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Theorem rusgrnumwwlkl1 41279
Description: In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.)
Hypothesis
Ref Expression
rusgrnumwwlkl1.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
rusgrnumwwlkl1 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ (1 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 𝐾)
Distinct variable groups:   𝑤,𝐺   𝑤,𝐾   𝑤,𝑃   𝑤,𝑉

Proof of Theorem rusgrnumwwlkl1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 1nn0 11061 . . . . . . . . 9 1 ∈ ℕ0
2 iswwlksn 41148 . . . . . . . . 9 (1 ∈ ℕ0 → (𝑤 ∈ (1 WWalkSN 𝐺) ↔ (𝑤 ∈ (WWalkS‘𝐺) ∧ (#‘𝑤) = (1 + 1))))
31, 2ax-mp 5 . . . . . . . 8 (𝑤 ∈ (1 WWalkSN 𝐺) ↔ (𝑤 ∈ (WWalkS‘𝐺) ∧ (#‘𝑤) = (1 + 1)))
4 rusgrnumwwlkl1.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
5 eqid 2514 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
64, 5iswwlks 41146 . . . . . . . . 9 (𝑤 ∈ (WWalkS‘𝐺) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
76anbi1i 726 . . . . . . . 8 ((𝑤 ∈ (WWalkS‘𝐺) ∧ (#‘𝑤) = (1 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = (1 + 1)))
83, 7bitri 262 . . . . . . 7 (𝑤 ∈ (1 WWalkSN 𝐺) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = (1 + 1)))
98a1i 11 . . . . . 6 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (𝑤 ∈ (1 WWalkSN 𝐺) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = (1 + 1))))
109anbi1d 736 . . . . 5 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → ((𝑤 ∈ (1 WWalkSN 𝐺) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃)))
11 1p1e2 10887 . . . . . . . . . . 11 (1 + 1) = 2
1211eqeq2i 2526 . . . . . . . . . 10 ((#‘𝑤) = (1 + 1) ↔ (#‘𝑤) = 2)
1312a1i 11 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → ((#‘𝑤) = (1 + 1) ↔ (#‘𝑤) = 2))
1413anbi2d 735 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = (1 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = 2)))
15 3anass 1034 . . . . . . . . . . . 12 ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
1615a1i 11 . . . . . . . . . . 11 (((𝐺 RegUSGraph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))))
17 fveq2 5986 . . . . . . . . . . . . . . . 16 (𝑤 = ∅ → (#‘𝑤) = (#‘∅))
18 hash0 12881 . . . . . . . . . . . . . . . 16 (#‘∅) = 0
1917, 18syl6eq 2564 . . . . . . . . . . . . . . 15 (𝑤 = ∅ → (#‘𝑤) = 0)
20 2ne0 10866 . . . . . . . . . . . . . . . . 17 2 ≠ 0
2120nesymi 2743 . . . . . . . . . . . . . . . 16 ¬ 0 = 2
22 eqeq1 2518 . . . . . . . . . . . . . . . 16 ((#‘𝑤) = 0 → ((#‘𝑤) = 2 ↔ 0 = 2))
2321, 22mtbiri 315 . . . . . . . . . . . . . . 15 ((#‘𝑤) = 0 → ¬ (#‘𝑤) = 2)
2419, 23syl 17 . . . . . . . . . . . . . 14 (𝑤 = ∅ → ¬ (#‘𝑤) = 2)
2524necon2ai 2715 . . . . . . . . . . . . 13 ((#‘𝑤) = 2 → 𝑤 ≠ ∅)
2625adantl 480 . . . . . . . . . . . 12 (((𝐺 RegUSGraph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → 𝑤 ≠ ∅)
2726biantrurd 527 . . . . . . . . . . 11 (((𝐺 RegUSGraph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))))
28 oveq1 6432 . . . . . . . . . . . . . . . . 17 ((#‘𝑤) = 2 → ((#‘𝑤) − 1) = (2 − 1))
29 2m1e1 10888 . . . . . . . . . . . . . . . . 17 (2 − 1) = 1
3028, 29syl6eq 2564 . . . . . . . . . . . . . . . 16 ((#‘𝑤) = 2 → ((#‘𝑤) − 1) = 1)
3130oveq2d 6441 . . . . . . . . . . . . . . 15 ((#‘𝑤) = 2 → (0..^((#‘𝑤) − 1)) = (0..^1))
3231adantl 480 . . . . . . . . . . . . . 14 (((𝐺 RegUSGraph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → (0..^((#‘𝑤) − 1)) = (0..^1))
3332raleqdv 3025 . . . . . . . . . . . . 13 (((𝐺 RegUSGraph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^1){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
34 fzo01 12282 . . . . . . . . . . . . . . 15 (0..^1) = {0}
3534raleqi 3023 . . . . . . . . . . . . . 14 (∀𝑖 ∈ (0..^1){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ {0} {(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺))
36 c0ex 9787 . . . . . . . . . . . . . . 15 0 ∈ V
37 fveq2 5986 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑤𝑖) = (𝑤‘0))
38 oveq1 6432 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
39 0p1e1 10885 . . . . . . . . . . . . . . . . . . 19 (0 + 1) = 1
4038, 39syl6eq 2564 . . . . . . . . . . . . . . . . . 18 (𝑖 = 0 → (𝑖 + 1) = 1)
4140fveq2d 5990 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑤‘(𝑖 + 1)) = (𝑤‘1))
4237, 41preq12d 4123 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → {(𝑤𝑖), (𝑤‘(𝑖 + 1))} = {(𝑤‘0), (𝑤‘1)})
