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Theorem crctcshwlkn0 26769
 Description: Cyclically shifting the indices of a circuit ⟨𝐹, 𝑃⟩ results in a walk ⟨𝐻, 𝑄⟩. (Contributed by AV, 10-Mar-2021.)
Hypotheses
Ref Expression
crctcsh.v 𝑉 = (Vtx‘𝐺)
crctcsh.i 𝐼 = (iEdg‘𝐺)
crctcsh.d (𝜑𝐹(Circuits‘𝐺)𝑃)
crctcsh.n 𝑁 = (#‘𝐹)
crctcsh.s (𝜑𝑆 ∈ (0..^𝑁))
crctcsh.h 𝐻 = (𝐹 cyclShift 𝑆)
crctcsh.q 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
Assertion
Ref Expression
crctcshwlkn0 ((𝜑𝑆 ≠ 0) → 𝐻(Walks‘𝐺)𝑄)
Distinct variable groups:   𝑥,𝑁   𝑥,𝑃   𝑥,𝑆   𝜑,𝑥   𝑥,𝐹   𝑥,𝐼   𝑥,𝑉
Allowed substitution hints:   𝑄(𝑥)   𝐺(𝑥)   𝐻(𝑥)

Proof of Theorem crctcshwlkn0
Dummy variables 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crctcsh.h . . . . 5 𝐻 = (𝐹 cyclShift 𝑆)
2 crctcsh.d . . . . . . 7 (𝜑𝐹(Circuits‘𝐺)𝑃)
3 crctiswlk 26747 . . . . . . 7 (𝐹(Circuits‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
4 crctcsh.i . . . . . . . 8 𝐼 = (iEdg‘𝐺)
54wlkf 26566 . . . . . . 7 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
62, 3, 53syl 18 . . . . . 6 (𝜑𝐹 ∈ Word dom 𝐼)
7 cshwcl 13590 . . . . . 6 (𝐹 ∈ Word dom 𝐼 → (𝐹 cyclShift 𝑆) ∈ Word dom 𝐼)
86, 7syl 17 . . . . 5 (𝜑 → (𝐹 cyclShift 𝑆) ∈ Word dom 𝐼)
91, 8syl5eqel 2734 . . . 4 (𝜑𝐻 ∈ Word dom 𝐼)
109adantr 480 . . 3 ((𝜑𝑆 ≠ 0) → 𝐻 ∈ Word dom 𝐼)
112, 3syl 17 . . . . . . . 8 (𝜑𝐹(Walks‘𝐺)𝑃)
12 crctcsh.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
1312wlkp 26568 . . . . . . . . 9 (𝐹(Walks‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
14 simpll 805 . . . . . . . . . . . 12 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ 𝑥 ≤ (𝑁𝑆)) → 𝑃:(0...(#‘𝐹))⟶𝑉)
15 crctcsh.s . . . . . . . . . . . . . . 15 (𝜑𝑆 ∈ (0..^𝑁))
16 elfznn0 12471 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℕ0)
1716adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑥 ∈ ℕ0)
18 elfzonn0 12552 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℕ0)
1918adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑆 ∈ ℕ0)
2017, 19nn0addcld 11393 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 + 𝑆) ∈ ℕ0)
2120adantr 480 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ∈ ℕ0)
22 crctcsh.n . . . . . . . . . . . . . . . . . 18 𝑁 = (#‘𝐹)
23 elfz3nn0 12472 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
2422, 23syl5eqelr 2735 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (0...𝑁) → (#‘𝐹) ∈ ℕ0)
2524ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (#‘𝐹) ∈ ℕ0)
26 elfzelz 12380 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℤ)
2726zred 11520 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℝ)
2827adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑥 ∈ ℝ)
29 elfzoelz 12509 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℤ)
3029zred 11520 . . . . . . . . . . . . . . . . . . . 20 (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℝ)
3130adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑆 ∈ ℝ)
32 elfzel2 12378 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
3332zred 11520 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (0...𝑁) → 𝑁 ∈ ℝ)
3433adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
35 leaddsub 10542 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑥 + 𝑆) ≤ 𝑁𝑥 ≤ (𝑁𝑆)))
3628, 31, 34, 35syl3anc 1366 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) ≤ 𝑁𝑥 ≤ (𝑁𝑆)))
3736biimpar 501 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ≤ 𝑁)
3837, 22syl6breq 4726 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ≤ (#‘𝐹))
3921, 25, 383jca 1261 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ (𝑥 + 𝑆) ≤ (#‘𝐹)))
4015, 39sylanl1 683 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ (𝑥 + 𝑆) ≤ (#‘𝐹)))
41 elfz2nn0 12469 . . . . . . . . . . . . . 14 ((𝑥 + 𝑆) ∈ (0...(#‘𝐹)) ↔ ((𝑥 + 𝑆) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ (𝑥 + 𝑆) ≤ (#‘𝐹)))
4240, 41sylibr 224 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ∈ (0...(#‘𝐹)))
4342adantll 750 . . . . . . . . . . . 12 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ∈ (0...(#‘𝐹)))
4414, 43ffvelrnd 6400 . . . . . . . . . . 11 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑃‘(𝑥 + 𝑆)) ∈ 𝑉)
45 simpll 805 . . . . . . . . . . . 12 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → 𝑃:(0...(#‘𝐹))⟶𝑉)
46 elfzoel2 12508 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
47 zaddcl 11455 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑥 + 𝑆) ∈ ℤ)
4847adantrr 753 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑥 + 𝑆) ∈ ℤ)
49 simprr 811 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
