Theorem vtxdun 26377

Description: The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 19-Feb-2021.)

Hypotheses

Ref	Expression
vtxdun.i	|- I = (iEdg ` G )
vtxdun.j	|- J = (iEdg ` H )
vtxdun.vg	|- V = (Vtx ` G )
vtxdun.vh	|- ( ph -> (Vtx ` H ) = V )
vtxdun.vu	|- ( ph -> (Vtx ` U ) = V )
vtxdun.d	|- ( ph -> ( dom I i^i dom J ) = (/) )
vtxdun.fi	|- ( ph -> Fun I )
vtxdun.fj	|- ( ph -> Fun J )
vtxdun.n	|- ( ph -> N e. V )
vtxdun.u	|- ( ph -> (iEdg ` U ) = ( I u. J ) )

Assertion

Ref	Expression
vtxdun	|- ( ph -> ( (VtxDeg ` U ) ` N ) = ( ( (VtxDeg ` G ) ` N ) +e ( (VtxDeg ` H ) ` N ) ) )

Proof of Theorem vtxdun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 2921	. . . . . . . 8 |- { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } = { x | ( x e. dom (iEdg ` U ) /\ N e. ( (iEdg ` U ) ` x ) ) }
2		vtxdun.u	. . . . . . . . . . . . . . 15 |- ( ph -> (iEdg ` U ) = ( I u. J ) )
3	2	dmeqd 5326	. . . . . . . . . . . . . 14 |- ( ph -> dom (iEdg ` U ) = dom ( I u. J ) )
4		dmun 5331	. . . . . . . . . . . . . 14 |- dom ( I u. J ) = ( dom I u. dom J )
5	3, 4	syl6eq 2672	. . . . . . . . . . . . 13 |- ( ph -> dom (iEdg ` U ) = ( dom I u. dom J ) )
6	5	eleq2d 2687	. . . . . . . . . . . 12 |- ( ph -> ( x e. dom (iEdg ` U ) <-> x e. ( dom I u. dom J ) ) )
7		elun 3753	. . . . . . . . . . . 12 |- ( x e. ( dom I u. dom J ) <-> ( x e. dom I \/ x e. dom J ) )
8	6, 7	syl6bb 276	. . . . . . . . . . 11 |- ( ph -> ( x e. dom (iEdg ` U ) <-> ( x e. dom I \/ x e. dom J ) ) )
9	8	anbi1d 741	. . . . . . . . . 10 |- ( ph -> ( ( x e. dom (iEdg ` U ) /\ N e. ( (iEdg ` U ) ` x ) ) <-> ( ( x e. dom I \/ x e. dom J ) /\ N e. ( (iEdg ` U ) ` x ) ) ) )
10		andir 912	. . . . . . . . . 10 |- ( ( ( x e. dom I \/ x e. dom J ) /\ N e. ( (iEdg ` U ) ` x ) ) <-> ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) )
11	9, 10	syl6bb 276	. . . . . . . . 9 |- ( ph -> ( ( x e. dom (iEdg ` U ) /\ N e. ( (iEdg ` U ) ` x ) ) <-> ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) ) )
12	11	abbidv 2741	. . . . . . . 8 |- ( ph -> { x | ( x e. dom (iEdg ` U ) /\ N e. ( (iEdg ` U ) ` x ) ) } = { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) } )
13	1, 12	syl5eq 2668	. . . . . . 7 |- ( ph -> { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } = { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) } )
14		unab 3894	. . . . . . . . 9 |- ( { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } u. { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } ) = { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) }
15	14	eqcomi 2631	. . . . . . . 8 |- { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) } = ( { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } u. { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } )
16	15	a1i 11	. . . . . . 7 |- ( ph -> { x | ( ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) \/ ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) ) } = ( { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } u. { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } ) )
17		df-rab 2921	. . . . . . . . 9 |- { x e. dom I | N e. ( (iEdg ` U ) ` x ) } = { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) }
18	2	fveq1d 6193	. . . . . . . . . . . . 13 |- ( ph -> ( (iEdg ` U ) ` x ) = ( ( I u. J ) ` x ) )
19	18	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom I ) -> ( (iEdg ` U ) ` x ) = ( ( I u. J ) ` x ) )
20		vtxdun.fi	. . . . . . . . . . . . . . 15 |- ( ph -> Fun I )
21		funfn 5918	. . . . . . . . . . . . . . 15 |- ( Fun I <-> I Fn dom I )
22	20, 21	sylib 208	. . . . . . . . . . . . . 14 |- ( ph -> I Fn dom I )
23	22	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom I ) -> I Fn dom I )
24		vtxdun.fj	. . . . . . . . . . . . . . 15 |- ( ph -> Fun J )
