Theorem cusgrsize 26350

Description: The size of a finite complete simple graph with n vertices ( n e. NN0) is ( n _C 2 ) (" n choose 2") resp. ( ( ( n - 1 ) * n ) / 2 ), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)

Hypotheses

Ref	Expression
cusgrsizeindb0.v	|- V = (Vtx ` G )
cusgrsizeindb0.e	|- E = (Edg ` G )

Assertion

Ref	Expression
cusgrsize	|- ( ( G e. ComplUSGraph /\ V e. Fin ) -> ( # ` E ) = ( ( # ` V ) _C 2 ) )

Proof of Theorem cusgrsize

Dummy variables e f n v c w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusgrsizeindb0.e	. . . . 5 |- E = (Edg ` G )
2		edgval 25941	. . . . 5 |- (Edg ` G ) = ran (iEdg ` G )
3	1, 2	eqtri 2644	. . . 4 |- E = ran (iEdg ` G )
4	3	a1i 11	. . 3 |- ( ( G e. ComplUSGraph /\ V e. Fin ) -> E = ran (iEdg ` G ) )
5	4	fveq2d 6195	. 2 |- ( ( G e. ComplUSGraph /\ V e. Fin ) -> ( # ` E ) = ( # ` ran (iEdg ` G ) ) )
6		cusgrsizeindb0.v	. . . . 5 |- V = (Vtx ` G )
7	6	opeq1i 4405	. . . 4 |- <. V , (iEdg ` G ) >. = <. (Vtx ` G ) , (iEdg ` G ) >.
8		cusgrop 26334	. . . 4 |- ( G e. ComplUSGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. ComplUSGraph )
9	7, 8	syl5eqel 2705	. . 3 |- ( G e. ComplUSGraph -> <. V , (iEdg ` G ) >. e. ComplUSGraph )
10		fvex 6201	. . . 4 |- (iEdg ` G ) e. _V
11		fvex 6201	. . . . 5 |- (Edg ` <. v , e >. ) e. _V
12		rabexg 4812	. . . . . 6 |- ( (Edg ` <. v , e >. ) e. _V -> { c e. (Edg ` <. v , e >. ) | n e/ c } e. _V )
13	12	resiexd 6480	. . . . 5 |- ( (Edg ` <. v , e >. ) e. _V -> ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) e. _V )
14	11, 13	ax-mp 5	. . . 4 |- ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) e. _V
15		rneq 5351	. . . . . 6 |- ( e = (iEdg ` G ) -> ran e = ran (iEdg ` G ) )
16	15	fveq2d 6195	. . . . 5 |- ( e = (iEdg ` G ) -> ( # ` ran e ) = ( # ` ran (iEdg ` G ) ) )
17		fveq2 6191	. . . . . 6 |- ( v = V -> ( # ` v ) = ( # ` V ) )
18	17	oveq1d 6665	. . . . 5 |- ( v = V -> ( ( # ` v ) _C 2 ) = ( ( # ` V ) _C 2 ) )
19	16, 18	eqeqan12rd 2640	. . . 4 |- ( ( v = V /\ e = (iEdg ` G ) ) -> ( ( # ` ran e ) = ( ( # ` v ) _C 2 ) <-> ( # ` ran (iEdg ` G ) ) = ( ( # ` V ) _C 2 ) ) )
20		rneq 5351	. . . . . 6 |- ( e = f -> ran e = ran f )
21	20	fveq2d 6195	. . . . 5 |- ( e = f -> ( # ` ran e ) = ( # ` ran f ) )
22		fveq2 6191	. . . . . 6 |- ( v = w -> ( # ` v ) = ( # ` w ) )
23	22	oveq1d 6665	. . . . 5 |- ( v = w -> ( ( # ` v ) _C 2 ) = ( ( # ` w ) _C 2 ) )
24	21, 23	eqeqan12rd 2640	. . . 4 |- ( ( v = w /\ e = f ) -> ( ( # ` ran e ) = ( ( # ` v ) _C 2 ) <-> ( # ` ran f ) = ( ( # ` w ) _C 2 ) ) )
25		vex 3203	. . . . . . 7 |- v e. _V
26		vex 3203	. . . . . . 7 |- e e. _V
27	25, 26	opvtxfvi 25889	. . . . . 6 |- (Vtx ` <. v , e >. ) = v
28	27	eqcomi 2631	. . . . 5 |- v = (Vtx ` <. v , e >. )
29		eqid 2622	. . . . 5 |- (Edg ` <. v , e >. ) = (Edg ` <. v , e >. )
30		eqid 2622	. . . . 5 |- { c e. (Edg ` <. v , e >. ) | n e/ c } = { c e. (Edg ` <. v , e >. ) | n e/ c }
31		eqid 2622	. . . . 5 |- <. ( v \ { n } ) , ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) >. = <. ( v \ { n } ) , ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) >.
