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Theorem crctcsh1wlkn0 41022
Description: Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a 1-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
crctcsh1wlkn0 ((𝜑𝑆 ≠ 0) → 𝐻(1Walks‘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 crctcsh1wlkn0
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 crctis1wlk 41000 . . . . . . 7 (𝐹(CircuitS‘𝐺)𝑃𝐹(1Walks‘𝐺)𝑃)
4 crctcsh.i . . . . . . . 8 𝐼 = (iEdg‘𝐺)
541wlkf 40817 . . . . . . 7 (𝐹(1Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
62, 3, 53syl 18 . . . . . 6 (𝜑𝐹 ∈ Word dom 𝐼)
7 cshwcl 13337 . . . . . 6 (𝐹 ∈ Word dom 𝐼 → (𝐹 cyclShift 𝑆) ∈ Word dom 𝐼)
86, 7syl 17 . . . . 5 (𝜑 → (𝐹 cyclShift 𝑆) ∈ Word dom 𝐼)
91, 8syl5eqel 2687 . . . 4 (𝜑𝐻 ∈ Word dom 𝐼)
109adantr 479 . . 3 ((𝜑𝑆 ≠ 0) → 𝐻 ∈ Word dom 𝐼)
112, 3syl 17 . . . . . . . 8 (𝜑𝐹(1Walks‘𝐺)𝑃)
12 crctcsh.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
13121wlkp 40819 . . . . . . . . 9 (𝐹(1Walks‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
14 simpll 785 . . . . . . . . . . . 12 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ 𝑥 ≤ (𝑁𝑆)) → 𝑃:(0...(#‘𝐹))⟶𝑉)
15 crctcsh.s . . . . . . . . . . . . . . 15 (𝜑𝑆 ∈ (0..^𝑁))
16 elfznn0 12253 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℕ0)
1716adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑥 ∈ ℕ0)
18 elfzonn0 12331 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℕ0)
1918adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑆 ∈ ℕ0)
2017, 19nn0addcld 11198 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 + 𝑆) ∈ ℕ0)
2120adantr 479 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ∈ ℕ0)
22 crctcsh.n . . . . . . . . . . . . . . . . . 18 𝑁 = (#‘𝐹)
23 elfz3nn0 12254 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
2422, 23syl5eqelr 2688 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (0...𝑁) → (#‘𝐹) ∈ ℕ0)
2524ad2antlr 758 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (#‘𝐹) ∈ ℕ0)
26 elfzelz 12164 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℤ)
2726zred 11310 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℝ)
2827adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑥 ∈ ℝ)
29 elfzoelz 12290 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℤ)
3029zred 11310 . . . . . . . . . . . . . . . . . . . 20 (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℝ)
3130adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑆 ∈ ℝ)
32 elfzel2 12162 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
3332zred 11310 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (0...𝑁) → 𝑁 ∈ ℝ)
3433adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
35 leaddsub 10349 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑥 + 𝑆) ≤ 𝑁𝑥 ≤ (𝑁𝑆)))
3628, 31, 34, 35syl3anc 1317 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) ≤ 𝑁𝑥 ≤ (𝑁𝑆)))
3736biimpar 500 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ≤ 𝑁)
3837, 22syl6breq 4614 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ≤ (#‘𝐹))
3921, 25, 383jca 1234 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ (𝑥 + 𝑆) ≤ (#‘𝐹)))
4015, 39sylanl1 679 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ (𝑥 + 𝑆) ≤ (#‘𝐹)))
41 elfz2nn0 12251 . . . . . . . . . . . . . 14 ((𝑥 + 𝑆) ∈ (0...(#‘𝐹)) ↔ ((𝑥 + 𝑆) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ (𝑥 + 𝑆) ≤ (#‘𝐹)))
4240, 41sylibr 222 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ∈ (0...(#‘𝐹)))
4342adantll 745 . . . . . . . . . . . 12 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑥 + 𝑆) ∈ (0...(#‘𝐹)))
4414, 43ffvelrnd 6249 . . . . . . . . . . 11 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ 𝑥 ≤ (𝑁𝑆)) → (𝑃‘(𝑥 + 𝑆)) ∈ 𝑉)
45 simpll 785 . . . . . . . . . . . 12 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → 𝑃:(0...(#‘𝐹))⟶𝑉)
46 elfzoel2 12289 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
47 zaddcl 11246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑥 + 𝑆) ∈ ℤ)
4847adantrr 748 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑥 + 𝑆) ∈ ℤ)
49 simprr 791 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
5048, 49zsubcld 11315 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℤ)
5150adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℤ)
52 zsubcl 11248 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑁𝑆) ∈ ℤ)
5352ancoms 467 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁𝑆) ∈ ℤ)
5453zred 11310 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁𝑆) ∈ ℝ)
55 zre 11210 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