25		funfn 5918	. . . . . . . . . . . . . . 15 |- ( Fun J <-> J Fn dom J )
26	24, 25	sylib 208	. . . . . . . . . . . . . 14 |- ( ph -> J Fn dom J )
27	26	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom I ) -> J Fn dom J )
28		vtxdun.d	. . . . . . . . . . . . . 14 |- ( ph -> ( dom I i^i dom J ) = (/) )
29	28	anim1i 592	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom I ) -> ( ( dom I i^i dom J ) = (/) /\ x e. dom I ) )
30		fvun1 6269	. . . . . . . . . . . . 13 |- ( ( I Fn dom I /\ J Fn dom J /\ ( ( dom I i^i dom J ) = (/) /\ x e. dom I ) ) -> ( ( I u. J ) ` x ) = ( I ` x ) )
31	23, 27, 29, 30	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom I ) -> ( ( I u. J ) ` x ) = ( I ` x ) )
32	19, 31	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ph /\ x e. dom I ) -> ( (iEdg ` U ) ` x ) = ( I ` x ) )
33	32	eleq2d 2687	. . . . . . . . . 10 |- ( ( ph /\ x e. dom I ) -> ( N e. ( (iEdg ` U ) ` x ) <-> N e. ( I ` x ) ) )
34	33	rabbidva 3188	. . . . . . . . 9 |- ( ph -> { x e. dom I | N e. ( (iEdg ` U ) ` x ) } = { x e. dom I | N e. ( I ` x ) } )
35	17, 34	syl5eqr 2670	. . . . . . . 8 |- ( ph -> { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } = { x e. dom I | N e. ( I ` x ) } )
36		df-rab 2921	. . . . . . . . 9 |- { x e. dom J | N e. ( (iEdg ` U ) ` x ) } = { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) }
37	18	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom J ) -> ( (iEdg ` U ) ` x ) = ( ( I u. J ) ` x ) )
38	22	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom J ) -> I Fn dom I )
39	26	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom J ) -> J Fn dom J )
40	28	anim1i 592	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. dom J ) -> ( ( dom I i^i dom J ) = (/) /\ x e. dom J ) )
41		fvun2 6270	. . . . . . . . . . . . 13 |- ( ( I Fn dom I /\ J Fn dom J /\ ( ( dom I i^i dom J ) = (/) /\ x e. dom J ) ) -> ( ( I u. J ) ` x ) = ( J ` x ) )
42	38, 39, 40, 41	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ph /\ x e. dom J ) -> ( ( I u. J ) ` x ) = ( J ` x ) )
43	37, 42	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ph /\ x e. dom J ) -> ( (iEdg ` U ) ` x ) = ( J ` x ) )
44	43	eleq2d 2687	. . . . . . . . . 10 |- ( ( ph /\ x e. dom J ) -> ( N e. ( (iEdg ` U ) ` x ) <-> N e. ( J ` x ) ) )
45	44	rabbidva 3188	. . . . . . . . 9 |- ( ph -> { x e. dom J | N e. ( (iEdg ` U ) ` x ) } = { x e. dom J | N e. ( J ` x ) } )
46	36, 45	syl5eqr 2670	. . . . . . . 8 |- ( ph -> { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } = { x e. dom J | N e. ( J ` x ) } )
47	35, 46	uneq12d 3768	. . . . . . 7 |- ( ph -> ( { x | ( x e. dom I /\ N e. ( (iEdg ` U ) ` x ) ) } u. { x | ( x e. dom J /\ N e. ( (iEdg ` U ) ` x ) ) } ) = ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) )
48	13, 16, 47	3eqtrd 2660	. . . . . 6 |- ( ph -> { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } = ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) )
49	48	fveq2d 6195	. . . . 5 |- ( ph -> ( # ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) = ( # ` ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) ) )
50		vtxdun.i	. . . . . . . . . 10 |- I = (iEdg ` G )
51		fvex 6201	. . . . . . . . . 10 |- (iEdg ` G ) e. _V
52	50, 51	eqeltri 2697	. . . . . . . . 9 |- I e. _V
53	52	dmex 7099	. . . . . . . 8 |- dom I e. _V
54	53	rabex 4813	. . . . . . 7 |- { x e. dom I | N e. ( I ` x ) } e. _V
55	54	a1i 11	. . . . . 6 |- ( ph -> { x e. dom I | N e. ( I ` x ) } e. _V )
56		vtxdun.j	. . . . . . . . . 10 |- J = (iEdg ` H )
57		fvex 6201	. . . . . . . . . 10 |- (iEdg ` H ) e. _V
58	56, 57	eqeltri 2697	. . . . . . . . 9 |- J e. _V
59	58	dmex 7099	. . . . . . . 8 |- dom J e. _V
60	59	rabex 4813	. . . . . . 7 |- { x e. dom J | N e. ( J ` x ) } e. _V
61	60	a1i 11	. . . . . 6 |- ( ph -> { x e. dom J | N e. ( J ` x ) } e. _V )