32	28, 29, 30, 31	cusgrres 26344	. . . 4 |- ( ( <. v , e >. e. ComplUSGraph /\ n e. v ) -> <. ( v \ { n } ) , ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) >. e. ComplUSGraph )
33		rneq 5351	. . . . . . 7 |- ( f = ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) -> ran f = ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) )
34	33	fveq2d 6195	. . . . . 6 |- ( f = ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) -> ( # ` ran f ) = ( # ` ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) )
35	34	adantl 482	. . . . 5 |- ( ( w = ( v \ { n } ) /\ f = ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) -> ( # ` ran f ) = ( # ` ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) )
36		fveq2 6191	. . . . . . 7 |- ( w = ( v \ { n } ) -> ( # ` w ) = ( # ` ( v \ { n } ) ) )
37	36	adantr 481	. . . . . 6 |- ( ( w = ( v \ { n } ) /\ f = ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) -> ( # ` w ) = ( # ` ( v \ { n } ) ) )
38	37	oveq1d 6665	. . . . 5 |- ( ( w = ( v \ { n } ) /\ f = ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) -> ( ( # ` w ) _C 2 ) = ( ( # ` ( v \ { n } ) ) _C 2 ) )
39	35, 38	eqeq12d 2637	. . . 4 |- ( ( w = ( v \ { n } ) /\ f = ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) -> ( ( # ` ran f ) = ( ( # ` w ) _C 2 ) <-> ( # ` ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) = ( ( # ` ( v \ { n } ) ) _C 2 ) ) )
40		edgopval 25944	. . . . . . . . 9 |- ( ( v e. _V /\ e e. _V ) -> (Edg ` <. v , e >. ) = ran e )
41	25, 26, 40	mp2an 708	. . . . . . . 8 |- (Edg ` <. v , e >. ) = ran e
42	41	a1i 11	. . . . . . 7 |- ( ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = 0 ) -> (Edg ` <. v , e >. ) = ran e )
43	42	eqcomd 2628	. . . . . 6 |- ( ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = 0 ) -> ran e = (Edg ` <. v , e >. ) )
44	43	fveq2d 6195	. . . . 5 |- ( ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = 0 ) -> ( # ` ran e ) = ( # ` (Edg ` <. v , e >. ) ) )
45		cusgrusgr 26315	. . . . . . 7 |- ( <. v , e >. e. ComplUSGraph -> <. v , e >. e. USGraph )
46		usgruhgr 26078	. . . . . . 7 |- ( <. v , e >. e. USGraph -> <. v , e >. e. UHGraph )
47	45, 46	syl 17	. . . . . 6 |- ( <. v , e >. e. ComplUSGraph -> <. v , e >. e. UHGraph )
48	28, 29	cusgrsizeindb0 26345	. . . . . 6 |- ( ( <. v , e >. e. UHGraph /\ ( # ` v ) = 0 ) -> ( # ` (Edg ` <. v , e >. ) ) = ( ( # ` v ) _C 2 ) )
49	47, 48	sylan 488	. . . . 5 |- ( ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = 0 ) -> ( # ` (Edg ` <. v , e >. ) ) = ( ( # ` v ) _C 2 ) )
50	44, 49	eqtrd 2656	. . . 4 |- ( ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = 0 ) -> ( # ` ran e ) = ( ( # ` v ) _C 2 ) )
51		rnresi 5479	. . . . . . . . . 10 |- ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) = { c e. (Edg ` <. v , e >. ) | n e/ c }
52	51	fveq2i 6194	. . . . . . . . 9 |- ( # ` ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) = ( # ` { c e. (Edg ` <. v , e >. ) | n e/ c } )
53	41	a1i 11	. . . . . . . . . . 11 |- ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) -> (Edg ` <. v , e >. ) = ran e )
54	53	rabeqdv 3194	. . . . . . . . . 10 |- ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) -> { c e. (Edg ` <. v , e >. ) | n e/ c } = { c e. ran e | n e/ c } )
