vtxdun - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > vtxdun		Structured version Visualization version GIF version

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	⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
vtxdun.vu	⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
vtxdun.d	⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)
vtxdun.fi	⊢ (𝜑 → Fun 𝐼)
vtxdun.fj	⊢ (𝜑 → Fun 𝐽)
vtxdun.n	⊢ (𝜑 → 𝑁 ∈ 𝑉)
vtxdun.u	⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))

**Assertion**
Ref	Expression
vtxdun	⊢ (𝜑 → ((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 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
2		vtxdun.u	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐼 ∪ 𝐽))
3	2	dmeqd 5326	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (iEdg‘𝑈) = dom (𝐼 ∪ 𝐽))
4		dmun 5331	. . . . . . . . . . . . . 14 ⊢ dom (𝐼 ∪ 𝐽) = (dom 𝐼 ∪ dom 𝐽)
5	3, 4	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (iEdg‘𝑈) = (dom 𝐼 ∪ dom 𝐽))
6	5	eleq2d 2687	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ 𝑥 ∈ (dom 𝐼 ∪ dom 𝐽)))
7		elun 3753	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (dom 𝐼 ∪ dom 𝐽) ↔ (𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽))
8	6, 7	syl6bb 276	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom (iEdg‘𝑈) ↔ (𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽)))
9	8	anbi1d 741	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
10		andir 912	. . . . . . . . . 10 ⊢ (((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))))
11	9, 10	syl6bb 276	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ↔ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))))
12	11	abbidv 2741	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))})
13	1, 12	syl5eq 2668	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))})
14		unab 3894	. . . . . . . . 9 ⊢ ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))}
15	14	eqcomi 2631	. . . . . . . 8 ⊢ {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))})
16	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)) ∨ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}))
17		df-rab 2921	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
18	2	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((iEdg‘𝑈)‘𝑥) = ((𝐼 ∪ 𝐽)‘𝑥))
19	18	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = ((𝐼 ∪ 𝐽)‘𝑥))
20		vtxdun.fi	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐼)
21		funfn 5918	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
22	20, 21	sylib 208	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 Fn dom 𝐼)
23	22	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
24		vtxdun.fj	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐽)
25		funfn 5918	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐽 ↔ 𝐽 Fn dom 𝐽)
26	24, 25	sylib 208	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 Fn dom 𝐽)
27	26	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → 𝐽 Fn dom 𝐽)
28		vtxdun.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)
29	28	anim1i 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼))
30		fvun1 6269	. . . . . . . . . . . . 13 ⊢ ((𝐼 Fn dom 𝐼 ∧ 𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐼)) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐼‘𝑥))
31	23, 27, 29, 30	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐼‘𝑥))
32	19, 31	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → ((iEdg‘𝑈)‘𝑥) = (𝐼‘𝑥))
33	32	eleq2d 2687	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐼‘𝑥)))
34	33	rabbidva 3188	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)})
35	17, 34	syl5eqr 2670	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)})
36		df-rab 2921	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}
37	18	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = ((𝐼 ∪ 𝐽)‘𝑥))
38	22	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → 𝐼 Fn dom 𝐼)
39	26	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → 𝐽 Fn dom 𝐽)
40	28	anim1i 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽))
41		fvun2 6270	. . . . . . . . . . . . 13 ⊢ ((𝐼 Fn dom 𝐼 ∧ 𝐽 Fn dom 𝐽 ∧ ((dom 𝐼 ∩ dom 𝐽) = ∅ ∧ 𝑥 ∈ dom 𝐽)) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐽‘𝑥))
42	38, 39, 40, 41	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((𝐼 ∪ 𝐽)‘𝑥) = (𝐽‘𝑥))
43	37, 42	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → ((iEdg‘𝑈)‘𝑥) = (𝐽‘𝑥))
44	43	eleq2d 2687	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → (𝑁 ∈ ((iEdg‘𝑈)‘𝑥) ↔ 𝑁 ∈ (𝐽‘𝑥)))
45	44	rabbidva 3188	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})
46	36, 45	syl5eqr 2670	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} = {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})
47	35, 46	uneq12d 3768	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥))}) = ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}))
48	13, 16, 47	3eqtrd 2660	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)} = ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}))
49	48	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = (#‘({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})))
