-vtx 28798  df-iedg 28799  df-edg 28848  df-uhgr 28858  df-ushgr 28859  df-upgr 28882  df-umgr 28883  df-uspgr 28950  df-usgr 28951  df-fusgr 29117  df-nbgr 29133  df-vtxdg 29267  df-rgr 29358  df-rusgr 29359  df-wlks 29400  df-wlkson 29401  df-trls 29493  df-trlson 29494  df-pths 29517  df-spths 29518  df-pthson 29519  df-spthson 29520  df-wwlks 29628  df-wwlksn 29629  df-wwlksnon 29630  df-wspthsn 29631  df-wspthsnon 29632  df-clwwlk 29779  df-clwwlkn 29822  df-clwwlknon 29885  df-conngr 29984  df-frgr 30056
This theorem is referenced by:  frgrregord13  30193
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