Theorem frgrreggt1 27251

Description: If a finite nonempty friendship graph is K-regular with K > 1, then K must be 2. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.)

Hypothesis

Ref	Expression
frgrreggt1.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
frgrreggt1	|- ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( ( G RegUSGraph K /\ 1 < K ) -> K = 2 ) )

Proof of Theorem frgrreggt1

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . 6 |- ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> G e. FriendGraph )
2	1	anim1i 592	. . . . 5 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( G e. FriendGraph /\ G RegUSGraph K ) )
3	2	ancomd 467	. . . 4 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( G RegUSGraph K /\ G e. FriendGraph ) )
4		simp3 1063	. . . . . 6 |- ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> V =/= (/) )
5		simp2 1062	. . . . . 6 |- ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> V e. Fin )
6	4, 5	jca 554	. . . . 5 |- ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( V =/= (/) /\ V e. Fin ) )
7	6	adantr 481	. . . 4 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( V =/= (/) /\ V e. Fin ) )
8		frgrreggt1.v	. . . . 5 |- V = (Vtx ` G )
9	8	numclwwlk7lem 27247	. . . 4 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) ) -> K e. NN0 )
10	3, 7, 9	syl2anc 693	. . 3 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> K e. NN0 )
11		2z 11409	. . . . . . . . . 10 |- 2 e. ZZ
12	11	a1i 11	. . . . . . . . 9 |- ( ( K e. NN0 /\ 2 < K ) -> 2 e. ZZ )
13		nn0z 11400	. . . . . . . . . . 11 |- ( K e. NN0 -> K e. ZZ )
14	13	adantr 481	. . . . . . . . . 10 |- ( ( K e. NN0 /\ 2 < K ) -> K e. ZZ )
15		peano2zm 11420	. . . . . . . . . 10 |- ( K e. ZZ -> ( K - 1 ) e. ZZ )
16	14, 15	syl 17	. . . . . . . . 9 |- ( ( K e. NN0 /\ 2 < K ) -> ( K - 1 ) e. ZZ )
17		zltlem1 11430	. . . . . . . . . . 11 |- ( ( 2 e. ZZ /\ K e. ZZ ) -> ( 2 < K <-> 2 <_ ( K - 1 ) ) )
18	11, 13, 17	sylancr 695	. . . . . . . . . 10 |- ( K e. NN0 -> ( 2 < K <-> 2 <_ ( K - 1 ) ) )
19	18	biimpa 501	. . . . . . . . 9 |- ( ( K e. NN0 /\ 2 < K ) -> 2 <_ ( K - 1 ) )
20		eluz2 11693	. . . . . . . . 9 |- ( ( K - 1 ) e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ ( K - 1 ) e. ZZ /\ 2 <_ ( K - 1 ) ) )
21	12, 16, 19, 20	syl3anbrc 1246	. . . . . . . 8 |- ( ( K e. NN0 /\ 2 < K ) -> ( K - 1 ) e. ( ZZ>= ` 2 ) )
22		exprmfct 15416	. . . . . . . 8 |- ( ( K - 1 ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( K - 1 ) )
23	21, 22	syl 17	. . . . . . 7 |- ( ( K e. NN0 /\ 2 < K ) -> E. p e. Prime p || ( K - 1 ) )
24	5	anim1i 592	. . . . . . . . . . . . . . 15 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( V e. Fin /\ G RegUSGraph K ) )
25	24	ancomd 467	. . . . . . . . . . . . . 14 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( G RegUSGraph K /\ V e. Fin ) )
26	8	finrusgrfusgr 26461	. . . . . . . . . . . . . 14 |- ( ( G RegUSGraph K /\ V e. Fin ) -> G e. FinUSGraph )
27	25, 26	syl 17	. . . . . . . . . . . . 13 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> G e. FinUSGraph )
28	27	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ p || ( K - 1 ) ) /\ ( K e. NN0 /\ 2 < K ) /\ ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) ) -> G e. FinUSGraph )