56 ltnle 9964 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑁𝑆) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑁𝑆) < 𝑥 ↔ ¬ 𝑥 ≤ (𝑁𝑆)))
5754, 55, 56syl2anr 493 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁𝑆) < 𝑥 ↔ ¬ 𝑥 ≤ (𝑁𝑆)))
58 zre 11210 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
5958adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
60 zre 11210 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑆 ∈ ℤ → 𝑆 ∈ ℝ)
6160adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑆 ∈ ℝ)
6255adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑥 ∈ ℝ)
63 ltsubadd 10343 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑁𝑆) < 𝑥𝑁 < (𝑥 + 𝑆)))
6459, 61, 62, 63syl2an23an 1378 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁𝑆) < 𝑥𝑁 < (𝑥 + 𝑆)))
6559adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℝ)
6648zred 11310 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑥 + 𝑆) ∈ ℝ)
6765, 66posdifd 10459 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 < (𝑥 + 𝑆) ↔ 0 < ((𝑥 + 𝑆) − 𝑁)))
68 0red 9893 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 0 ∈ ℝ)
6950zred 11310 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℝ)
70 ltle 9973 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((0 ∈ ℝ ∧ ((𝑥 + 𝑆) − 𝑁) ∈ ℝ) → (0 < ((𝑥 + 𝑆) − 𝑁) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7168, 69, 70syl2anc 690 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (0 < ((𝑥 + 𝑆) − 𝑁) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7267, 71sylbid 228 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 < (𝑥 + 𝑆) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7364, 72sylbid 228 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁𝑆) < 𝑥 → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7457, 73sylbird 248 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (¬ 𝑥 ≤ (𝑁𝑆) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7574imp 443 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → 0 ≤ ((𝑥 + 𝑆) − 𝑁))
7651, 75jca 552 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
7776exp31 627 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℤ → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑥 ≤ (𝑁𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
7826, 77syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (0...𝑁) → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑥 ≤ (𝑁𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
7978com12 32 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (0...𝑁) → (¬ 𝑥 ≤ (𝑁𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
8029, 46, 79syl2anc 690 . . . . . . . . . . . . . . . . . 18 (𝑆 ∈ (0..^𝑁) → (𝑥 ∈ (0...𝑁) → (¬ 𝑥 ≤ (𝑁𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
8180imp31 446 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
82 elnn0z 11219 . . . . . . . . . . . . . . . . 17 (((𝑥 + 𝑆) − 𝑁) ∈ ℕ0 ↔ (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
8381, 82sylibr 222 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℕ0)
8424ad2antlr 758 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (#‘𝐹) ∈ ℕ0)
85 elfzo0 12327 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (0..^𝑁) ↔ (𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
86 elfz2nn0 12251 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (0...𝑁) ↔ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁))
87 nn0re 11144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑆 ∈ ℕ0𝑆 ∈ ℝ)
88873ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆 ∈ ℝ)
89 nn0re 11144 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℕ0𝑥 ∈ ℝ)
90893ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁) → 𝑥 ∈ ℝ)
9188, 90anim12ci 588 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → (𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ))
92 nnre 10870 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
9392, 92jca 552 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
94933ad2ant2 1075 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
9594adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
9691, 95jca 552 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → ((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
97 simpr3 1061 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → 𝑥𝑁)
98 ltle 9973 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁𝑆𝑁))
9987, 92, 98syl2an 492 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ ℕ0𝑁 ∈ ℕ) → (𝑆 < 𝑁𝑆𝑁))
100993impia 1252 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆𝑁)