50		vtxdun.i	. . . . . . . . . 10 ⊢ 𝐼 = (iEdg‘𝐺)
51		fvex 6201	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) ∈ V
52	50, 51	eqeltri 2697	. . . . . . . . 9 ⊢ 𝐼 ∈ V
53	52	dmex 7099	. . . . . . . 8 ⊢ dom 𝐼 ∈ V
54	53	rabex 4813	. . . . . . 7 ⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ V
55	54	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ V)
56		vtxdun.j	. . . . . . . . . 10 ⊢ 𝐽 = (iEdg‘𝐻)
57		fvex 6201	. . . . . . . . . 10 ⊢ (iEdg‘𝐻) ∈ V
58	56, 57	eqeltri 2697	. . . . . . . . 9 ⊢ 𝐽 ∈ V
59	58	dmex 7099	. . . . . . . 8 ⊢ dom 𝐽 ∈ V
60	59	rabex 4813	. . . . . . 7 ⊢ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ V
61	60	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ V)
62		ssrab2 3687	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ⊆ dom 𝐼
63		ssrab2 3687	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ⊆ dom 𝐽
64		ss2in 3840	. . . . . . . . 9 ⊢ (({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ⊆ dom 𝐼 ∧ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ⊆ dom 𝐽) → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽))
65	62, 63, 64	mp2an 708	. . . . . . . 8 ⊢ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ (dom 𝐼 ∩ dom 𝐽)
66	65, 28	syl5sseq 3653	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ ∅)
67		ss0 3974	. . . . . . 7 ⊢ (({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) = ∅)
68	66, 67	syl 17	. . . . . 6 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) = ∅)
69		hashunx 13175	. . . . . 6 ⊢ (({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ V ∧ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ V ∧ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∩ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) = ∅) → (#‘({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) = ((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})))
70	55, 61, 68, 69	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (#‘({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∪ {𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) = ((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})))
71	49, 70	eqtrd 2656	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) = ((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})))
72		df-rab 2921	. . . . . . . 8 ⊢ {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
73	8	anbi1d 741	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
74		andir 912	. . . . . . . . . 10 ⊢ (((𝑥 ∈ dom 𝐼 ∨ 𝑥 ∈ dom 𝐽) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})))
75	73, 74	syl6bb 276	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ↔ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))))
76	75	abbidv 2741	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom (iEdg‘𝑈) ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))})
77	72, 76	syl5eq 2668	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))})
78		unab 3894	. . . . . . . . 9 ⊢ ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}) = {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))}
79	78	eqcomi 2631	. . . . . . . 8 ⊢ {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})})
80	79	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∣ ((𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}) ∨ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁}))} = ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}))
81		df-rab 2921	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
82	32	eqeq1d 2624	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐼) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐼‘𝑥) = {𝑁}))
83	82	rabbidva 3188	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})
84	81, 83	syl5eqr 2670	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})
85		df-rab 2921	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}
86	43	eqeq1d 2624	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐽) → (((iEdg‘𝑈)‘𝑥) = {𝑁} ↔ (𝐽‘𝑥) = {𝑁}))
87	86	rabbidva 3188	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})
88	85, 87	syl5eqr 2670	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})
89	84, 88	uneq12d 3768	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ (𝑥 ∈ dom 𝐼 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})} ∪ {𝑥 ∣ (𝑥 ∈ dom 𝐽 ∧ ((iEdg‘𝑈)‘𝑥) = {𝑁})}) = ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))
90	77, 80, 89	3eqtrd 2660	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}} = ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))
91	90	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = (#‘({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
92	53	rabex 4813	. . . . . . 7 ⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ V
93	92	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ V)
94	59	rabex 4813	. . . . . . 7 ⊢ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ V
95	94	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ V)
96		ssrab2 3687	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ⊆ dom 𝐼
97		ssrab2 3687	. . . . . . . . 9 ⊢ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ⊆ dom 𝐽