29		simp1l 1085	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ p || ( K - 1 ) ) /\ ( K e. NN0 /\ 2 < K ) /\ ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) ) -> p e. Prime )
30		numclwwlk8 27250	. . . . . . . . . . . 12 |- ( ( G e. FinUSGraph /\ p e. Prime ) -> ( ( # ` ( p ClWWalksN G ) ) mod p ) = 0 )
31	28, 29, 30	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( p e. Prime /\ p || ( K - 1 ) ) /\ ( K e. NN0 /\ 2 < K ) /\ ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) ) -> ( ( # ` ( p ClWWalksN G ) ) mod p ) = 0 )
32	3	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ p || ( K - 1 ) ) /\ ( K e. NN0 /\ 2 < K ) /\ ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) ) -> ( G RegUSGraph K /\ G e. FriendGraph ) )
33		pm3.22 465	. . . . . . . . . . . . . . 15 |- ( ( V e. Fin /\ V =/= (/) ) -> ( V =/= (/) /\ V e. Fin ) )
34	33	3adant1 1079	. . . . . . . . . . . . . 14 |- ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( V =/= (/) /\ V e. Fin ) )
35	34	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( V =/= (/) /\ V e. Fin ) )
36	35	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ p || ( K - 1 ) ) /\ ( K e. NN0 /\ 2 < K ) /\ ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) ) -> ( V =/= (/) /\ V e. Fin ) )
37		simp1 1061	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ p || ( K - 1 ) ) /\ ( K e. NN0 /\ 2 < K ) /\ ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) ) -> ( p e. Prime /\ p || ( K - 1 ) ) )
38	8	numclwwlk7 27249	. . . . . . . . . . . 12 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) /\ ( p e. Prime /\ p || ( K - 1 ) ) ) -> ( ( # ` ( p ClWWalksN G ) ) mod p ) = 1 )
39	32, 36, 37, 38	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( p e. Prime /\ p || ( K - 1 ) ) /\ ( K e. NN0 /\ 2 < K ) /\ ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) ) -> ( ( # ` ( p ClWWalksN G ) ) mod p ) = 1 )
40		eqeq1 2626	. . . . . . . . . . . 12 |- ( ( ( # ` ( p ClWWalksN G ) ) mod p ) = 0 -> ( ( ( # ` ( p ClWWalksN G ) ) mod p ) = 1 <-> 0 = 1 ) )
41		ax-1ne0 10005	. . . . . . . . . . . . . 14 |- 1 =/= 0
42	41	nesymi 2851	. . . . . . . . . . . . 13 |- -. 0 = 1
43	42	pm2.21i 116	. . . . . . . . . . . 12 |- ( 0 = 1 -> K = 2 )
44	40, 43	syl6bi 243	. . . . . . . . . . 11 |- ( ( ( # ` ( p ClWWalksN G ) ) mod p ) = 0 -> ( ( ( # ` ( p ClWWalksN G ) ) mod p ) = 1 -> K = 2 ) )
45	31, 39, 44	sylc 65	. . . . . . . . . 10 |- ( ( ( p e. Prime /\ p || ( K - 1 ) ) /\ ( K e. NN0 /\ 2 < K ) /\ ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) ) -> K = 2 )
46	45	a1d 25	. . . . . . . . 9 |- ( ( ( p e. Prime /\ p || ( K - 1 ) ) /\ ( K e. NN0 /\ 2 < K ) /\ ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) ) -> ( 1 < K -> K = 2 ) )
47	46	3exp 1264	. . . . . . . 8 |- ( ( p e. Prime /\ p || ( K - 1 ) ) -> ( ( K e. NN0 /\ 2 < K ) -> ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( 1 < K -> K = 2 ) ) ) )
48	47	rexlimiva 3028	. . . . . . 7 |- ( E. p e. Prime p || ( K - 1 ) -> ( ( K e. NN0 /\ 2 < K ) -> ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( 1 < K -> K = 2 ) ) ) )
49	23, 48	mpcom 38	. . . . . 6 |- ( ( K e. NN0 /\ 2 < K ) -> ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( 1 < K -> K = 2 ) ) )
50	49	expcom 451	. . . . 5 |- ( 2 < K -> ( K e. NN0 -> ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( 1 < K -> K = 2 ) ) ) )