101100adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → 𝑆𝑁)
10296, 97, 101jca32 555 . . . . . . . . . . . . . . . . . . . . 21 (((𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ0𝑁 ∈ ℕ0𝑥𝑁)) → (((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) ∧ (𝑥𝑁𝑆𝑁)))
10385, 86, 102syl2anb 494 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) ∧ (𝑥𝑁𝑆𝑁)))
104 le2add 10355 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) → ((𝑥𝑁𝑆𝑁) → (𝑥 + 𝑆) ≤ (𝑁 + 𝑁)))
105104imp 443 . . . . . . . . . . . . . . . . . . . 20 ((((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) ∧ (𝑥𝑁𝑆𝑁)) → (𝑥 + 𝑆) ≤ (𝑁 + 𝑁))
106103, 105syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 + 𝑆) ≤ (𝑁 + 𝑁))
10766, 65, 653jca 1234 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
108107ex 448 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℤ → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
10926, 108syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ (0...𝑁) → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
110109com12 32 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (0...𝑁) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
11129, 46, 110syl2anc 690 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (0..^𝑁) → (𝑥 ∈ (0...𝑁) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
112111imp 443 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
113 lesubadd 10345 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝑥 + 𝑆) − 𝑁) ≤ 𝑁 ↔ (𝑥 + 𝑆) ≤ (𝑁 + 𝑁)))
114112, 113syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (((𝑥 + 𝑆) − 𝑁) ≤ 𝑁 ↔ (𝑥 + 𝑆) ≤ (𝑁 + 𝑁)))
115106, 114mpbird 245 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) − 𝑁) ≤ 𝑁)
116115adantr 479 . . . . . . . . . . . . . . . . 17 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ≤ 𝑁)
117116, 22syl6breq 4614 . . . . . . . . . . . . . . . 16 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹))
11883, 84, 1173jca 1234 . . . . . . . . . . . . . . 15 (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹)))
11915, 118sylanl1 679 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹)))
120 elfz2nn0 12251 . . . . . . . . . . . . . 14 (((𝑥 + 𝑆) − 𝑁) ∈ (0...(#‘𝐹)) ↔ (((𝑥 + 𝑆) − 𝑁) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹)))
121119, 120sylibr 222 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ (0...(#‘𝐹)))
122121adantll 745 . . . . . . . . . . . 12 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ (0...(#‘𝐹)))
12345, 122ffvelrnd 6249 . . . . . . . . . . 11 (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) ∧ ¬ 𝑥 ≤ (𝑁𝑆)) → (𝑃‘((𝑥 + 𝑆) − 𝑁)) ∈ 𝑉)
12444, 123ifclda 4065 . . . . . . . . . 10 ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑𝑥 ∈ (0...𝑁))) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)
125124exp32 628 . . . . . . . . 9 (𝑃:(0...(#‘𝐹))⟶𝑉 → (𝜑 → (𝑥 ∈ (0...𝑁) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)))
12613, 125syl 17 . . . . . . . 8 (𝐹(1Walks‘𝐺)𝑃 → (𝜑 → (𝑥 ∈ (0...𝑁) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)))
12711, 126mpcom 37 . . . . . . 7 (𝜑 → (𝑥 ∈ (0...𝑁) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉))
128127imp 443 . . . . . 6 ((𝜑𝑥 ∈ (0...𝑁)) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)
129 crctcsh.q . . . . . 6 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
130128, 129fmptd 6273 . . . . 5 (𝜑𝑄:(0...𝑁)⟶𝑉)
13112, 4, 2, 22, 15, 1crctcshlem2 41019 . . . . . . 7 (𝜑 → (#‘𝐻) = 𝑁)
132131oveq2d 6539 . . . . . 6 (𝜑 → (0...(#‘𝐻)) = (0...𝑁))
133132feq2d 5926 . . . . 5 (𝜑 → (𝑄:(0...(#‘𝐻))⟶𝑉𝑄:(0...𝑁)⟶𝑉))
134130, 133mpbird 245 . . . 4 (𝜑𝑄:(0...(#‘𝐻))⟶𝑉)
135134adantr 479 . . 3 ((𝜑𝑆 ≠ 0) → 𝑄:(0...(#‘𝐻))⟶𝑉)
136 wlkv 40813 . . . . . . . . 9 (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
137136simp1d 1065 . . . . . . . 8 (𝐹(1Walks‘𝐺)𝑃𝐺 ∈ V)
13812, 4is1wlkg 40814 . . . . . . . 8 (𝐺 ∈ V → (𝐹(1Walks‘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))))))
139137, 138syl 17 . . . . . . 7 (𝐹(1Walks‘𝐺)𝑃 → (𝐹(1Walks‘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))))))
140139ibi 254 . . . . . 6 (𝐹(1Walks‘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)))))
1412, 3, 1403syl 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)))))
142141adantr 479 . . . 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)))))
14322eqcomi 2614 . . . . . . . . . 10 (#‘𝐹) = 𝑁
144143oveq2i 6534 . . . . . . . . 9 (0..^(#‘𝐹)) = (0..^𝑁)
145144raleqi 3114 . . . . . . . 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))))
