Theorem vdgrf 26425

Description: The vertex degree function is a function from vertices to nonnegative integers or plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)

Assertion

Ref	Expression
vdgrf	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑉 VDeg 𝐸):𝑉⟶(ℕ0 ∪ {+∞}))

Proof of Theorem vdgrf

Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0addcl 11205	. . . . . . 7 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ ℕ0)
2		elun1 3742	. . . . . . 7 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ ℕ0 → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
3	1, 2	syl 17	. . . . . 6 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
4		nn0re 11178	. . . . . . 7 ⊢ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 → (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℝ)
5		nn0re 11178	. . . . . . 7 ⊢ ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0 → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℝ)
6		rexadd 11937	. . . . . . . 8 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℝ ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℝ) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})))
7	6	eleq1d 2672	. . . . . . 7 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℝ ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℝ) → (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}) ↔ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
8	4, 5, 7	syl2an 493	. . . . . 6 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) → (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}) ↔ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
9	3, 8	mpbird 246	. . . . 5 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
10	9	a1d 25	. . . 4 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) → (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
11		ianor 508	. . . . 5 ⊢ (¬ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) ↔ (¬ (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0))
12		df-nel 2783	. . . . . . . . . 10 ⊢ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∉ ℕ0 ↔ ¬ (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0)
13		rabexg 4739	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑋 → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V)
14	13	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V)
15	14	ad2antrl 760	. . . . . . . . . . . 12 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∉ ℕ0 ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V)
16		rabexg 4739	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑋 → {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V)
17	16	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V)
18	17	ad2antrl 760	. . . . . . . . . . . 12 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∉ ℕ0 ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V)
19		simpl 472	. . . . . . . . . . . 12 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∉ ℕ0 ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∉ ℕ0)
20		hashinfxadd 13035	. . . . . . . . . . . 12 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V ∧ {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V ∧ (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∉ ℕ0) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞)
21	15, 18, 19, 20	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∉ ℕ0 ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞)
22	21	ex 449	. . . . . . . . . 10 ⊢ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∉ ℕ0 → (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞))
23	12, 22	sylbir 224	. . . . . . . . 9 ⊢ (¬ (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 → (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞))
24		df-nel 2783	. . . . . . . . . 10 ⊢ ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0 ↔ ¬ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0)
25	17	ad2antrl 760	. . . . . . . . . . . 12 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0 ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V)
26	14	ad2antrl 760	. . . . . . . . . . . 12 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0 ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V)
27		simpl 472	. . . . . . . . . . . 12 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0 ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0)
28		hashxrcl 13010	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℝ*)
29		hashxrcl 13010	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V → (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℝ*)
30	28, 29	anim12ci 589	. . . . . . . . . . . . . . 15 ⊢ (({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V ∧ {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℝ* ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℝ*))
31	30	3adant3 1074	. . . . . . . . . . . . . 14 ⊢ (({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V ∧ {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℝ* ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℝ*))
32		xaddcom 11945	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℝ* ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℝ*) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)})))
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V ∧ {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)})))
34		hashinfxadd 13035	. . . . . . . . . . . . 13 ⊢ (({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V ∧ {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0) → ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)})) = +∞)
35	33, 34	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}} ∈ V ∧ {𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)} ∈ V ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞)
36	25, 26, 27, 35	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0 ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞)
37	36	ex 449	. . . . . . . . . 10 ⊢ ((#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∉ ℕ0 → (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞))
38	24, 37	sylbir 224	. . . . . . . . 9 ⊢ (¬ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0 → (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞))
39	23, 38	jaoi 393	. . . . . . . 8 ⊢ ((¬ (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) → (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞))
40	39	imp 444	. . . . . . 7 ⊢ (((¬ (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) = +∞)
41		pnfex 9972	. . . . . . . . . 10 ⊢ +∞ ∈ V
42	41	snid 4155	. . . . . . . . 9 ⊢ +∞ ∈ {+∞}
43	42	olci 405	. . . . . . . 8 ⊢ (+∞ ∈ ℕ0 ∨ +∞ ∈ {+∞})
44		elun 3715	. . . . . . . 8 ⊢ (+∞ ∈ (ℕ0 ∪ {+∞}) ↔ (+∞ ∈ ℕ0 ∨ +∞ ∈ {+∞}))
45	43, 44	mpbir 220	. . . . . . 7 ⊢ +∞ ∈ (ℕ0 ∪ {+∞})
46	40, 45	syl6eqel 2696	. . . . . 6 ⊢ (((¬ (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) ∧ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉)) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
47	46	ex 449	. . . . 5 ⊢ ((¬ (#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) → (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
48	11, 47	sylbi 206	. . . 4 ⊢ (¬ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}) ∈ ℕ0) → (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
49	10, 48	pm2.61i 175	. . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑢 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
50		eqid 2610	. . 3 ⊢ (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))) = (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}})))
51	49, 50	fmptd 6292	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))):𝑉⟶(ℕ0 ∪ {+∞}))
52		vdgrfval 26422	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑉 VDeg 𝐸) = (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))))
53	52	feq1d 5943	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → ((𝑉 VDeg 𝐸):𝑉⟶(ℕ0 ∪ {+∞}) ↔ (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))):𝑉⟶(ℕ0 ∪ {+∞})))
54	51, 53	mpbird 246	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑉 VDeg 𝐸):𝑉⟶(ℕ0 ∪ {+∞}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173   ∪ cun 3538  {csn 4125   ↦ cmpt 4643   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814   + caddc 9818  +∞cpnf 9950  ℝ^*cxr 9952  ℕ₀cn0 11169   +_𝑒 cxad 11820  #chash 12979   VDeg cvdg 26420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-hash 12980  df-vdgr 26421

This theorem is referenced by:  vdgrnn0pnf  26436  vdgfrgragt2  26554