Theorem finsumvtxdg2size 26446

Description: The sum of the degrees of all vertices of a finite pseudograph of finite size is twice the size of the pseudograph. See equation (1) in section I.1 in [Bollobas] p. 4. Here, the "proof" is simply the statement "Since each edge has two endvertices, the sum of the degrees is exactly twice the number of edges". The formal proof of this theorem (for pseudographs) is much more complicated, taking also the used auxiliary theorems into account. The proof for a (finite) simple graph (see fusgr1th 26447) would be shorter, but nevertheless still laborious. Although this theorem would hold also for infinite pseudographs and pseudographs of infinite size, the proof of this most general version (see theorem "sumvtxdg2size" below) would require many more auxiliary theorems (e.g., the extension of the sum sum_ over an arbitrary set).

I dedicate this theorem and its proof to Norman Megill, who deceased too early on December 9, 2021. This proof is an example for the rigor which was the main motivation for Norman Megill to invent and develop Metamath, see section 1.1.6 "Rigor" on page 19 of the Metamath book: "... it is usually assumed in mathematical literature that the person reading the proof is a mathematician familiar with the specialty being described, and that the missing steps are obvious to such a reader or at least the reader is capable of filling them in." I filled in the missing steps of Bollobas' proof as Norm would have liked it... (Contributed by Alexander van der Vekens, 19-Dec-2021.)

Hypotheses

Ref	Expression
sumvtxdg2size.v	|- V = (Vtx ` G )
sumvtxdg2size.i	|- I = (iEdg ` G )
sumvtxdg2size.d	|- D = (VtxDeg ` G )

Assertion

Ref	Expression
finsumvtxdg2size	|- ( ( G e. UPGraph /\ V e. Fin /\ I e. Fin ) -> sum_ v e. V ( D ` v ) = ( 2 x. ( # ` I ) ) )

Distinct variable groups:   v, G   v, V

Allowed substitution hints:   D( v)   I( v)

