Theorem cusgrsizeinds 26348

Description: Part 1 of induction step in cusgrsize 26350. The size of a complete simple graph with n vertices is ( n - 1 ) plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)

Hypotheses

Ref	Expression
cusgrsizeindb0.v	|- V = (Vtx ` G )
cusgrsizeindb0.e	|- E = (Edg ` G )
cusgrsizeinds.f	|- F = { e e. E | N e/ e }

Assertion

Ref	Expression
cusgrsizeinds	|- ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) -> ( # ` E ) = ( ( ( # ` V ) - 1 ) + ( # ` F ) ) )

Distinct variable groups:   e, E   e, G   e, N   e, V

Allowed substitution hint:   F( e)

Proof of Theorem cusgrsizeinds

Step	Hyp	Ref	Expression
1		cusgrusgr 26315	. . . 4 |- ( G e. ComplUSGraph -> G e. USGraph )
2		cusgrsizeindb0.v	. . . . . . . 8 |- V = (Vtx ` G )
3	2	isfusgr 26210	. . . . . . 7 |- ( G e. FinUSGraph <-> ( G e. USGraph /\ V e. Fin ) )
4		fusgrfis 26222	. . . . . . 7 |- ( G e. FinUSGraph -> (Edg ` G ) e. Fin )
5	3, 4	sylbir 225	. . . . . 6 |- ( ( G e. USGraph /\ V e. Fin ) -> (Edg ` G ) e. Fin )
6	5	a1d 25	. . . . 5 |- ( ( G e. USGraph /\ V e. Fin ) -> ( N e. V -> (Edg ` G ) e. Fin ) )
7	6	ex 450	. . . 4 |- ( G e. USGraph -> ( V e. Fin -> ( N e. V -> (Edg ` G ) e. Fin ) ) )
8	1, 7	syl 17	. . 3 |- ( G e. ComplUSGraph -> ( V e. Fin -> ( N e. V -> (Edg ` G ) e. Fin ) ) )
9	8	3imp 1256	. 2 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) -> (Edg ` G ) e. Fin )
10		eqid 2622	. . . . . . 7 |- { e e. E | N e. e } = { e e. E | N e. e }
11		cusgrsizeinds.f	. . . . . . 7 |- F = { e e. E | N e/ e }
12	10, 11	elnelun 3964	. . . . . 6 |- ( { e e. E | N e. e } u. F ) = E
13	12	eqcomi 2631	. . . . 5 |- E = ( { e e. E | N e. e } u. F )
14	13	fveq2i 6194	. . . 4 |- ( # ` E ) = ( # ` ( { e e. E | N e. e } u. F ) )
15	14	a1i 11	. . 3 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) /\ (Edg ` G ) e. Fin ) -> ( # ` E ) = ( # ` ( { e e. E | N e. e } u. F ) ) )
16		cusgrsizeindb0.e	. . . . . . . 8 |- E = (Edg ` G )
17	16	eqcomi 2631	. . . . . . 7 |- (Edg ` G ) = E
18	17	eleq1i 2692	. . . . . 6 |- ( (Edg ` G ) e. Fin <-> E e. Fin )
19		rabfi 8185	. . . . . 6 |- ( E e. Fin -> { e e. E | N e. e } e. Fin )
20	18, 19	sylbi 207	. . . . 5 |- ( (Edg ` G ) e. Fin -> { e e. E | N e. e } e. Fin )
21	20	adantl 482	. . . 4 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) /\ (Edg ` G ) e. Fin ) -> { e e. E | N e. e } e. Fin )
22	1	anim1i 592	. . . . . . . 8 |- ( ( G e. ComplUSGraph /\ V e. Fin ) -> ( G e. USGraph /\ V e. Fin ) )
23	22, 3	sylibr 224	. . . . . . 7 |- ( ( G e. ComplUSGraph /\ V e. Fin ) -> G e. FinUSGraph )
24	2, 16, 11	usgrfilem 26219	. . . . . . 7 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( E e. Fin <-> F e. Fin ) )
25	23, 24	stoic3 1701	. . . . . 6 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) -> ( E e. Fin <-> F e. Fin ) )
26	18, 25	syl5bb 272	. . . . 5 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) -> ( (Edg ` G ) e. Fin <-> F e. Fin ) )
27	26	biimpa 501	. . . 4 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) /\ (Edg ` G ) e. Fin ) -> F e. Fin )
28	10, 11	elneldisj 3963	. . . . 5 |- ( { e e. E | N e. e } i^i F ) = (/)
29	28	a1i 11	. . . 4 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) /\ (Edg ` G ) e. Fin ) -> ( { e e. E | N e. e } i^i F ) = (/) )
30		hashun 13171	. . . 4 |- ( ( { e e. E | N e. e } e. Fin /\ F e. Fin /\ ( { e e. E | N e. e } i^i F ) = (/) ) -> ( # ` ( { e e. E | N e. e } u. F ) ) = ( ( # ` { e e. E | N e. e } ) + ( # ` F ) ) )
31	21, 27, 29, 30	syl3anc 1326	. . 3 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) /\ (Edg ` G ) e. Fin ) -> ( # ` ( { e e. E | N e. e } u. F ) ) = ( ( # ` { e e. E | N e. e } ) + ( # ` F ) ) )
32	2, 16	cusgrsizeindslem 26347	. . . . 5 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) -> ( # ` { e e. E | N e. e } ) = ( ( # ` V ) - 1 ) )
33	32	adantr 481	. . . 4 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) /\ (Edg ` G ) e. Fin ) -> ( # ` { e e. E | N e. e } ) = ( ( # ` V ) - 1 ) )
34	33	oveq1d 6665	. . 3 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) /\ (Edg ` G ) e. Fin ) -> ( ( # ` { e e. E | N e. e } ) + ( # ` F ) ) = ( ( ( # ` V ) - 1 ) + ( # ` F ) ) )
35	15, 31, 34	3eqtrd 2660	. 2 |- ( ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) /\ (Edg ` G ) e. Fin ) -> ( # ` E ) = ( ( ( # ` V ) - 1 ) + ( # ` F ) ) )
36	9, 35	mpdan 702	1 |- ( ( G e. ComplUSGraph /\ V e. Fin /\ N e. V ) -> ( # ` E ) = ( ( ( # ` V ) - 1 ) + ( # ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   e/ wnel 2897   {crab 2916   u. cun 3572   i^i cin 3573   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   + caddc 9939   - cmin 10266   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FinUSGraph cfusgr 26208  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-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  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:  cusgrsize2inds  26349