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Theorem rusgra0edg 26482
 Description: Special case for graphs without edges: There are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
Hypotheses
Ref Expression
rusgranumwlk.w 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
rusgranumwlk.l 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
Assertion
Ref Expression
rusgra0edg ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)
Distinct variable groups:   𝐸,𝑐,𝑛   𝑁,𝑐,𝑛   𝑉,𝑐,𝑛   𝑣,𝑁,𝑤   𝑃,𝑛,𝑣,𝑤   𝑣,𝑉   𝑛,𝑊,𝑣,𝑤   𝑤,𝑉,𝑐   𝑣,𝐸,𝑤
Allowed substitution hints:   𝑃(𝑐)   𝐿(𝑤,𝑣,𝑛,𝑐)   𝑊(𝑐)

Proof of Theorem rusgra0edg
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 rusisusgra 26458 . . 3 (⟨𝑉, 𝐸⟩ RegUSGrph 0 → 𝑉 USGrph 𝐸)
2 id 22 . . 3 (𝑃𝑉𝑃𝑉)
3 nnnn0 11176 . . 3 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
4 rusgranumwlk.w . . . 4 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
5 rusgranumwlk.l . . . 4 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
64, 5rusgranumwlklem4 26479 . . 3 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
71, 2, 3, 6syl3an 1360 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
8 df-rab 2905 . . . . 5 {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∣ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃)}
9 usgrav 25867 . . . . . . . . . . . . . 14 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
101, 9syl 17 . . . . . . . . . . . . 13 (⟨𝑉, 𝐸⟩ RegUSGrph 0 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
1110, 3anim12i 588 . . . . . . . . . . . 12 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑁 ∈ ℕ) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
12113adant2 1073 . . . . . . . . . . 11 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
13 df-3an 1033 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
1412, 13sylibr 223 . . . . . . . . . 10 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
15 iswwlkn 26212 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
16 iswwlk 26211 . . . . . . . . . . . . 13 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑤 ∈ (𝑉 WWalks 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
17163adant3 1074 . . . . . . . . . . . 12 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ (𝑉 WWalks 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
1817anbi1d 737 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
1915, 18bitrd 267 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
2014, 19syl 17 . . . . . . . . 9 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
2120anbi1d 737 . . . . . . . 8 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃)))
22 oveq1 6556 . . . . . . . . . . . . . . . . 17 ((#‘𝑤) = (𝑁 + 1) → ((#‘𝑤) − 1) = ((𝑁 + 1) − 1))
23 nncn 10905 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
24 1cnd 9935 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 1 ∈ ℂ)
2523, 24pncand 10272 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
26253ad2ant3 1077 . . . . . . . . . . . . . . . . 17 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → ((𝑁 + 1) − 1) = 𝑁)
2722, 26sylan9eqr 2666 . . . . . . . . . . . . . . . 16 (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → ((#‘𝑤) − 1) = 𝑁)
2827oveq2d 6565 . . . . . . . . . . . . . . 15 (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → (0..^((#‘𝑤) − 1)) = (0..^𝑁))
2928raleqdv 3121 . . . . . . . . . . . . . 14 (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))
30 rusgrargra 26457 . . . . . . . . . . . . . . . . 17 (⟨𝑉, 𝐸⟩ RegUSGrph 0 → ⟨𝑉, 𝐸⟩ RegGrph 0)
31 0eusgraiff0rgra 26466 . . . . . . . . . . . . . . . . . 18 (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegGrph 0 ↔ 𝐸 = ∅))
32 rneq 5272 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐸 = ∅ → ran 𝐸 = ran ∅)
33 rn0 5298 . . . . . . . . . . . . . . . . . . . . . . . . 25 ran ∅ = ∅
3432, 33syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐸 = ∅ → ran 𝐸 = ∅)
3534eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐸 = ∅ → ({(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ∅))
36 noel 3878 . . . . . . . . . . . . . . . . . . . . . . . 24 ¬ {(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ∅
3736bifal 1488 . . . . . . . . . . . . . . . . . . . . . . 23 ({(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ∅ ↔ ⊥)
3835, 37syl6bb 275 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸 = ∅ → ({(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
3938adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸 = ∅ ∧ (𝑃𝑉𝑁 ∈ ℕ)) → ({(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
4039ralbidv 2969 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 = ∅ ∧ (𝑃𝑉𝑁 ∈ ℕ)) → (∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁)⊥))
41 fal 1482 . . . . . . . . . . . . . . . . . . . . . 22 ¬ ⊥
4241ralf0 4030 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑖 ∈ (0..^𝑁)⊥ ↔ (0..^𝑁) = ∅)
4342a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 = ∅ ∧ (𝑃𝑉𝑁 ∈ ℕ)) → (∀𝑖 ∈ (0..^𝑁)⊥ ↔ (0..^𝑁) = ∅))
44 0nn0 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℕ0
4544a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 0 ∈ ℕ0)
46 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ)