5048, 49zsubcld 11525 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℤ)
5150adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℤ)
52 zsubcl 11457 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑁𝑆) ∈ ℤ)
5352ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁𝑆) ∈ ℤ)
5453zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁𝑆) ∈ ℝ)
55 zre 11419 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
56 ltnle 10155 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁𝑆) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑁𝑆) < 𝑥 ↔ ¬ 𝑥 ≤ (𝑁𝑆)))
5754, 55, 56syl2anr 494 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁𝑆) < 𝑥 ↔ ¬ 𝑥 ≤ (𝑁𝑆)))
58 zre 11419 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
5958adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
60 zre 11419 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑆 ∈ ℤ → 𝑆 ∈ ℝ)
6160adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑆 ∈ ℝ)
6255adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑥 ∈ ℝ)
63 ltsubadd 10536 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑁𝑆) < 𝑥𝑁 < (𝑥 + 𝑆)))
6459, 61, 62, 63syl2an23an 1427 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁𝑆) < 𝑥𝑁 < (𝑥 + 𝑆)))
6559adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℝ)
6648zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑥 + 𝑆) ∈ ℝ)
6765, 66posdifd 10652 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 < (𝑥 + 𝑆) ↔ 0 < ((𝑥 + 𝑆) − 𝑁)))
68 0red 10079 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 0 ∈ ℝ)
6950zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℝ)
70 ltle 10164 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ ((𝑥 + 𝑆) − 𝑁) ∈ ℝ) → (0 < ((𝑥 + 𝑆) − 𝑁) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7168, 69, 70syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (0 < ((𝑥 + 𝑆) − 𝑁) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7267, 71sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 < (𝑥 + 𝑆) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7364, 72sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁𝑆) < 𝑥 → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7457, 73sylbird 250 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (¬ 𝑥 ≤ (𝑁𝑆) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7574imp 444 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → 0 ≤ ((𝑥 + 𝑆) − 𝑁))
7651, 75jca 553 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7776exp31 629 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℤ → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑥 ≤ (𝑁𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
7877, 26syl11 33 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (0...𝑁) → (¬ 𝑥 ≤ (𝑁𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
7929, 46, 78syl2anc 694 . . . . . . . . . . . . . . . . . 18 (𝑆 ∈ (0..^𝑁) → (𝑥 ∈ (0...𝑁) → (¬ 𝑥 ≤ (𝑁𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
8079imp31 447 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
81 elnn0z 11428 . . . . . . . . . . . . . . . . 17 (((𝑥 + 𝑆) − 𝑁) ∈ ℕ0 ↔ (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
8280, 81sylibr 224 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℕ0)
8324ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (#‘𝐹) ∈ ℕ0)
84 elfzo0 12548 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (0..^𝑁) ↔ (𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
85 elfz2nn0 12469 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (0...𝑁) ↔ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁))
86 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑆 ∈ ℕ0𝑆 ∈ ℝ)
87863ad2ant1 1102 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆 ∈ ℝ)
88 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℕ0𝑥 ∈ ℝ)
89883ad2ant1 1102 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁) → 𝑥 ∈ ℝ)
9087, 89anim12ci 590 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → (𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ))
91 nnre 11065 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
9291, 91jca 553 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
93923ad2ant2 1103 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
9493adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
9590, 94jca 553 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → ((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
96 simpr3 1089 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → 𝑥𝑁)
97 ltle 10164 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁𝑆𝑁))
9886, 91, 97syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ ℕ0𝑁 ∈ ℕ) → (𝑆 < 𝑁𝑆𝑁))