62		ssrab2 3687	. . . . . . . . 9 |- { x e. dom I | N e. ( I ` x ) } C_ dom I
63		ssrab2 3687	. . . . . . . . 9 |- { x e. dom J | N e. ( J ` x ) } C_ dom J
64		ss2in 3840	. . . . . . . . 9 |- ( ( { x e. dom I | N e. ( I ` x ) } C_ dom I /\ { x e. dom J | N e. ( J ` x ) } C_ dom J ) -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ ( dom I i^i dom J ) )
65	62, 63, 64	mp2an 708	. . . . . . . 8 |- ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ ( dom I i^i dom J )
66	65, 28	syl5sseq 3653	. . . . . . 7 |- ( ph -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ (/) )
67		ss0 3974	. . . . . . 7 |- ( ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) C_ (/) -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) = (/) )
68	66, 67	syl 17	. . . . . 6 |- ( ph -> ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) = (/) )
69		hashunx 13175	. . . . . 6 |- ( ( { x e. dom I | N e. ( I ` x ) } e. _V /\ { x e. dom J | N e. ( J ` x ) } e. _V /\ ( { x e. dom I | N e. ( I ` x ) } i^i { x e. dom J | N e. ( J ` x ) } ) = (/) ) -> ( # ` ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) ) = ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom J | N e. ( J ` x ) } ) ) )
70	55, 61, 68, 69	syl3anc 1326	. . . . 5 |- ( ph -> ( # ` ( { x e. dom I | N e. ( I ` x ) } u. { x e. dom J | N e. ( J ` x ) } ) ) = ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom J | N e. ( J ` x ) } ) ) )
71	49, 70	eqtrd 2656	. . . 4 |- ( ph -> ( # ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) = ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom J | N e. ( J ` x ) } ) ) )
72		df-rab 2921	. . . . . . . 8 |- { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom (iEdg ` U ) /\ ( (iEdg ` U ) ` x ) = { N } ) }
73	8	anbi1d 741	. . . . . . . . . 10 |- ( ph -> ( ( x e. dom (iEdg ` U ) /\ ( (iEdg ` U ) ` x ) = { N } ) <-> ( ( x e. dom I \/ x e. dom J ) /\ ( (iEdg ` U ) ` x ) = { N } ) ) )
74		andir 912	. . . . . . . . . 10 |- ( ( ( x e. dom I \/ x e. dom J ) /\ ( (iEdg ` U ) ` x ) = { N } ) <-> ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) )
75	73, 74	syl6bb 276	. . . . . . . . 9 |- ( ph -> ( ( x e. dom (iEdg ` U ) /\ ( (iEdg ` U ) ` x ) = { N } ) <-> ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) ) )
76	75	abbidv 2741	. . . . . . . 8 |- ( ph -> { x | ( x e. dom (iEdg ` U ) /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) } )
77	72, 76	syl5eq 2668	. . . . . . 7 |- ( ph -> { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } = { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) } )
78		unab 3894	. . . . . . . . 9 |- ( { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } u. { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } ) = { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) }
79	78	eqcomi 2631	. . . . . . . 8 |- { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) } = ( { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } u. { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } )
80	79	a1i 11	. . . . . . 7 |- ( ph -> { x | ( ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) \/ ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) ) } = ( { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } u. { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } ) )
81		df-rab 2921	. . . . . . . . 9 |- { x e. dom I | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) }
82	32	eqeq1d 2624	. . . . . . . . . 10 |- ( ( ph /\ x e. dom I ) -> ( ( (iEdg ` U ) ` x ) = { N } <-> ( I ` x ) = { N } ) )
83	82	rabbidva 3188	. . . . . . . . 9 |- ( ph -> { x e. dom I | ( (iEdg ` U ) ` x ) = { N } } = { x e. dom I | ( I ` x ) = { N } } )
84	81, 83	syl5eqr 2670	. . . . . . . 8 |- ( ph -> { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x e. dom I | ( I ` x ) = { N } } )
