Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cusgrrusgr | Structured version Visualization version GIF version |
Description: A complete simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
cusgrrusgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
cusgrrusgr | ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 RegUSGraph ((#‘𝑉) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cusgrusgr 26315 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph ) | |
2 | 1 | 3ad2ant1 1082 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 ∈ USGraph ) |
3 | hashnncl 13157 | . . . . 5 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) ∈ ℕ ↔ 𝑉 ≠ ∅)) | |
4 | nnm1nn0 11334 | . . . . . 6 ⊢ ((#‘𝑉) ∈ ℕ → ((#‘𝑉) − 1) ∈ ℕ0) | |
5 | 4 | nn0xnn0d 11372 | . . . . 5 ⊢ ((#‘𝑉) ∈ ℕ → ((#‘𝑉) − 1) ∈ ℕ0*) |
6 | 3, 5 | syl6bir 244 | . . . 4 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → ((#‘𝑉) − 1) ∈ ℕ0*)) |
7 | 6 | imp 445 | . . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) − 1) ∈ ℕ0*) |
8 | 7 | 3adant1 1079 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) − 1) ∈ ℕ0*) |
9 | cusgrcplgr 26316 | . . . . . 6 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph) | |
10 | 9 | 3ad2ant1 1082 | . . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 ∈ ComplGraph) |
11 | cusgrrusgr.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | 11 | nbcplgr 26330 | . . . . 5 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑣 ∈ 𝑉) → (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) |
13 | 10, 12 | sylan 488 | . . . 4 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) → (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) |
14 | 13 | ralrimiva 2966 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∀𝑣 ∈ 𝑉 (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) |
15 | 2 | anim1i 592 | . . . . . . . 8 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉)) |
16 | 15 | adantr 481 | . . . . . . 7 ⊢ ((((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) ∧ (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) → (𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉)) |
17 | 11 | hashnbusgrvd 26424 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣)) |
18 | 16, 17 | syl 17 | . . . . . 6 ⊢ ((((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) ∧ (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) → (#‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣)) |
19 | fveq2 6191 | . . . . . . 7 ⊢ ((𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣}) → (#‘(𝐺 NeighbVtx 𝑣)) = (#‘(𝑉 ∖ {𝑣}))) | |
20 | hashdifsn 13202 | . . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝑣 ∈ 𝑉) → (#‘(𝑉 ∖ {𝑣})) = ((#‘𝑉) − 1)) | |
21 | 20 | 3ad2antl2 1224 | . . . . . . 7 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) → (#‘(𝑉 ∖ {𝑣})) = ((#‘𝑉) − 1)) |
22 | 19, 21 | sylan9eqr 2678 | . . . . . 6 ⊢ ((((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) ∧ (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) → (#‘(𝐺 NeighbVtx 𝑣)) = ((#‘𝑉) − 1)) |
23 | 18, 22 | eqtr3d 2658 | . . . . 5 ⊢ ((((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) ∧ (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) → ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1)) |
24 | 23 | ex 450 | . . . 4 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) → ((𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣}) → ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1))) |
25 | 24 | ralimdva 2962 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣}) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1))) |
26 | 14, 25 | mpd 15 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1)) |
27 | simp1 1061 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 ∈ ComplUSGraph) | |
28 | ovex 6678 | . . 3 ⊢ ((#‘𝑉) − 1) ∈ V | |
29 | eqid 2622 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
30 | 11, 29 | isrusgr0 26462 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ ((#‘𝑉) − 1) ∈ V) → (𝐺 RegUSGraph ((#‘𝑉) − 1) ↔ (𝐺 ∈ USGraph ∧ ((#‘𝑉) − 1) ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1)))) |
31 | 27, 28, 30 | sylancl 694 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 RegUSGraph ((#‘𝑉) − 1) ↔ (𝐺 ∈ USGraph ∧ ((#‘𝑉) − 1) ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1)))) |
32 | 2, 8, 26, 31 | mpbir3and 1245 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 RegUSGraph ((#‘𝑉) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 ∅c0 3915 {csn 4177 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 1c1 9937 − cmin 10266 ℕcn 11020 ℕ0*cxnn0 11363 #chash 13117 Vtxcvtx 25874 USGraph cusgr 26044 NeighbVtx cnbgr 26224 ComplGraphccplgr 26226 ComplUSGraphccusgr 26227 VtxDegcvtxdg 26361 RegUSGraph crusgr 26452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-xadd 11947 df-fz 12327 df-hash 13118 df-edg 25940 df-uhgr 25953 df-ushgr 25954 df-upgr 25977 df-umgr 25978 df-uspgr 26045 df-usgr 26046 df-nbgr 26228 df-uvtxa 26230 df-cplgr 26231 df-cusgr 26232 df-vtxdg 26362 df-rgr 26453 df-rusgr 26454 |
This theorem is referenced by: cusgrm1rusgr 26478 |
Copyright terms: Public domain | W3C validator |