Proof of Theorem finsumvtxdg2size

Dummy variables e k n f i w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgrop 25989	. . . 4 |- ( G e. UPGraph -> <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph )
2		fvex 6201	. . . . . 6 |- (iEdg ` G ) e. _V
3		fvex 6201	. . . . . . 7 |- (iEdg ` <. k , e >. ) e. _V
4	3	resex 5443	. . . . . 6 |- ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. _V
5		eleq1 2689	. . . . . . . 8 |- ( e = (iEdg ` G ) -> ( e e. Fin <-> (iEdg ` G ) e. Fin ) )
6	5	adantl 482	. . . . . . 7 |- ( ( k = (Vtx ` G ) /\ e = (iEdg ` G ) ) -> ( e e. Fin <-> (iEdg ` G ) e. Fin ) )
7		simpl 473	. . . . . . . . 9 |- ( ( k = (Vtx ` G ) /\ e = (iEdg ` G ) ) -> k = (Vtx ` G ) )
8		oveq12 6659	. . . . . . . . . . 11 |- ( ( k = (Vtx ` G ) /\ e = (iEdg ` G ) ) -> ( kVtxDeg e ) = ( (Vtx ` G )VtxDeg (iEdg ` G ) ) )
9	8	fveq1d 6193	. . . . . . . . . 10 |- ( ( k = (Vtx ` G ) /\ e = (iEdg ` G ) ) -> ( ( kVtxDeg e ) ` v ) = ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) )
10	9	adantr 481	. . . . . . . . 9 |- ( ( ( k = (Vtx ` G ) /\ e = (iEdg ` G ) ) /\ v e. k ) -> ( ( kVtxDeg e ) ` v ) = ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) )
11	7, 10	sumeq12dv 14437	. . . . . . . 8 |- ( ( k = (Vtx ` G ) /\ e = (iEdg ` G ) ) -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = sum_ v e. (Vtx ` G ) ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) )
12		fveq2 6191	. . . . . . . . . 10 |- ( e = (iEdg ` G ) -> ( # ` e ) = ( # ` (iEdg ` G ) ) )
13	12	oveq2d 6666	. . . . . . . . 9 |- ( e = (iEdg ` G ) -> ( 2 x. ( # ` e ) ) = ( 2 x. ( # ` (iEdg ` G ) ) ) )
14	13	adantl 482	. . . . . . . 8 |- ( ( k = (Vtx ` G ) /\ e = (iEdg ` G ) ) -> ( 2 x. ( # ` e ) ) = ( 2 x. ( # ` (iEdg ` G ) ) ) )
15	11, 14	eqeq12d 2637	. . . . . . 7 |- ( ( k = (Vtx ` G ) /\ e = (iEdg ` G ) ) -> ( sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) <-> sum_ v e. (Vtx ` G ) ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) = ( 2 x. ( # ` (iEdg ` G ) ) ) ) )
16	6, 15	imbi12d 334	. . . . . 6 |- ( ( k = (Vtx ` G ) /\ e = (iEdg ` G ) ) -> ( ( e e. Fin -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) <-> ( (iEdg ` G ) e. Fin -> sum_ v e. (Vtx ` G ) ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) = ( 2 x. ( # ` (iEdg ` G ) ) ) ) ) )
17		eleq1 2689	. . . . . . . 8 |- ( e = f -> ( e e. Fin <-> f e. Fin ) )
18	17	adantl 482	. . . . . . 7 |- ( ( k = w /\ e = f ) -> ( e e. Fin <-> f e. Fin ) )
19		simpl 473	. . . . . . . . 9 |- ( ( k = w /\ e = f ) -> k = w )
20		oveq12 6659	. . . . . . . . . . . 12 |- ( ( k = w /\ e = f ) -> ( kVtxDeg e ) = ( wVtxDeg f ) )
21		df-ov 6653	. . . . . . . . . . . 12 |- ( wVtxDeg f ) = (VtxDeg ` <. w , f >. )
22	20, 21	syl6eq 2672	. . . . . . . . . . 11 |- ( ( k = w /\ e = f ) -> ( kVtxDeg e ) = (VtxDeg ` <. w , f >. ) )
23	22	fveq1d 6193	. . . . . . . . . 10 |- ( ( k = w /\ e = f ) -> ( ( kVtxDeg e ) ` v ) = ( (VtxDeg ` <. w , f >. ) ` v ) )
24	23	adantr 481	. . . . . . . . 9 |- ( ( ( k = w /\ e = f ) /\ v e. k ) -> ( ( kVtxDeg e ) ` v ) = ( (VtxDeg ` <. w , f >. ) ` v ) )
25	19, 24	sumeq12dv 14437	. . . . . . . 8 |- ( ( k = w /\ e = f ) -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = sum_ v e. w ( (VtxDeg ` <. w , f >. ) ` v ) )
26		fveq2 6191	. . . . . . . . . 10 |- ( e = f -> ( # ` e ) = ( # ` f ) )
27	26	oveq2d 6666	. . . . . . . . 9 |- ( e = f -> ( 2 x. ( # ` e ) ) = ( 2 x. ( # ` f ) ) )
28	27	adantl 482	. . . . . . . 8 |- ( ( k = w /\ e = f ) -> ( 2 x. ( # ` e ) ) = ( 2 x. ( # ` f ) ) )
29	25, 28	eqeq12d 2637	. . . . . . 7 |- ( ( k = w /\ e = f ) -> ( sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) <-> sum_ v e. w ( (VtxDeg ` <. w , f >. ) ` v ) = ( 2 x. ( # ` f ) ) ) )