51	50	com23 86	. . . 4 |- ( 2 < K -> ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( K e. NN0 -> ( 1 < K -> K = 2 ) ) ) )
52		1red 10055	. . . . . . . . 9 |- ( K e. NN0 -> 1 e. RR )
53		nn0re 11301	. . . . . . . . 9 |- ( K e. NN0 -> K e. RR )
54	52, 53	ltnled 10184	. . . . . . . 8 |- ( K e. NN0 -> ( 1 < K <-> -. K <_ 1 ) )
55		1e2m1 11136	. . . . . . . . . . 11 |- 1 = ( 2 - 1 )
56	55	a1i 11	. . . . . . . . . 10 |- ( K e. NN0 -> 1 = ( 2 - 1 ) )
57	56	breq2d 4665	. . . . . . . . 9 |- ( K e. NN0 -> ( K <_ 1 <-> K <_ ( 2 - 1 ) ) )
58	57	notbid 308	. . . . . . . 8 |- ( K e. NN0 -> ( -. K <_ 1 <-> -. K <_ ( 2 - 1 ) ) )
59		zltlem1 11430	. . . . . . . . . . 11 |- ( ( K e. ZZ /\ 2 e. ZZ ) -> ( K < 2 <-> K <_ ( 2 - 1 ) ) )
60	13, 11, 59	sylancl 694	. . . . . . . . . 10 |- ( K e. NN0 -> ( K < 2 <-> K <_ ( 2 - 1 ) ) )
61	60	bicomd 213	. . . . . . . . 9 |- ( K e. NN0 -> ( K <_ ( 2 - 1 ) <-> K < 2 ) )
62	61	notbid 308	. . . . . . . 8 |- ( K e. NN0 -> ( -. K <_ ( 2 - 1 ) <-> -. K < 2 ) )
63	54, 58, 62	3bitrd 294	. . . . . . 7 |- ( K e. NN0 -> ( 1 < K <-> -. K < 2 ) )
64		2re 11090	. . . . . . . . 9 |- 2 e. RR
65		lttri3 10121	. . . . . . . . . 10 |- ( ( K e. RR /\ 2 e. RR ) -> ( K = 2 <-> ( -. K < 2 /\ -. 2 < K ) ) )
66	65	biimprd 238	. . . . . . . . 9 |- ( ( K e. RR /\ 2 e. RR ) -> ( ( -. K < 2 /\ -. 2 < K ) -> K = 2 ) )
67	53, 64, 66	sylancl 694	. . . . . . . 8 |- ( K e. NN0 -> ( ( -. K < 2 /\ -. 2 < K ) -> K = 2 ) )
68	67	expd 452	. . . . . . 7 |- ( K e. NN0 -> ( -. K < 2 -> ( -. 2 < K -> K = 2 ) ) )
69	63, 68	sylbid 230	. . . . . 6 |- ( K e. NN0 -> ( 1 < K -> ( -. 2 < K -> K = 2 ) ) )
70	69	com3r 87	. . . . 5 |- ( -. 2 < K -> ( K e. NN0 -> ( 1 < K -> K = 2 ) ) )
71	70	a1d 25	. . . 4 |- ( -. 2 < K -> ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( K e. NN0 -> ( 1 < K -> K = 2 ) ) ) )
72	51, 71	pm2.61i 176	. . 3 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( K e. NN0 -> ( 1 < K -> K = 2 ) ) )
73	10, 72	mpd 15	. 2 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ G RegUSGraph K ) -> ( 1 < K -> K = 2 ) )
74	73	expimpd 629	1 |- ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( ( G RegUSGraph K /\ 1 < K ) -> K = 2 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   - cmin 10266   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   mod cmo 12668   #chash 13117   || cdvds 14983   Primecprime 15385  Vtxcvtx 25874   FinUSGraph cfusgr 26208   RegUSGraph crusgr 26452   ClWWalksN cclwwlksn 26876   FriendGraph cfrgr 27120

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-inf2 8538  ax-ac2 9285  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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  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-disj 4621  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-se 5074  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-isom 5897  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-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-ac 8939  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-reps 13306  df-csh 13535  df-s2 13593  df-s3 13594  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-nbgr 26228  df-vtxdg 26362  df-rgr 26453  df-rusgr 26454  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsn 26725  df-wspthsnon 26726  df-clwwlks 26877  df-clwwlksn 26878  df-frgr 27121

This theorem is referenced by:  frgrreg  27252