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Theorem wwlknredwwlkn 26254
 Description: For each walk (as word) of length at least 1 there is a shorter walk (as word). (Contributed by Alexander van der Vekens, 22-Aug-2018.)
Assertion
Ref Expression
wwlknredwwlkn (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))
Distinct variable groups:   𝑦,𝐸   𝑦,𝑁   𝑦,𝑉   𝑦,𝑊

Proof of Theorem wwlknredwwlkn
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2611 . . 3 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = (𝑊 substr ⟨0, (𝑁 + 1)⟩))
2 wwlknimp 26215 . . . 4 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
3 simprl 790 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → 𝑊 ∈ Word 𝑉)
4 peano2nn0 11210 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
5 peano2nn0 11210 . . . . . . . . . . . . 13 ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ0)
64, 5syl 17 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ0)
7 id 22 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
8 nn0p1nn 11209 . . . . . . . . . . . . . 14 ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ)
94, 8syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ)
10 nn0re 11178 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
11 id 22 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ → 𝑁 ∈ ℝ)
12 peano2re 10088 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
13 peano2re 10088 . . . . . . . . . . . . . . . . 17 ((𝑁 + 1) ∈ ℝ → ((𝑁 + 1) + 1) ∈ ℝ)
1412, 13syl 17 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℝ → ((𝑁 + 1) + 1) ∈ ℝ)
1511, 12, 143jca 1235 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℝ → (𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ))
1610, 15syl 17 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ))
1710ltp1d 10833 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0𝑁 < (𝑁 + 1))
18 nn0re 11178 . . . . . . . . . . . . . . . 16 ((𝑁 + 1) ∈ ℕ0 → (𝑁 + 1) ∈ ℝ)
194, 18syl 17 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℝ)
2019ltp1d 10833 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0 → (𝑁 + 1) < ((𝑁 + 1) + 1))
21 lttr 9993 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ) → ((𝑁 < (𝑁 + 1) ∧ (𝑁 + 1) < ((𝑁 + 1) + 1)) → 𝑁 < ((𝑁 + 1) + 1)))
2221imp 444 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ) ∧ (𝑁 < (𝑁 + 1) ∧ (𝑁 + 1) < ((𝑁 + 1) + 1))) → 𝑁 < ((𝑁 + 1) + 1))
2316, 17, 20, 22syl12anc 1316 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 < ((𝑁 + 1) + 1))
24 elfzo0 12376 . . . . . . . . . . . . 13 (𝑁 ∈ (0..^((𝑁 + 1) + 1)) ↔ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) + 1) ∈ ℕ ∧ 𝑁 < ((𝑁 + 1) + 1)))
257, 9, 23, 24syl3anbrc 1239 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝑁 ∈ (0..^((𝑁 + 1) + 1)))
26 fargshiftlem 26162 . . . . . . . . . . . 12 ((((𝑁 + 1) + 1) ∈ ℕ0𝑁 ∈ (0..^((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
276, 25, 26syl2anc 691 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
2827adantr 480 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
29 oveq2 6557 . . . . . . . . . . . . 13 ((#‘𝑊) = ((𝑁 + 1) + 1) → (1...(#‘𝑊)) = (1...((𝑁 + 1) + 1)))
3029eleq2d 2673 . . . . . . . . . . . 12 ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ∈ (1...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
3130adantl 481 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑁 + 1) ∈ (1...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
3231adantl 481 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → ((𝑁 + 1) ∈ (1...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
3328, 32mpbird 246 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...(#‘𝑊)))
343, 33jca 553 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1))) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
35343adantr3 1215 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
36 swrd0fvlsw 13295 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))) → ( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑊‘((𝑁 + 1) − 1)))
3735, 36syl 17 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → ( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑊‘((𝑁 + 1) − 1)))
38 lsw 13204 . . . . . . . 8 (𝑊 ∈ Word 𝑉 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
39383ad2ant1 1075 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
4039adantl 481 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
4137, 40preq12d 4220 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} = {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘((#‘𝑊) − 1))})
42 oveq1 6556 . . . . . . . . . . 11 ((#‘𝑊) = ((𝑁 + 1) + 1) → ((#‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
43423ad2ant2 1076 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → ((#‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
4443adantl 481 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → ((#‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