99983impia 1280 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆𝑁)
10099adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → 𝑆𝑁)
10195, 96, 100jca32 557 . . . . . . . . . . . . . . . . . . . . 21 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → (((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) ∧ (𝑥𝑁𝑆𝑁)))
10284, 85, 101syl2anb 495 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) ∧ (𝑥𝑁𝑆𝑁)))
103 le2add 10548 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) → ((𝑥𝑁𝑆𝑁) → (𝑥 + 𝑆) ≤ (𝑁 + 𝑁)))
104103imp 444 . . . . . . . . . . . . . . . . . . . 20 ((((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) ∧ (𝑥𝑁𝑆𝑁)) → (𝑥 + 𝑆) ≤ (𝑁 + 𝑁))
105102, 104syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 + 𝑆) ≤ (𝑁 + 𝑁))
10666, 65, 653jca 1261 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
107106ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ℤ → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
108107, 26syl11 33 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (0...𝑁) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
10929, 46, 108syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (0..^𝑁) → (𝑥 ∈ (0...𝑁) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
110109imp 444 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
111 lesubadd 10538 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝑥 + 𝑆) − 𝑁) ≤ 𝑁 ↔ (𝑥 + 𝑆) ≤ (𝑁 + 𝑁)))
112110, 111syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (((𝑥 + 𝑆) − 𝑁) ≤ 𝑁 ↔ (𝑥 + 𝑆) ≤ (𝑁 + 𝑁)))
113105, 112mpbird 247 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) − 𝑁) ≤ 𝑁)
114113adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ≤ 𝑁)
115114, 22syl6breq 4726 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹))
11682, 83, 1153jca 1261 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹)))
11715, 116sylanl1 683 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹)))
118 elfz2nn0 12469 . . . . . . . . . . . . . 14 (((𝑥 + 𝑆) − 𝑁) ∈ (0...(#‘𝐹)) ↔ (((𝑥 + 𝑆) − 𝑁) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹)))
119117, 118sylibr 224 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ (0...(#‘𝐹)))
120119adantll 750 . . . . . . . . . . . 12 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ (0...(#‘𝐹)))
12145, 120ffvelrnd 6400 . . . . . . . . . . 11 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (𝑃‘((𝑥 + 𝑆) − 𝑁)) ∈ 𝑉)
12244, 121ifclda 4153 . . . . . . . . . 10 ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)
123122exp32 630 . . . . . . . . 9 (𝑃:(0...(#‘𝐹))⟶𝑉 → (𝜑 → (𝑥 ∈ (0...𝑁) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)))
12413, 123syl 17 . . . . . . . 8 (𝐹(Walks‘𝐺)𝑃 → (𝜑 → (𝑥 ∈ (0...𝑁) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)))
12511, 124mpcom 38 . . . . . . 7 (𝜑 → (𝑥 ∈ (0...𝑁) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉))
126125imp 444 . . . . . 6 ((𝜑𝑥 ∈ (0...𝑁)) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)
127 crctcsh.q . . . . . 6 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
128126, 127fmptd 6425 . . . . 5 (𝜑𝑄:(0...𝑁)⟶𝑉)
12912, 4, 2, 22, 15, 1crctcshlem2 26766 . . . . . . 7 (𝜑 → (#‘𝐻) = 𝑁)
130129oveq2d 6706 . . . . . 6 (𝜑 → (0...(#‘𝐻)) = (0...𝑁))
131130feq2d 6069 . . . . 5 (𝜑 → (𝑄:(0...(#‘𝐻))⟶𝑉𝑄:(0...𝑁)⟶𝑉))
132128, 131mpbird 247 . . . 4 (𝜑𝑄:(0...(#‘𝐻))⟶𝑉)
133132adantr 480 . . 3 ((𝜑𝑆 ≠ 0) → 𝑄:(0...(#‘𝐻))⟶𝑉)
13412, 4wlkprop 26563 . . . . . 6 (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))))
1352, 3, 1343syl 18 . . . . 5 (𝜑 → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))))
136135adantr 480 . . . 4 ((𝜑𝑆 ≠ 0) → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))))
13722eqcomi 2660 . . . . . . . . . 10 (#‘𝐹) = 𝑁
138137oveq2i 6701 . . . . . . . . 9 (0..^(#‘𝐹)) = (0..^𝑁)
139138raleqi 3172 . . . . . . . 8 (∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) ↔ ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))
140 fzo1fzo0n0 12558 . . . . . . . . . . . . . . 15 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ (0..^𝑁) ∧ 𝑆 ≠ 0))
141140simplbi2 654 . . . . . . . . . . . . . 14 (𝑆 ∈ (0..^𝑁) → (𝑆 ≠ 0 → 𝑆 ∈ (1..^𝑁)))
14215, 141syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑆 ≠ 0 → 𝑆 ∈ (1..^𝑁)))
143142imp 444 . . . . . . . . . . . 12 ((𝜑𝑆 ≠ 0) → 𝑆 ∈ (1..^𝑁))
144143ad2antlr 763 . . . . . . . . . . 11 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))) → 𝑆 ∈ (1..^𝑁))
145 simplll 813 . . . . . . . . . . 11 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))) → 𝐹 ∈ Word dom 𝐼)