98		ss2in 3840	. . . . . . . . 9 ⊢ (({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ⊆ dom 𝐼 ∧ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ⊆ dom 𝐽) → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽))
99	96, 97, 98	mp2an 708	. . . . . . . 8 ⊢ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ (dom 𝐼 ∩ dom 𝐽)
100	99, 28	syl5sseq 3653	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ ∅)
101		ss0 3974	. . . . . . 7 ⊢ (({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ⊆ ∅ → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) = ∅)
102	100, 101	syl 17	. . . . . 6 ⊢ (𝜑 → ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) = ∅)
103		hashunx 13175	. . . . . 6 ⊢ (({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ V ∧ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ V ∧ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∩ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) = ∅) → (#‘({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})) = ((#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
104	93, 95, 102, 103	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (#‘({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∪ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})) = ((#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
105	91, 104	eqtrd 2656	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}}) = ((#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
106	71, 105	oveq12d 6668	. . 3 ⊢ (𝜑 → ((#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})) = (((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) +_𝑒 ((#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))))
107		hashxnn0 13127	. . . . 5 ⊢ ({𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)} ∈ V → (#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) ∈ ℕ₀^*)
108	55, 107	syl 17	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) ∈ ℕ₀^*)
109		hashxnn0 13127	. . . . 5 ⊢ ({𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)} ∈ V → (#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ∈ ℕ₀^*)
110	61, 109	syl 17	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) ∈ ℕ₀^*)
111		hashxnn0 13127	. . . . 5 ⊢ ({𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}} ∈ V → (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) ∈ ℕ₀^*)
112	93, 111	syl 17	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) ∈ ℕ₀^*)
113		hashxnn0 13127	. . . . 5 ⊢ ({𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}} ∈ V → (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ∈ ℕ₀^*)
114	95, 113	syl 17	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}) ∈ ℕ₀^*)
115	108, 110, 112, 114	xnn0add4d 12134	. . 3 ⊢ (𝜑 → (((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)})) +_𝑒 ((#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))) = (((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})) +_𝑒 ((#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))))
116	106, 115	eqtrd 2656	. 2 ⊢ (𝜑 → ((#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})) = (((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})) +_𝑒 ((#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))))
117		vtxdun.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑉)
118		vtxdun.vu	. . . 4 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉)
119	117, 118	eleqtrrd 2704	. . 3 ⊢ (𝜑 → 𝑁 ∈ (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 ∈ (Vtx‘𝑈) → ((VtxDeg‘𝑈)‘𝑁) = ((#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ ((iEdg‘𝑈)‘𝑥) = {𝑁}})))
124	119, 123	syl 17	. 2 ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = ((#‘{𝑥 ∈ dom (iEdg‘𝑈) ∣ 𝑁 ∈ ((iEdg‘𝑈)‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ 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 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑁) = ((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})))
128	117, 127	syl 17	. . 3 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = ((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})))
129		vtxdun.vh	. . . . 5 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
130	117, 129	eleqtrrd 2704	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (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 ∈ (Vtx‘𝐻) → ((VtxDeg‘𝐻)‘𝑁) = ((#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
134	130, 133	syl 17	. . 3 ⊢ (𝜑 → ((VtxDeg‘𝐻)‘𝑁) = ((#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}})))
135	128, 134	oveq12d 6668	. 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝑁) +_𝑒 ((VtxDeg‘𝐻)‘𝑁)) = (((#‘{𝑥 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) = {𝑁}})) +_𝑒 ((#‘{𝑥 ∈ dom 𝐽 ∣ 𝑁 ∈ (𝐽‘𝑥)}) +_𝑒 (#‘{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) = {𝑁}}))))
136	116, 124, 135	3eqtr4d 2666	1 ⊢ (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +_𝑒 ((VtxDeg‘𝐻)‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 {crab 2916 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 dom cdm 5114 Fun wfun 5882 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ℕ₀^*cxnn0 11363 +_𝑒 cxad 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

W3C validator