30	18, 29	imbi12d 334	. . . . . 6 |- ( ( k = w /\ e = f ) -> ( ( e e. Fin -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) <-> ( f e. Fin -> sum_ v e. w ( (VtxDeg ` <. w , f >. ) ` v ) = ( 2 x. ( # ` f ) ) ) ) )
31		vex 3203	. . . . . . . . 9 |- k e. _V
32		vex 3203	. . . . . . . . 9 |- e e. _V
33	31, 32	opvtxfvi 25889	. . . . . . . 8 |- (Vtx ` <. k , e >. ) = k
34	33	eqcomi 2631	. . . . . . 7 |- k = (Vtx ` <. k , e >. )
35		eqid 2622	. . . . . . 7 |- (iEdg ` <. k , e >. ) = (iEdg ` <. k , e >. )
36		eqid 2622	. . . . . . 7 |- { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } = { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) }
37		eqid 2622	. . . . . . 7 |- <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. = <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >.
38	34, 35, 36, 37	upgrres 26198	. . . . . 6 |- ( ( <. k , e >. e. UPGraph /\ n e. k ) -> <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. e. UPGraph )
39		eleq1 2689	. . . . . . . 8 |- ( f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) -> ( f e. Fin <-> ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin ) )
40	39	adantl 482	. . . . . . 7 |- ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) -> ( f e. Fin <-> ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin ) )
41		simpl 473	. . . . . . . . 9 |- ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) -> w = ( k \ { n } ) )
42		opeq12 4404	. . . . . . . . . . . 12 |- ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) -> <. w , f >. = <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. )
43	42	fveq2d 6195	. . . . . . . . . . 11 |- ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) -> (VtxDeg ` <. w , f >. ) = (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) )
44	43	fveq1d 6193	. . . . . . . . . 10 |- ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) -> ( (VtxDeg ` <. w , f >. ) ` v ) = ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) )
45	44	adantr 481	. . . . . . . . 9 |- ( ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) /\ v e. w ) -> ( (VtxDeg ` <. w , f >. ) ` v ) = ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) )
46	41, 45	sumeq12dv 14437	. . . . . . . 8 |- ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) -> sum_ v e. w ( (VtxDeg ` <. w , f >. ) ` v ) = sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) )
47		fveq2 6191	. . . . . . . . . 10 |- ( f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) -> ( # ` f ) = ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) )
48	47	oveq2d 6666	. . . . . . . . 9 |- ( f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) -> ( 2 x. ( # ` f ) ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) )
49	48	adantl 482	. . . . . . . 8 |- ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) -> ( 2 x. ( # ` f ) ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) )
50	46, 49	eqeq12d 2637	. . . . . . 7 |- ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) -> ( sum_ v e. w ( (VtxDeg ` <. w , f >. ) ` v ) = ( 2 x. ( # ` f ) ) <-> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) )
51	40, 50	imbi12d 334	. . . . . 6 |- ( ( w = ( k \ { n } ) /\ f = ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) -> ( ( f e. Fin -> sum_ v e. w ( (VtxDeg ` <. w , f >. ) ` v ) = ( 2 x. ( # ` f ) ) ) <-> ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) ) )
52		hasheq0 13154	. . . . . . . . 9 |- ( k e. _V -> ( ( # ` k ) = 0 <-> k = (/) ) )
53	31, 52	ax-mp 5	. . . . . . . 8 |- ( ( # ` k ) = 0 <-> k = (/) )
54		2t0e0 11183	. . . . . . . . . 10 |- ( 2 x. 0 ) = 0