146 wkslem1 26559 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) ↔ if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
147146cbvralv 3201 . . . . . . . . . . . . 13 (∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) ↔ ∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
148147biimpi 206 . . . . . . . . . . . 12 (∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
149148adantl 481 . . . . . . . . . . 11 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
150 crctprop 26743 . . . . . . . . . . . . . 14 (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))
151137fveq2i 6232 . . . . . . . . . . . . . . . . . 18 (𝑃‘(#‘𝐹)) = (𝑃𝑁)
152151eqeq2i 2663 . . . . . . . . . . . . . . . . 17 ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃𝑁))
153152biimpi 206 . . . . . . . . . . . . . . . 16 ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑃‘0) = (𝑃𝑁))
154153eqcomd 2657 . . . . . . . . . . . . . . 15 ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑃𝑁) = (𝑃‘0))
155154adantl 481 . . . . . . . . . . . . . 14 ((𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑃𝑁) = (𝑃‘0))
1562, 150, 1553syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝑃𝑁) = (𝑃‘0))
157156ad2antrl 764 . . . . . . . . . . . 12 (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) → (𝑃𝑁) = (𝑃‘0))
158157adantr 480 . . . . . . . . . . 11 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))) → (𝑃𝑁) = (𝑃‘0))
159144, 127, 1, 22, 145, 149, 158crctcshwlkn0lem7 26764 . . . . . . . . . 10 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))) → ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
160129oveq2d 6706 . . . . . . . . . . . . 13 (𝜑 → (0..^(#‘𝐻)) = (0..^𝑁))
161160raleqdv 3174 . . . . . . . . . . . 12 (𝜑 → (∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
162161ad2antrl 764 . . . . . . . . . . 11 (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) → (∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
163162adantr 480 . . . . . . . . . 10 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))) → (∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
164159, 163mpbird 247 . . . . . . . . 9 ((((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
165164ex 449 . . . . . . . 8 (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) → (∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
166139, 165syl5bi 232 . . . . . . 7 (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) → (∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
167166ex 449 . . . . . 6 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) → ((𝜑𝑆 ≠ 0) → (∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))))
168167com23 86 . . . . 5 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) → (∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) → ((𝜑𝑆 ≠ 0) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))))
1691683impia 1280 . . . 4 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)))) → ((𝜑𝑆 ≠ 0) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
170136, 169mpcom 38 . . 3 ((𝜑𝑆 ≠ 0) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
17110, 133, 1703jca 1261 . 2 ((𝜑𝑆 ≠ 0) → (𝐻 ∈ Word dom 𝐼𝑄:(0...(#‘𝐻))⟶𝑉 ∧ ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
17212, 4, 2, 22, 15, 1, 127crctcshlem3 26767 . . . 4 (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
173172adantr 480 . . 3 ((𝜑𝑆 ≠ 0) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
17412, 4iswlk 26562 . . 3 ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V) → (𝐻(Walks‘𝐺)𝑄 ↔ (𝐻 ∈ Word dom 𝐼𝑄:(0...(#‘𝐻))⟶𝑉 ∧ ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))))
175173, 174syl 17 . 2 ((𝜑𝑆 ≠ 0) → (𝐻(Walks‘𝐺)𝑄 ↔ (𝐻 ∈ Word dom 𝐼𝑄:(0...(#‘𝐻))⟶𝑉 ∧ ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))))
176171, 175mpbird 247 1 ((𝜑𝑆 ≠ 0) → 𝐻(Walks‘𝐺)𝑄)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383  if-wif 1032   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  Vcvv 3231   ⊆ wss 3607  ifcif 4119  {csn 4210  {cpr 4212   class class class wbr 4685   ↦ cmpt 4762  dom cdm 5143  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112   ≤ cle 10113   − cmin 10304  ℕcn 11058  ℕ0cn0 11330  ℤcz 11415  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323   cyclShift ccsh 13580  Vtxcvtx 25919  iEdgciedg 25920  Walkscwlks 26548  Trailsctrls 26643  Circuitsccrcts 26735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-hash 13158  df-word 13331  df-concat 13333  df-substr 13335  df-csh 13581  df-wlks 26551  df-trls 26645  df-crcts 26737 This theorem is referenced by:  crctcshwlk  26770
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