Theorem cusgrrusgr 26477

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.)

Hypothesis

Ref	Expression
cusgrrusgr.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
cusgrrusgr	|- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> G RegUSGraph ( ( # ` V ) - 1 ) )

Proof of Theorem cusgrrusgr

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusgrusgr 26315	. . 3 |- ( G e. ComplUSGraph -> G e. USGraph )
2	1	3ad2ant1 1082	. 2 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> G e. USGraph )
3		hashnncl 13157	. . . . 5 |- ( V e. Fin -> ( ( # ` V ) e. NN <-> V =/= (/) ) )
4		nnm1nn0 11334	. . . . . 6 |- ( ( # ` V ) e. NN -> ( ( # ` V ) - 1 ) e. NN0 )
5	4	nn0xnn0d 11372	. . . . 5 |- ( ( # ` V ) e. NN -> ( ( # ` V ) - 1 ) e. NN0* )
6	3, 5	syl6bir 244	. . . 4 |- ( V e. Fin -> ( V =/= (/) -> ( ( # ` V ) - 1 ) e. NN0* ) )
7	6	imp 445	. . 3 |- ( ( V e. Fin /\ V =/= (/) ) -> ( ( # ` V ) - 1 ) e. NN0* )
8	7	3adant1 1079	. 2 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> ( ( # ` V ) - 1 ) e. NN0* )
9		cusgrcplgr 26316	. . . . . 6 |- ( G e. ComplUSGraph -> G e. ComplGraph )
10	9	3ad2ant1 1082	. . . . 5 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> G e. ComplGraph )
11		cusgrrusgr.v	. . . . . 6 |- V = (Vtx ` G )
12	11	nbcplgr 26330	. . . . 5 |- ( ( G e. ComplGraph /\ v e. V ) -> ( G NeighbVtx v ) = ( V \ { v } ) )
13	10, 12	sylan 488	. . . 4 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) /\ v e. V ) -> ( G NeighbVtx v ) = ( V \ { v } ) )
14	13	ralrimiva 2966	. . 3 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> A. v e. V ( G NeighbVtx v ) = ( V \ { v } ) )
15	2	anim1i 592	. . . . . . . 8 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) /\ v e. V ) -> ( G e. USGraph /\ v e. V ) )
16	15	adantr 481	. . . . . . 7 |- ( ( ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) /\ v e. V ) /\ ( G NeighbVtx v ) = ( V \ { v } ) ) -> ( G e. USGraph /\ v e. V ) )
17	11	hashnbusgrvd 26424	. . . . . . 7 |- ( ( G e. USGraph /\ v e. V ) -> ( # ` ( G NeighbVtx v ) ) = ( (VtxDeg ` G ) ` v ) )
18	16, 17	syl 17	. . . . . 6 |- ( ( ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) /\ v e. 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 e. Fin /\ v e. V ) -> ( # ` ( V \ { v } ) ) = ( ( # ` V ) - 1 ) )
21	20	3ad2antl2 1224	. . . . . . 7 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) /\ v e. V ) -> ( # ` ( V \ { v } ) ) = ( ( # ` V ) - 1 ) )
22	19, 21	sylan9eqr 2678	. . . . . 6 |- ( ( ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) /\ v e. V ) /\ ( G NeighbVtx v ) = ( V \ { v } ) ) -> ( # ` ( G NeighbVtx v ) ) = ( ( # ` V ) - 1 ) )
23	18, 22	eqtr3d 2658	. . . . 5 |- ( ( ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) /\ v e. V ) /\ ( G NeighbVtx v ) = ( V \ { v } ) ) -> ( (VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) )
24	23	ex 450	. . . 4 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) /\ v e. V ) -> ( ( G NeighbVtx v ) = ( V \ { v } ) -> ( (VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) )
25	24	ralimdva 2962	. . 3 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> ( A. v e. V ( G NeighbVtx v ) = ( V \ { v } ) -> A. v e. V ( (VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) )
26	14, 25	mpd 15	. 2 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> A. v e. V ( (VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) )
27		simp1 1061	. . 3 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> G e. ComplUSGraph )
28		ovex 6678	. . 3 |- ( ( # ` V ) - 1 ) e. _V
29		eqid 2622	. . . 4 |- (VtxDeg ` G ) = (VtxDeg ` G )
30	11, 29	isrusgr0 26462	. . 3 |- ( ( G e. ComplUSGraph /\ ( ( # ` V ) - 1 ) e. _V ) -> ( G RegUSGraph ( ( # ` V ) - 1 ) <-> ( G e. USGraph /\ ( ( # ` V ) - 1 ) e. NN0* /\ A. v e. V ( (VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) ) )
31	27, 28, 30	sylancl 694	. 2 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> ( G RegUSGraph ( ( # ` V ) - 1 ) <-> ( G e. USGraph /\ ( ( # ` V ) - 1 ) e. NN0* /\ A. v e. V ( (VtxDeg ` G ) ` v ) = ( ( # ` V ) - 1 ) ) ) )
32	2, 8, 26, 31	mpbir3and 1245	1 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ V =/= (/) ) -> G RegUSGraph ( ( # ` V ) - 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.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   NNcn 11020  NN0*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