85		df-rab 2921	. . . . . . . . 9 |- { x e. dom J | ( (iEdg ` U ) ` x ) = { N } } = { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) }
86	43	eqeq1d 2624	. . . . . . . . . 10 |- ( ( ph /\ x e. dom J ) -> ( ( (iEdg ` U ) ` x ) = { N } <-> ( J ` x ) = { N } ) )
87	86	rabbidva 3188	. . . . . . . . 9 |- ( ph -> { x e. dom J | ( (iEdg ` U ) ` x ) = { N } } = { x e. dom J | ( J ` x ) = { N } } )
88	85, 87	syl5eqr 2670	. . . . . . . 8 |- ( ph -> { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } = { x e. dom J | ( J ` x ) = { N } } )
89	84, 88	uneq12d 3768	. . . . . . 7 |- ( ph -> ( { x | ( x e. dom I /\ ( (iEdg ` U ) ` x ) = { N } ) } u. { x | ( x e. dom J /\ ( (iEdg ` U ) ` x ) = { N } ) } ) = ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) )
90	77, 80, 89	3eqtrd 2660	. . . . . 6 |- ( ph -> { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } = ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) )
91	90	fveq2d 6195	. . . . 5 |- ( ph -> ( # ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) = ( # ` ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) ) )
92	53	rabex 4813	. . . . . . 7 |- { x e. dom I | ( I ` x ) = { N } } e. _V
93	92	a1i 11	. . . . . 6 |- ( ph -> { x e. dom I | ( I ` x ) = { N } } e. _V )
94	59	rabex 4813	. . . . . . 7 |- { x e. dom J | ( J ` x ) = { N } } e. _V
95	94	a1i 11	. . . . . 6 |- ( ph -> { x e. dom J | ( J ` x ) = { N } } e. _V )
96		ssrab2 3687	. . . . . . . . 9 |- { x e. dom I | ( I ` x ) = { N } } C_ dom I
97		ssrab2 3687	. . . . . . . . 9 |- { x e. dom J | ( J ` x ) = { N } } C_ dom J
98		ss2in 3840	. . . . . . . . 9 |- ( ( { x e. dom I | ( I ` x ) = { N } } C_ dom I /\ { x e. dom J | ( J ` x ) = { N } } C_ dom J ) -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ ( dom I i^i dom J ) )
99	96, 97, 98	mp2an 708	. . . . . . . 8 |- ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ ( dom I i^i dom J )
100	99, 28	syl5sseq 3653	. . . . . . 7 |- ( ph -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ (/) )
101		ss0 3974	. . . . . . 7 |- ( ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) C_ (/) -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) = (/) )
102	100, 101	syl 17	. . . . . 6 |- ( ph -> ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) = (/) )
103		hashunx 13175	. . . . . 6 |- ( ( { x e. dom I | ( I ` x ) = { N } } e. _V /\ { x e. dom J | ( J ` x ) = { N } } e. _V /\ ( { x e. dom I | ( I ` x ) = { N } } i^i { x e. dom J | ( J ` x ) = { N } } ) = (/) ) -> ( # ` ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) ) = ( ( # ` { x e. dom I | ( I ` x ) = { N } } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) )
104	93, 95, 102, 103	syl3anc 1326	. . . . 5 |- ( ph -> ( # ` ( { x e. dom I | ( I ` x ) = { N } } u. { x e. dom J | ( J ` x ) = { N } } ) ) = ( ( # ` { x e. dom I | ( I ` x ) = { N } } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) )
105	91, 104	eqtrd 2656	. . . 4 |- ( ph -> ( # ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) = ( ( # ` { x e. dom I | ( I ` x ) = { N } } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) )
106	71, 105	oveq12d 6668	. . 3 |- ( ph -> ( ( # ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) ) = ( ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom J | N e. ( J ` x ) } ) ) +e ( ( # ` { x e. dom I | ( I ` x ) = { N } } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) ) )
107		hashxnn0 13127	. . . . 5 |- ( { x e. dom I | N e. ( I ` x ) } e. _V -> ( # ` { x e. dom I | N e. ( I ` x ) } ) e. NN0* )
108	55, 107	syl 17	. . . 4 |- ( ph -> ( # ` { x e. dom I | N e. ( I ` x ) } ) e. NN0* )
109		hashxnn0 13127	. . . . 5 |- ( { x e. dom J | N e. ( J ` x ) } e. _V -> ( # ` { x e. dom J | N e. ( J ` x ) } ) e. NN0* )
110	61, 109	syl 17	. . . 4 |- ( ph -> ( # ` { x e. dom J | N e. ( J ` x ) } ) e. NN0* )
