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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 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → (𝑉 VDeg 𝐸):𝑉⟶(ℕ0 ∪ {+∞}))

Proof of Theorem vdgrf
Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0addcl 11205 . . . . . . 7 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ ℕ0)
2 elun1 3742 . . . . . . 7 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ ℕ0 → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
31, 2syl 17 . . . . . 6 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
4 nn0re 11178 . . . . . . 7 ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 → (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℝ)
5 nn0re 11178 . . . . . . 7 ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0 → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℝ)
6 rexadd 11937 . . . . . . . 8 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℝ ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℝ) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})))
76eleq1d 2672 . . . . . . 7 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℝ ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℝ) → (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}) ↔ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
84, 5, 7syl2an 493 . . . . . 6 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) → (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}) ↔ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
93, 8mpbird 246 . . . . 5 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
109a1d 25 . . . 4 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) → (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
11 ianor 508 . . . . 5 (¬ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) ↔ (¬ (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0))
12 df-nel 2783 . . . . . . . . . 10 ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∉ ℕ0 ↔ ¬ (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0)
13 rabexg 4739 . . . . . . . . . . . . . 14 (𝐴𝑋 → {𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V)
14133ad2ant3 1077 . . . . . . . . . . . . 13 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → {𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V)
1514ad2antrl 760 . . . . . . . . . . . 12 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∉ ℕ0 ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → {𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V)
16 rabexg 4739 . . . . . . . . . . . . . 14 (𝐴𝑋 → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V)
17163ad2ant3 1077 . . . . . . . . . . . . 13 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V)
1817ad2antrl 760 . . . . . . . . . . . 12 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∉ ℕ0 ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V)
19 simpl 472 . . . . . . . . . . . 12 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∉ ℕ0 ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∉ ℕ0)
20 hashinfxadd 13035 . . . . . . . . . . . 12 (({𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V ∧ {𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V ∧ (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∉ ℕ0) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞)
2115, 18, 19, 20syl3anc 1318 . . . . . . . . . . 11 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∉ ℕ0 ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞)
2221ex 449 . . . . . . . . . 10 ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∉ ℕ0 → (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞))
2312, 22sylbir 224 . . . . . . . . 9 (¬ (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 → (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞))
24 df-nel 2783 . . . . . . . . . 10 ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0 ↔ ¬ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0)
2517ad2antrl 760 . . . . . . . . . . . 12 (((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0 ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → {𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V)
2614ad2antrl 760 . . . . . . . . . . . 12 (((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0 ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → {𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V)
27 simpl 472 . . . . . . . . . . . 12 (((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0 ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0)
28 hashxrcl 13010 . . . . . . . . . . . . . . . 16 ({𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V → (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℝ*)
29 hashxrcl 13010 . . . . . . . . . . . . . . . 16 ({𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V → (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℝ*)
3028, 29anim12ci 589 . . . . . . . . . . . . . . 15 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V ∧ {𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℝ* ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℝ*))
31303adant3 1074 . . . . . . . . . . . . . 14 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V ∧ {𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℝ* ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℝ*))
32 xaddcom 11945 . . . . . . . . . . . . . 14 (((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℝ* ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℝ*) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) +𝑒 (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)})))
3331, 32syl 17 . . . . . . . . . . . . 13 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V ∧ {𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) +𝑒 (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)})))
34 hashinfxadd 13035 . . . . . . . . . . . . 13 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V ∧ {𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0) → ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) +𝑒 (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)})) = +∞)
3533, 34eqtrd 2644 . . . . . . . . . . . 12 (({𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}} ∈ V ∧ {𝑥𝐴𝑢 ∈ (𝐸𝑥)} ∈ V ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞)
3625, 26, 27, 35syl3anc 1318 . . . . . . . . . . 11 (((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0 ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞)
3736ex 449 . . . . . . . . . 10 ((#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∉ ℕ0 → (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞))
3824, 37sylbir 224 . . . . . . . . 9 (¬ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0 → (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞))
3923, 38jaoi 393 . . . . . . . 8 ((¬ (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) → (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞))
4039imp 444 . . . . . . 7 (((¬ (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) = +∞)
41 pnfex 9972 . . . . . . . . . 10 +∞ ∈ V
4241snid 4155 . . . . . . . . 9 +∞ ∈ {+∞}
4342olci 405 . . . . . . . 8 (+∞ ∈ ℕ0 ∨ +∞ ∈ {+∞})
44 elun 3715 . . . . . . . 8 (+∞ ∈ (ℕ0 ∪ {+∞}) ↔ (+∞ ∈ ℕ0 ∨ +∞ ∈ {+∞}))
4543, 44mpbir 220 . . . . . . 7 +∞ ∈ (ℕ0 ∪ {+∞})
4640, 45syl6eqel 2696 . . . . . 6 (((¬ (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) ∧ ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉)) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
4746ex 449 . . . . 5 ((¬ (#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∨ ¬ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) → (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
4811, 47sylbi 206 . . . 4 (¬ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) ∈ ℕ0 ∧ (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}) ∈ ℕ0) → (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞})))
4910, 48pm2.61i 175 . . 3 (((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑢𝑉) → ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})) ∈ (ℕ0 ∪ {+∞}))
50 eqid 2610 . . 3 (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}})))
5149, 50fmptd 6292 . 2 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))):𝑉⟶(ℕ0 ∪ {+∞}))
52 vdgrfval 26422 . . 3 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → (𝑉 VDeg 𝐸) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))))
5352feq1d 5943 . 2 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → ((𝑉 VDeg 𝐸):𝑉⟶(ℕ0 ∪ {+∞}) ↔ (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))):𝑉⟶(ℕ0 ∪ {+∞})))
5451, 53mpbird 246 1 ((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → (𝑉 VDeg 𝐸):𝑉⟶(ℕ0 ∪ {+∞}))
 Colors of variables: wff setvar class 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
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