55	54	fveq2d 6195	. . . . . . . . 9 |- ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) -> ( # ` { c e. (Edg ` <. v , e >. ) | n e/ c } ) = ( # ` { c e. ran e | n e/ c } ) )
56	52, 55	syl5eq 2668	. . . . . . . 8 |- ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) -> ( # ` ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) = ( # ` { c e. ran e | n e/ c } ) )
57	56	eqeq1d 2624	. . . . . . 7 |- ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) -> ( ( # ` ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) = ( ( # ` ( v \ { n } ) ) _C 2 ) <-> ( # ` { c e. ran e | n e/ c } ) = ( ( # ` ( v \ { n } ) ) _C 2 ) ) )
58	57	biimpd 219	. . . . . 6 |- ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) -> ( ( # ` ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) = ( ( # ` ( v \ { n } ) ) _C 2 ) -> ( # ` { c e. ran e | n e/ c } ) = ( ( # ` ( v \ { n } ) ) _C 2 ) ) )
59	58	imdistani 726	. . . . 5 |- ( ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) /\ ( # ` ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) = ( ( # ` ( v \ { n } ) ) _C 2 ) ) -> ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) /\ ( # ` { c e. ran e | n e/ c } ) = ( ( # ` ( v \ { n } ) ) _C 2 ) ) )
60	41	eqcomi 2631	. . . . . . 7 |- ran e = (Edg ` <. v , e >. )
61		eqid 2622	. . . . . . 7 |- { c e. ran e | n e/ c } = { c e. ran e | n e/ c }
62	28, 60, 61	cusgrsize2inds 26349	. . . . . 6 |- ( ( y + 1 ) e. NN0 -> ( ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) -> ( ( # ` { c e. ran e | n e/ c } ) = ( ( # ` ( v \ { n } ) ) _C 2 ) -> ( # ` ran e ) = ( ( # ` v ) _C 2 ) ) ) )
63	62	imp31 448	. . . . 5 |- ( ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) /\ ( # ` { c e. ran e | n e/ c } ) = ( ( # ` ( v \ { n } ) ) _C 2 ) ) -> ( # ` ran e ) = ( ( # ` v ) _C 2 ) )
64	59, 63	syl 17	. . . 4 |- ( ( ( ( y + 1 ) e. NN0 /\ ( <. v , e >. e. ComplUSGraph /\ ( # ` v ) = ( y + 1 ) /\ n e. v ) ) /\ ( # ` ran ( _I |` { c e. (Edg ` <. v , e >. ) | n e/ c } ) ) = ( ( # ` ( v \ { n } ) ) _C 2 ) ) -> ( # ` ran e ) = ( ( # ` v ) _C 2 ) )
65	10, 14, 19, 24, 32, 39, 50, 64	opfi1ind 13284	. . 3 |- ( ( <. V , (iEdg ` G ) >. e. ComplUSGraph /\ V e. Fin ) -> ( # ` ran (iEdg ` G ) ) = ( ( # ` V ) _C 2 ) )
66	9, 65	sylan 488	. 2 |- ( ( G e. ComplUSGraph /\ V e. Fin ) -> ( # ` ran (iEdg ` G ) ) = ( ( # ` V ) _C 2 ) )
67	5, 66	eqtrd 2656	1 |- ( ( G e. ComplUSGraph /\ V e. Fin ) -> ( # ` E ) = ( ( # ` V ) _C 2 ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   e/ wnel 2897   {crab 2916   _Vcvv 3200   \ cdif 3571   {csn 4177   <.cop 4183   _I cid 5023   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   NN0cn0 11292   _C cbc 13089   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   USGraph cusgr 26044  ComplUSGraphccusgr 26227

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-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-fac 13061  df-bc 13090  df-hash 13118  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-nbgr 26228  df-uvtxa 26230  df-cplgr 26231  df-cusgr 26232

This theorem is referenced by:  fusgrmaxsize  26360