111		hashxnn0 13127	. . . . 5 |- ( { x e. dom I | ( I ` x ) = { N } } e. _V -> ( # ` { x e. dom I | ( I ` x ) = { N } } ) e. NN0* )
112	93, 111	syl 17	. . . 4 |- ( ph -> ( # ` { x e. dom I | ( I ` x ) = { N } } ) e. NN0* )
113		hashxnn0 13127	. . . . 5 |- ( { x e. dom J | ( J ` x ) = { N } } e. _V -> ( # ` { x e. dom J | ( J ` x ) = { N } } ) e. NN0* )
114	95, 113	syl 17	. . . 4 |- ( ph -> ( # ` { x e. dom J | ( J ` x ) = { N } } ) e. NN0* )
115	108, 110, 112, 114	xnn0add4d 12134	. . 3 |- ( ph -> ( ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom J | N e. ( J ` x ) } ) ) +e ( ( # ` { x e. dom I | ( I ` x ) = { N } } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) ) = ( ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom I | ( I ` x ) = { N } } ) ) +e ( ( # ` { x e. dom J | N e. ( J ` x ) } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) ) )
116	106, 115	eqtrd 2656	. 2 |- ( ph -> ( ( # ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) ) = ( ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom I | ( I ` x ) = { N } } ) ) +e ( ( # ` { x e. dom J | N e. ( J ` x ) } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) ) )
117		vtxdun.n	. . . 4 |- ( ph -> N e. V )
118		vtxdun.vu	. . . 4 |- ( ph -> (Vtx ` U ) = V )
119	117, 118	eleqtrrd 2704	. . 3 |- ( ph -> N e. (Vtx ` U ) )
120		eqid 2622	. . . 4 |- (Vtx ` U ) = (Vtx ` U )
121		eqid 2622	. . . 4 |- (iEdg ` U ) = (iEdg ` U )
122		eqid 2622	. . . 4 |- dom (iEdg ` U ) = dom (iEdg ` U )
123	120, 121, 122	vtxdgval 26364	. . 3 |- ( N e. (Vtx ` U ) -> ( (VtxDeg ` U ) ` N ) = ( ( # ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) ) )
124	119, 123	syl 17	. 2 |- ( ph -> ( (VtxDeg ` U ) ` N ) = ( ( # ` { x e. dom (iEdg ` U ) | N e. ( (iEdg ` U ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` U ) | ( (iEdg ` U ) ` x ) = { N } } ) ) )
125		vtxdun.vg	. . . . 5 |- V = (Vtx ` G )
126		eqid 2622	. . . . 5 |- dom I = dom I
127	125, 50, 126	vtxdgval 26364	. . . 4 |- ( N e. V -> ( (VtxDeg ` G ) ` N ) = ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom I | ( I ` x ) = { N } } ) ) )
128	117, 127	syl 17	. . 3 |- ( ph -> ( (VtxDeg ` G ) ` N ) = ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom I | ( I ` x ) = { N } } ) ) )
129		vtxdun.vh	. . . . 5 |- ( ph -> (Vtx ` H ) = V )
130	117, 129	eleqtrrd 2704	. . . 4 |- ( ph -> N e. (Vtx ` H ) )
131		eqid 2622	. . . . 5 |- (Vtx ` H ) = (Vtx ` H )
132		eqid 2622	. . . . 5 |- dom J = dom J
133	131, 56, 132	vtxdgval 26364	. . . 4 |- ( N e. (Vtx ` H ) -> ( (VtxDeg ` H ) ` N ) = ( ( # ` { x e. dom J | N e. ( J ` x ) } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) )
134	130, 133	syl 17	. . 3 |- ( ph -> ( (VtxDeg ` H ) ` N ) = ( ( # ` { x e. dom J | N e. ( J ` x ) } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) )
135	128, 134	oveq12d 6668	. 2 |- ( ph -> ( ( (VtxDeg ` G ) ` N ) +e ( (VtxDeg ` H ) ` N ) ) = ( ( ( # ` { x e. dom I | N e. ( I ` x ) } ) +e ( # ` { x e. dom I | ( I ` x ) = { N } } ) ) +e ( ( # ` { x e. dom J | N e. ( J ` x ) } ) +e ( # ` { x e. dom J | ( J ` x ) = { N } } ) ) ) )
136	116, 124, 135	3eqtr4d 2666	1 |- ( ph -> ( (VtxDeg ` U ) ` N ) = ( ( (VtxDeg ` G ) ` N ) +e ( (VtxDeg ` H ) ` N ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650  NN0*cxnn0 11363   +ecxad 11944   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  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-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-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-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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-xadd 11947  df-hash 13118  df-vtxdg 26362

This theorem is referenced by:  vtxdfiun  26378  vtxduhgrun  26379  p1evtxdeqlem  26408