146 fzo1fzo0n0 12337 . . . . . . . . . . . . . . 15 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ (0..^𝑁) ∧ 𝑆 ≠ 0))
147146simplbi2 652 . . . . . . . . . . . . . 14 (𝑆 ∈ (0..^𝑁) → (𝑆 ≠ 0 → 𝑆 ∈ (1..^𝑁)))
14815, 147syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑆 ≠ 0 → 𝑆 ∈ (1..^𝑁)))
149148imp 443 . . . . . . . . . . . 12 ((𝜑𝑆 ≠ 0) → 𝑆 ∈ (1..^𝑁))
150149ad2antlr 758 . . . . . . . . . . 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))
151 simplll 793 . . . . . . . . . . 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)
152 1wlkslem1 40807 . . . . . . . . . . . . . 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)))))
153152cbvralv 3142 . . . . . . . . . . . . 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))))
154153biimpi 204 . . . . . . . . . . . 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))))
155154adantl 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)))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
156 crctprop 40996 . . . . . . . . . . . . . 14 (𝐹(CircuitS‘𝐺)𝑃 → (𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))
157143fveq2i 6087 . . . . . . . . . . . . . . . . . 18 (𝑃‘(#‘𝐹)) = (𝑃𝑁)
158157eqeq2i 2617 . . . . . . . . . . . . . . . . 17 ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃𝑁))
159158biimpi 204 . . . . . . . . . . . . . . . 16 ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑃‘0) = (𝑃𝑁))
160159eqcomd 2611 . . . . . . . . . . . . . . 15 ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑃𝑁) = (𝑃‘0))
161160adantl 480 . . . . . . . . . . . . . 14 ((𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑃𝑁) = (𝑃‘0))
1622, 156, 1613syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝑃𝑁) = (𝑃‘0))
163162ad2antrl 759 . . . . . . . . . . . 12 (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑𝑆 ≠ 0)) → (𝑃𝑁) = (𝑃‘0))
164163adantr 479 . . . . . . . . . . 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))
165150, 129, 1, 22, 151, 155, 164crctcsh1wlkn0lem7 41017 . . . . . . . . . 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))))
166131oveq2d 6539 . . . . . . . . . . . . 13 (𝜑 → (0..^(#‘𝐻)) = (0..^𝑁))
167166raleqdv 3116 . . . . . . . . . . . 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)))))
168167ad2antrl 759 . . . . . . . . . . 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)))))
169168adantr 479 . . . . . . . . . 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)))))
170165, 169mpbird 245 . . . . . . . . 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))))
171170ex 448 . . . . . . . 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)))))
172145, 171syl5bi 230 . . . . . . 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)))))
173172ex 448 . . . . . 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))))))
174173com23 83 . . . . 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))))))
1751743impia 1252 . . . 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)))))
176142, 175mpcom 37 . . 3 ((𝜑𝑆 ≠ 0) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
17710, 135, 1763jca 1234 . 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)))))
17812, 4, 2, 22, 15, 1, 129crctcshlem3 41020 . . . 4 (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
179178adantr 479 . . 3 ((𝜑𝑆 ≠ 0) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
18012, 4is1wlk 40811 . . 3 ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V) → (𝐻(1Walks‘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))))))
181179, 180syl 17 . 2 ((𝜑𝑆 ≠ 0) → (𝐻(1Walks‘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))))))
182177, 181mpbird 245 1 ((𝜑𝑆 ≠ 0) → 𝐻(1Walks‘𝐺)𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  if-wif 1005  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wral 2891  Vcvv 3168  wss 3535  ifcif 4031  {csn 4120  {cpr 4122   class class class wbr 4573  cmpt 4633  dom cdm 5024  wf 5782  cfv 5786  (class class class)co 6523  cr 9787  0cc0 9788  1c1 9789   + caddc 9791   < clt 9926  cle 9927  cmin 10113  cn 10863  0cn0 11135  cz 11206  ...cfz 12148  ..^cfzo 12285  #chash 12930  Word cword 13088   cyclShift ccsh 13327  Vtxcvtx 40227  iEdgciedg 40228  1Walksc1wlks 40794  TrailSctrls 40897  CircuitSccrcts 40988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-inf 8205  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-fl 12406  df-mod 12482  df-hash 12931  df-word 13096  df-concat 13098  df-substr 13100  df-csh 13328  df-1wlks 40798  df-trls 40899  df-crcts 40990
This theorem is referenced by:  crctcsh1wlk  41023
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