55	54	a1i 11	. . . . . . . . 9 |- ( ( <. k , e >. e. UPGraph /\ k = (/) ) -> ( 2 x. 0 ) = 0 )
56	31, 32	opiedgfvi 25890	. . . . . . . . . . . . 13 |- (iEdg ` <. k , e >. ) = e
57	56	eqcomi 2631	. . . . . . . . . . . 12 |- e = (iEdg ` <. k , e >. )
58		upgruhgr 25997	. . . . . . . . . . . . . 14 |- ( <. k , e >. e. UPGraph -> <. k , e >. e. UHGraph )
59	58	adantr 481	. . . . . . . . . . . . 13 |- ( ( <. k , e >. e. UPGraph /\ k = (/) ) -> <. k , e >. e. UHGraph )
60	34	eqeq1i 2627	. . . . . . . . . . . . . 14 |- ( k = (/) <-> (Vtx ` <. k , e >. ) = (/) )
61		uhgr0vb 25967	. . . . . . . . . . . . . 14 |- ( ( <. k , e >. e. UPGraph /\ (Vtx ` <. k , e >. ) = (/) ) -> ( <. k , e >. e. UHGraph <-> (iEdg ` <. k , e >. ) = (/) ) )
62	60, 61	sylan2b 492	. . . . . . . . . . . . 13 |- ( ( <. k , e >. e. UPGraph /\ k = (/) ) -> ( <. k , e >. e. UHGraph <-> (iEdg ` <. k , e >. ) = (/) ) )
63	59, 62	mpbid 222	. . . . . . . . . . . 12 |- ( ( <. k , e >. e. UPGraph /\ k = (/) ) -> (iEdg ` <. k , e >. ) = (/) )
64	57, 63	syl5eq 2668	. . . . . . . . . . 11 |- ( ( <. k , e >. e. UPGraph /\ k = (/) ) -> e = (/) )
65		hasheq0 13154	. . . . . . . . . . . 12 |- ( e e. _V -> ( ( # ` e ) = 0 <-> e = (/) ) )
66	32, 65	ax-mp 5	. . . . . . . . . . 11 |- ( ( # ` e ) = 0 <-> e = (/) )
67	64, 66	sylibr 224	. . . . . . . . . 10 |- ( ( <. k , e >. e. UPGraph /\ k = (/) ) -> ( # ` e ) = 0 )
68	67	oveq2d 6666	. . . . . . . . 9 |- ( ( <. k , e >. e. UPGraph /\ k = (/) ) -> ( 2 x. ( # ` e ) ) = ( 2 x. 0 ) )
69		sumeq1 14419	. . . . . . . . . . 11 |- ( k = (/) -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = sum_ v e. (/) ( ( kVtxDeg e ) ` v ) )
70		sum0 14452	. . . . . . . . . . 11 |- sum_ v e. (/) ( ( kVtxDeg e ) ` v ) = 0
71	69, 70	syl6eq 2672	. . . . . . . . . 10 |- ( k = (/) -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = 0 )
72	71	adantl 482	. . . . . . . . 9 |- ( ( <. k , e >. e. UPGraph /\ k = (/) ) -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = 0 )
73	55, 68, 72	3eqtr4rd 2667	. . . . . . . 8 |- ( ( <. k , e >. e. UPGraph /\ k = (/) ) -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) )
74	53, 73	sylan2b 492	. . . . . . 7 |- ( ( <. k , e >. e. UPGraph /\ ( # ` k ) = 0 ) -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) )
75	74	a1d 25	. . . . . 6 |- ( ( <. k , e >. e. UPGraph /\ ( # ` k ) = 0 ) -> ( e e. Fin -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) )
76		eleq1 2689	. . . . . . . . . . 11 |- ( ( y + 1 ) = ( # ` k ) -> ( ( y + 1 ) e. NN0 <-> ( # ` k ) e. NN0 ) )
77	76	eqcoms 2630	. . . . . . . . . 10 |- ( ( # ` k ) = ( y + 1 ) -> ( ( y + 1 ) e. NN0 <-> ( # ` k ) e. NN0 ) )
78	77	3ad2ant2 1083	. . . . . . . . 9 |- ( ( <. k , e >. e. UPGraph /\ ( # ` k ) = ( y + 1 ) /\ n e. k ) -> ( ( y + 1 ) e. NN0 <-> ( # ` k ) e. NN0 ) )
79		hashclb 13149	. . . . . . . . . . . 12 |- ( k e. _V -> ( k e. Fin <-> ( # ` k ) e. NN0 ) )
80	79	biimprd 238	. . . . . . . . . . 11 |- ( k e. _V -> ( ( # ` k ) e. NN0 -> k e. Fin ) )
81	31, 80	ax-mp 5	. . . . . . . . . 10 |- ( ( # ` k ) e. NN0 -> k e. Fin )
82		eqid 2622	. . . . . . . . . . . . . . 15 |- ( k \ { n } ) = ( k \ { n } )
83		eqid 2622	. . . . . . . . . . . . . . 15 |- { i e. dom e | n e/ ( e ` i ) } = { i e. dom e | n e/ ( e ` i ) }
84	56	dmeqi 5325	. . . . . . . . . . . . . . . . . 18 |- dom (iEdg ` <. k , e >. ) = dom e
85	84	rabeqi 3193	. . . . . . . . . . . . . . . . 17 |- { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } = { i e. dom e | n e/ ( (iEdg ` <. k , e >. ) ` i ) }
86		eqidd 2623	. . . . . . . . . . . . . . . . . . 19 |- ( i e. dom e -> n = n )
87	56	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( i e. dom e -> (iEdg ` <. k , e >. ) = e )
