Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 40641 | . . 3 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph ) | |
2 | 1 | 3ad2ant1 1075 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 ∈ USGraph ) |
3 | hashnncl 13018 | . . . . 5 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) ∈ ℕ ↔ 𝑉 ≠ ∅)) | |
4 | nnm1nn0 11211 | . . . . . 6 ⊢ ((#‘𝑉) ∈ ℕ → ((#‘𝑉) − 1) ∈ ℕ0) | |
5 | 4 | nn0xnn0d 11249 | . . . . 5 ⊢ ((#‘𝑉) ∈ ℕ → ((#‘𝑉) − 1) ∈ ℕ0*) |
6 | 3, 5 | syl6bir 243 | . . . 4 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → ((#‘𝑉) − 1) ∈ ℕ0*)) |
7 | 6 | imp 444 | . . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) − 1) ∈ ℕ0*) |
8 | 7 | 3adant1 1072 | . 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) − 1) ∈ ℕ0*) |
9 | cusgrcplgr 40642 | . . . . . 6 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph) | |
10 | 9 | 3ad2ant1 1075 | . . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 ∈ ComplGraph) |
11 | cusgrrusgr.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
12 | 11 | nbcplgr 40656 | . . . . 5 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑣 ∈ 𝑉) → (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) |
13 | 10, 12 | sylan 487 | . . . 4 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) → (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) |
14 | 13 | ralrimiva 2949 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∀𝑣 ∈ 𝑉 (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) |
15 | 2 | anim1i 590 | . . . . . . . 8 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉)) |
16 | 15 | adantr 480 | . . . . . . 7 ⊢ ((((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) ∧ (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) → (𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉)) |
17 | 11 | hashnbusgrvd 40744 | . . . . . . 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 6103 | . . . . . . 7 ⊢ ((𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣}) → (#‘(𝐺 NeighbVtx 𝑣)) = (#‘(𝑉 ∖ {𝑣}))) | |
20 | hashdifsn 13063 | . . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 𝑣 ∈ 𝑉) → (#‘(𝑉 ∖ {𝑣})) = ((#‘𝑉) − 1)) | |
21 | 20 | 3ad2antl2 1217 | . . . . . . 7 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) → (#‘(𝑉 ∖ {𝑣})) = ((#‘𝑉) − 1)) |
22 | 19, 21 | sylan9eqr 2666 | . . . . . 6 ⊢ ((((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) ∧ (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) → (#‘(𝐺 NeighbVtx 𝑣)) = ((#‘𝑉) − 1)) |
23 | 18, 22 | eqtr3d 2646 | . . . . 5 ⊢ ((((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) ∧ (𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣})) → ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1)) |
24 | 23 | ex 449 | . . . 4 ⊢ (((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝑣 ∈ 𝑉) → ((𝐺 NeighbVtx 𝑣) = (𝑉 ∖ {𝑣}) → ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1))) |
25 | 24 | ralimdva 2945 | . . 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 1054 | . . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 ∈ ComplUSGraph) | |
28 | ovex 6577 | . . 3 ⊢ ((#‘𝑉) − 1) ∈ V | |
29 | eqid 2610 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
30 | 11, 29 | isrusgr0 40766 | . . 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 693 | . 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 1238 | 1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 RegUSGraph ((#‘𝑉) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 1c1 9816 − cmin 10145 ℕcn 10897 ℕ0*cxnn0 11240 #chash 12979 Vtxcvtx 25673 USGraph cusgr 40379 NeighbVtx cnbgr 40550 ComplGraphccplgr 40552 ComplUSGraphccusgr 40553 VtxDegcvtxdg 40681 RegUSGraph crusgr 40756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-hash 12980 df-uhgr 25724 df-ushgr 25725 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-nbgr 40554 df-uvtxa 40556 df-cplgr 40557 df-cusgr 40558 df-vtxdg 40682 df-rgr 40757 df-rusgr 40758 |
This theorem is referenced by: cusgrm1rusgr 40782 |
Copyright terms: Public domain | W3C validator |