4342eleq1d 2576 . . . . . . . . . . . . . . 15 (𝑖 = 0 → ({(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
4436, 43ralsn 4072 . . . . . . . . . . . . . 14 (∀𝑖 ∈ {0} {(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))
4535, 44bitri 262 . . . . . . . . . . . . 13 (∀𝑖 ∈ (0..^1){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))
4633, 45syl6bb 274 . . . . . . . . . . . 12 (((𝐺 RegUSGraph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
4746anbi2d 735 . . . . . . . . . . 11 (((𝐺 RegUSGraph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))))
4816, 27, 473bitr2d 294 . . . . . . . . . 10 (((𝐺 RegUSGraph 𝐾𝑃𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))))
4948ex 448 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → ((#‘𝑤) = 2 → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))))
5049pm5.32rd 669 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = 2) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = 2)))
5114, 50bitrd 266 . . . . . . 7 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = (1 + 1)) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = 2)))
5251anbi1d 736 . . . . . 6 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = 2) ∧ (𝑤‘0) = 𝑃)))
53 anass 678 . . . . . 6 ((((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = 2) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)))
5452, 53syl6bb 274 . . . . 5 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))))
55 anass 678 . . . . . . 7 (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))))
56 ancom 464 . . . . . . . . 9 (({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
57 df-3an 1032 . . . . . . . . 9 (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
5856, 57bitr4i 265 . . . . . . . 8 (({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
5958anbi2i 725 . . . . . . 7 ((𝑤 ∈ Word 𝑉 ∧ ({(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))))
6055, 59bitri 262 . . . . . 6 (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))))
6160a1i 11 . . . . 5 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))))
6210, 54, 613bitrd 292 . . . 4 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → ((𝑤 ∈ (1 WWalkSN 𝐺) ∧ (𝑤‘0) = 𝑃) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))))
6362rabbidva2 3066 . . 3 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → {𝑤 ∈ (1 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))})
6463fveq2d 5990 . 2 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ (1 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}))
654rusgrnumwrdl2 40893 . 2 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)
6664, 65eqtrd 2548 1 ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ (1 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1938  wne 2684  wral 2800  {crab 2804  c0 3777  {csn 4028  {cpr 4030   class class class wbr 4481  cfv 5689  (class class class)co 6425  0cc0 9689  1c1 9690   + caddc 9692  cmin 10015  2c2 10823  0cn0 11045  ..^cfzo 12199  #chash 12844  Word cword 13001  Vtxcvtx 40336  Edgcedga 40458   RegUSGraph crusgr 40863  WWalkScwwlks 41135   WWalkSN cwwlksn 41136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-cnex 9745  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-mulcom 9753  ax-addass 9754  ax-mulass 9755  ax-distr 9756  ax-i2m1 9757  ax-1ne0 9758  ax-1rid 9759  ax-rnegex 9760  ax-rrecex 9761  ax-cnre 9762  ax-pre-lttri 9763  ax-pre-lttrn 9764  ax-pre-ltadd 9765  ax-pre-mulgt0 9766
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6832  df-1st 6932  df-2nd 6933  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-1o 7321  df-2o 7322  df-oadd 7325  df-er 7503  df-map 7620  df-pm 7621  df-en 7716  df-dom 7717  df-sdom 7718  df-fin 7719  df-card 8522  df-cda 8747  df-pnf 9829  df-mnf 9830  df-xr 9831  df-ltxr 9832  df-le 9833  df-sub 10017  df-neg 10018  df-nn 10774  df-2 10832  df-n0 11046  df-z 11117  df-uz 11424  df-xadd 11685  df-fz 12063  df-fzo 12200  df-hash 12845  df-word 13009  df-xnn0 40305  df-uhgr 40387  df-ushgr 40388  df-upgr 40415  df-umgr 40416  df-edga 40459  df-uspgr 40487  df-usgr 40488  df-nbgr 40661  df-vtxdg 40789  df-rgr 40864  df-rusgr 40865  df-wwlks 41140  df-wwlksn 41141
This theorem is referenced by:  rusgrnumwwlkb1  41282
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