47 nngt0 10926 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 0 < 𝑁)
4845, 46, 473jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (0 ∈ ℕ0𝑁 ∈ ℕ ∧ 0 < 𝑁))
4948ad2antll 761 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐸 = ∅ ∧ (𝑃𝑉𝑁 ∈ ℕ)) → (0 ∈ ℕ0𝑁 ∈ ℕ ∧ 0 < 𝑁))
50 elfzo0 12376 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ∈ (0..^𝑁) ↔ (0 ∈ ℕ0𝑁 ∈ ℕ ∧ 0 < 𝑁))
5149, 50sylibr 223 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐸 = ∅ ∧ (𝑃𝑉𝑁 ∈ ℕ)) → 0 ∈ (0..^𝑁))
52 fzon0 12356 . . . . . . . . . . . . . . . . . . . . . . 23 ((0..^𝑁) ≠ ∅ ↔ 0 ∈ (0..^𝑁))
5351, 52sylibr 223 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸 = ∅ ∧ (𝑃𝑉𝑁 ∈ ℕ)) → (0..^𝑁) ≠ ∅)
5453neneqd 2787 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸 = ∅ ∧ (𝑃𝑉𝑁 ∈ ℕ)) → ¬ (0..^𝑁) = ∅)
55 nbfal 1486 . . . . . . . . . . . . . . . . . . . . 21 (¬ (0..^𝑁) = ∅ ↔ ((0..^𝑁) = ∅ ↔ ⊥))
5654, 55sylib 207 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 = ∅ ∧ (𝑃𝑉𝑁 ∈ ℕ)) → ((0..^𝑁) = ∅ ↔ ⊥))
5740, 43, 563bitrd 293 . . . . . . . . . . . . . . . . . . 19 ((𝐸 = ∅ ∧ (𝑃𝑉𝑁 ∈ ℕ)) → (∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
5857ex 449 . . . . . . . . . . . . . . . . . 18 (𝐸 = ∅ → ((𝑃𝑉𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥)))
5931, 58syl6bi 242 . . . . . . . . . . . . . . . . 17 (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegGrph 0 → ((𝑃𝑉𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))))
601, 30, 59sylc 63 . . . . . . . . . . . . . . . 16 (⟨𝑉, 𝐸⟩ RegUSGrph 0 → ((𝑃𝑉𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥)))
61603impib 1254 . . . . . . . . . . . . . . 15 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
6261adantr 480 . . . . . . . . . . . . . 14 (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
6329, 62bitrd 267 . . . . . . . . . . . . 13 (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
64633anbi3d 1397 . . . . . . . . . . . 12 (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ⊥)))
65 df-3an 1033 . . . . . . . . . . . . 13 ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ⊥) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ⊥))
66 ancom 465 . . . . . . . . . . . . 13 (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ⊥) ↔ (⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)))
6765, 66bitri 263 . . . . . . . . . . . 12 ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ⊥) ↔ (⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)))
6864, 67syl6bb 275 . . . . . . . . . . 11 (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉))))
6968ex 449 . . . . . . . . . 10 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → ((#‘𝑤) = (𝑁 + 1) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)))))
7069pm5.32rd 670 . . . . . . . . 9 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ↔ ((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1))))
7170anbi1d 737 . . . . . . . 8 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ (((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃)))
72 anass 679 . . . . . . . . . 10 ((((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃)))
73 anass 679 . . . . . . . . . . 11 (((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃)) ↔ (⊥ ∧ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃))))
7441intnanr 952 . . . . . . . . . . . 12 ¬ (⊥ ∧ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃)))
7574bifal 1488 . . . . . . . . . . 11 ((⊥ ∧ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃))) ↔ ⊥)
7673, 75bitri 263 . . . . . . . . . 10 (((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃)) ↔ ⊥)
7772, 76bitri 263 . . . . . . . . 9 ((((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ⊥)
7877a1i 11 . . . . . . . 8 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → ((((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ⊥))
7921, 71, 783bitrd 293 . . . . . . 7 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃) ↔ ⊥))
8079abbidv 2728 . . . . . 6 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → {𝑤 ∣ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃)} = {𝑤 ∣ ⊥})
8141abf 3930 . . . . . 6 {𝑤 ∣ ⊥} = ∅
8280, 81syl6eq 2660 . . . . 5 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → {𝑤 ∣ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃)} = ∅)
838, 82syl5eq 2656 . . . 4 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} = ∅)
8483fveq2d 6107 . . 3 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (#‘∅))
85 hash0 13019 . . 3 (#‘∅) = 0
8684, 85syl6eq 2660 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = 0)
877, 86eqtrd 2644 1 ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ⊥wfal 1480   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859   Walks cwalk 26026   WWalks cwwlk 26205   WWalksN cwwlkn 26206   RegGrph crgra 26449   RegUSGrph crusgra 26450 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-fal 1481  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-rmo 2904  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-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  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-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by: (None)
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