4544fveq2d 6107 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1)))
4645preq2d 4219 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘((#‘𝑊) − 1))} = {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘(((𝑁 + 1) + 1) − 1))})
47 nn0cn 11179 . . . . . . . . . . 11 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
48 1cnd 9935 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4947, 48pncand 10272 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → ((𝑁 + 1) − 1) = 𝑁)
5049fveq2d 6107 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑊‘((𝑁 + 1) − 1)) = (𝑊𝑁))
514nn0cnd 11230 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
5251, 48pncand 10272 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
5352fveq2d 6107 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑊‘(((𝑁 + 1) + 1) − 1)) = (𝑊‘(𝑁 + 1)))
5450, 53preq12d 4220 . . . . . . . 8 (𝑁 ∈ ℕ0 → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘(((𝑁 + 1) + 1) − 1))} = {(𝑊𝑁), (𝑊‘(𝑁 + 1))})
5554adantr 480 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘(((𝑁 + 1) + 1) − 1))} = {(𝑊𝑁), (𝑊‘(𝑁 + 1))})
5646, 55eqtrd 2644 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘((#‘𝑊) − 1))} = {(𝑊𝑁), (𝑊‘(𝑁 + 1))})
57 fzonn0p1 12411 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ (0..^(𝑁 + 1)))
58 fveq2 6103 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → (𝑊𝑖) = (𝑊𝑁))
59 oveq1 6556 . . . . . . . . . . . . . 14 (𝑖 = 𝑁 → (𝑖 + 1) = (𝑁 + 1))
6059fveq2d 6107 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → (𝑊‘(𝑖 + 1)) = (𝑊‘(𝑁 + 1)))
6158, 60preq12d 4220 . . . . . . . . . . . 12 (𝑖 = 𝑁 → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊𝑁), (𝑊‘(𝑁 + 1))})
6261eleq1d 2672 . . . . . . . . . . 11 (𝑖 = 𝑁 → ({(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑊𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸))
6362rspcv 3278 . . . . . . . . . 10 (𝑁 ∈ (0..^(𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑊𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸))
6457, 63syl 17 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑊𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸))
6564com12 32 . . . . . . . 8 (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑁 ∈ ℕ0 → {(𝑊𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸))
66653ad2ant3 1077 . . . . . . 7 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ0 → {(𝑊𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸))
6766impcom 445 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → {(𝑊𝑁), (𝑊‘(𝑁 + 1))} ∈ ran 𝐸)
6856, 67eqeltrd 2688 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘((#‘𝑊) − 1))} ∈ ran 𝐸)
6941, 68eqeltrd 2688 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)) → {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} ∈ ran 𝐸)
702, 69sylan2 490 . . 3 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} ∈ ran 𝐸)
71 wwlknred 26251 . . . . 5 (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
7271imp 444 . . . 4 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
73 eqeq2 2621 . . . . . 6 (𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ↔ (𝑊 substr ⟨0, (𝑁 + 1)⟩) = (𝑊 substr ⟨0, (𝑁 + 1)⟩)))
74 fveq2 6103 . . . . . . . 8 (𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → ( lastS ‘𝑦) = ( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)))
7574preq1d 4218 . . . . . . 7 (𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → {( lastS ‘𝑦), ( lastS ‘𝑊)} = {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)})
7675eleq1d 2672 . . . . . 6 (𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → ({( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸 ↔ {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} ∈ ran 𝐸))
7773, 76anbi12d 743 . . . . 5 (𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∧ {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} ∈ ran 𝐸)))
7877adantl 481 . . . 4 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ 𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩)) → (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∧ {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} ∈ ran 𝐸)))
7972, 78rspcedv 3286 . . 3 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∧ {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} ∈ ran 𝐸) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))
801, 70, 79mp2and 711 . 2 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸))
8180ex 449 1 (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150   WWalksN cwwlkn 26206 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-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  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-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-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-substr 13158  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  wwlknredwwlkn0  26255
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