88	87	fveq1d 6193	. . . . . . . . . . . . . . . . . . 19 |- ( i e. dom e -> ( (iEdg ` <. k , e >. ) ` i ) = ( e ` i ) )
89	86, 88	neleq12d 2901	. . . . . . . . . . . . . . . . . 18 |- ( i e. dom e -> ( n e/ ( (iEdg ` <. k , e >. ) ` i ) <-> n e/ ( e ` i ) ) )
90	89	rabbiia 3185	. . . . . . . . . . . . . . . . 17 |- { i e. dom e | n e/ ( (iEdg ` <. k , e >. ) ` i ) } = { i e. dom e | n e/ ( e ` i ) }
91	85, 90	eqtri 2644	. . . . . . . . . . . . . . . 16 |- { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } = { i e. dom e | n e/ ( e ` i ) }
92	56, 91	reseq12i 5394	. . . . . . . . . . . . . . 15 |- ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) = ( e |` { i e. dom e | n e/ ( e ` i ) } )
93	34, 57, 82, 83, 92, 37	finsumvtxdg2sstep 26445	. . . . . . . . . . . . . 14 |- ( ( ( <. k , e >. e. UPGraph /\ n e. k ) /\ ( k e. Fin /\ e e. Fin ) ) -> ( ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) -> sum_ v e. k ( (VtxDeg ` <. k , e >. ) ` v ) = ( 2 x. ( # ` e ) ) ) )
94		df-ov 6653	. . . . . . . . . . . . . . . . . 18 |- ( kVtxDeg e ) = (VtxDeg ` <. k , e >. )
95	94	fveq1i 6192	. . . . . . . . . . . . . . . . 17 |- ( ( kVtxDeg e ) ` v ) = ( (VtxDeg ` <. k , e >. ) ` v )
96	95	a1i 11	. . . . . . . . . . . . . . . 16 |- ( v e. k -> ( ( kVtxDeg e ) ` v ) = ( (VtxDeg ` <. k , e >. ) ` v ) )
97	96	sumeq2i 14429	. . . . . . . . . . . . . . 15 |- sum_ v e. k ( ( kVtxDeg e ) ` v ) = sum_ v e. k ( (VtxDeg ` <. k , e >. ) ` v )
98	97	eqeq1i 2627	. . . . . . . . . . . . . 14 |- ( sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) <-> sum_ v e. k ( (VtxDeg ` <. k , e >. ) ` v ) = ( 2 x. ( # ` e ) ) )
99	93, 98	syl6ibr 242	. . . . . . . . . . . . 13 |- ( ( ( <. k , e >. e. UPGraph /\ n e. k ) /\ ( k e. Fin /\ e e. Fin ) ) -> ( ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) )
100	99	exp32 631	. . . . . . . . . . . 12 |- ( ( <. k , e >. e. UPGraph /\ n e. k ) -> ( k e. Fin -> ( e e. Fin -> ( ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) ) ) )
101	100	com34 91	. . . . . . . . . . 11 |- ( ( <. k , e >. e. UPGraph /\ n e. k ) -> ( k e. Fin -> ( ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) -> ( e e. Fin -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) ) ) )
102	101	3adant2 1080	. . . . . . . . . 10 |- ( ( <. k , e >. e. UPGraph /\ ( # ` k ) = ( y + 1 ) /\ n e. k ) -> ( k e. Fin -> ( ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) -> ( e e. Fin -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) ) ) )
103	81, 102	syl5 34	. . . . . . . . 9 |- ( ( <. k , e >. e. UPGraph /\ ( # ` k ) = ( y + 1 ) /\ n e. k ) -> ( ( # ` k ) e. NN0 -> ( ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) -> ( e e. Fin -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) ) ) )
104	78, 103	sylbid 230	. . . . . . . 8 |- ( ( <. k , e >. e. UPGraph /\ ( # ` k ) = ( y + 1 ) /\ n e. k ) -> ( ( y + 1 ) e. NN0 -> ( ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) -> ( e e. Fin -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) ) ) )
105	104	impcom 446	. . . . . . 7 |- ( ( ( y + 1 ) e. NN0 /\ ( <. k , e >. e. UPGraph /\ ( # ` k ) = ( y + 1 ) /\ n e. k ) ) -> ( ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) -> ( e e. Fin -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) ) )
106	105	imp 445	. . . . . 6 |- ( ( ( ( y + 1 ) e. NN0 /\ ( <. k , e >. e. UPGraph /\ ( # ` k ) = ( y + 1 ) /\ n e. k ) ) /\ ( ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) e. Fin -> sum_ v e. ( k \ { n } ) ( (VtxDeg ` <. ( k \ { n } ) , ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) >. ) ` v ) = ( 2 x. ( # ` ( (iEdg ` <. k , e >. ) |` { i e. dom (iEdg ` <. k , e >. ) | n e/ ( (iEdg ` <. k , e >. ) ` i ) } ) ) ) ) ) -> ( e e. Fin -> sum_ v e. k ( ( kVtxDeg e ) ` v ) = ( 2 x. ( # ` e ) ) ) )
107	2, 4, 16, 30, 38, 51, 75, 106	opfi1ind 13284	. . . . 5 |- ( ( <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph /\ (Vtx ` G ) e. Fin ) -> ( (iEdg ` G ) e. Fin -> sum_ v e. (Vtx ` G ) ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) = ( 2 x. ( # ` (iEdg ` G ) ) ) ) )
108	107	ex 450	. . . 4 |- ( <. (Vtx ` G ) , (iEdg ` G ) >. e. UPGraph -> ( (Vtx ` G ) e. Fin -> ( (iEdg ` G ) e. Fin -> sum_ v e. (Vtx ` G ) ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) = ( 2 x. ( # ` (iEdg ` G ) ) ) ) ) )
109	1, 108	syl 17	. . 3 |- ( G e. UPGraph -> ( (Vtx ` G ) e. Fin -> ( (iEdg ` G ) e. Fin -> sum_ v e. (Vtx ` G ) ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) = ( 2 x. ( # ` (iEdg ` G ) ) ) ) ) )
110		sumvtxdg2size.v	. . . . 5 |- V = (Vtx ` G )
111	110	eleq1i 2692	. . . 4 |- ( V e. Fin <-> (Vtx ` G ) e. Fin )
112	111	a1i 11	. . 3 |- ( G e. UPGraph -> ( V e. Fin <-> (Vtx ` G ) e. Fin ) )
113		sumvtxdg2size.i	. . . . . 6 |- I = (iEdg ` G )
114	113	eleq1i 2692	. . . . 5 |- ( I e. Fin <-> (iEdg ` G ) e. Fin )
115	114	a1i 11	. . . 4 |- ( G e. UPGraph -> ( I e. Fin <-> (iEdg ` G ) e. Fin ) )
116	110	a1i 11	. . . . . 6 |- ( G e. UPGraph -> V = (Vtx ` G ) )
117		sumvtxdg2size.d	. . . . . . . . 9 |- D = (VtxDeg ` G )
118		vtxdgop 26366	. . . . . . . . 9 |- ( G e. UPGraph -> (VtxDeg ` G ) = ( (Vtx ` G )VtxDeg (iEdg ` G ) ) )
119	117, 118	syl5eq 2668	. . . . . . . 8 |- ( G e. UPGraph -> D = ( (Vtx ` G )VtxDeg (iEdg ` G ) ) )
120	119	fveq1d 6193	. . . . . . 7 |- ( G e. UPGraph -> ( D ` v ) = ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) )
121	120	adantr 481	. . . . . 6 |- ( ( G e. UPGraph /\ v e. V ) -> ( D ` v ) = ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) )
122	116, 121	sumeq12dv 14437	. . . . 5 |- ( G e. UPGraph -> sum_ v e. V ( D ` v ) = sum_ v e. (Vtx ` G ) ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) )
123	113	fveq2i 6194	. . . . . . 7 |- ( # ` I ) = ( # ` (iEdg ` G ) )
124	123	oveq2i 6661	. . . . . 6 |- ( 2 x. ( # ` I ) ) = ( 2 x. ( # ` (iEdg ` G ) ) )
125	124	a1i 11	. . . . 5 |- ( G e. UPGraph -> ( 2 x. ( # ` I ) ) = ( 2 x. ( # ` (iEdg ` G ) ) ) )
126	122, 125	eqeq12d 2637	. . . 4 |- ( G e. UPGraph -> ( sum_ v e. V ( D ` v ) = ( 2 x. ( # ` I ) ) <-> sum_ v e. (Vtx ` G ) ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) = ( 2 x. ( # ` (iEdg ` G ) ) ) ) )
127	115, 126	imbi12d 334	. . 3 |- ( G e. UPGraph -> ( ( I e. Fin -> sum_ v e. V ( D ` v ) = ( 2 x. ( # ` I ) ) ) <-> ( (iEdg ` G ) e. Fin -> sum_ v e. (Vtx ` G ) ( ( (Vtx ` G )VtxDeg (iEdg ` G ) ) ` v ) = ( 2 x. ( # ` (iEdg ` G ) ) ) ) ) )
128	109, 112, 127	3imtr4d 283	. 2 |- ( G e. UPGraph -> ( V e. Fin -> ( I e. Fin -> sum_ v e. V ( D ` v ) = ( 2 x. ( # ` I ) ) ) ) )
129	128	3imp 1256	1 |- ( ( G e. UPGraph /\ V e. Fin /\ I e. Fin ) -> sum_ v e. V ( D ` v ) = ( 2 x. ( # ` I ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   e/ wnel 2897   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   2c2 11070   NN0cn0 11292   #chash 13117   sum_csu 14416  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951   UPGraph cupgr 25975  VtxDegcvtxdg 26361

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-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-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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  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-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-vtxdg 26362

This theorem is referenced by:  fusgr1th  26447  finsumvtxdgeven  26448