crctcsh1wlkn0 - Mathbox for Alexander van der Vekens

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Theorem crctcsh1wlkn0 41024

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	⊢ (𝜑 → 𝐹(CircuitS‘𝐺)𝑃)
crctcsh.n	⊢ 𝑁 = (#‘𝐹)
crctcsh.s	⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))
crctcsh.h	⊢ 𝐻 = (𝐹 cyclShift 𝑆)
crctcsh.q	⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))

**Assertion**
Ref	Expression
crctcsh1wlkn0	⊢ ((𝜑 ∧ 𝑆 ≠ 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.

Step	Hyp	Ref	Expression
1		crctcsh.h	. . . . 5 ⊢ 𝐻 = (𝐹 cyclShift 𝑆)
2		crctcsh.d	. . . . . . 7 ⊢ (𝜑 → 𝐹(CircuitS‘𝐺)𝑃)
3		crctis1wlk 41002	. . . . . . 7 ⊢ (𝐹(CircuitS‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃)
4		crctcsh.i	. . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺)
5	4	1wlkf 40819	. . . . . . 7 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
6	2, 3, 5	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
7		cshwcl 13395	. . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → (𝐹 cyclShift 𝑆) ∈ Word dom 𝐼)
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹 cyclShift 𝑆) ∈ Word dom 𝐼)
9	1, 8	syl5eqel 2692	. . . 4 ⊢ (𝜑 → 𝐻 ∈ Word dom 𝐼)
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝐻 ∈ Word dom 𝐼)
11	2, 3	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹(1Walks‘𝐺)𝑃)
12		crctcsh.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
13	12	1wlkp 40821	. . . . . . . . 9 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶𝑉)
14		simpll 786	. . . . . . . . . . . 12 ⊢ (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑 ∧ 𝑥 ∈ (0...𝑁))) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → 𝑃:(0...(#‘𝐹))⟶𝑉)
15		crctcsh.s	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))
16		elfznn0 12302	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℕ₀)
17	16	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑥 ∈ ℕ₀)
18		elfzonn0 12380	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℕ₀)
19	18	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑆 ∈ ℕ₀)
20	17, 19	nn0addcld 11232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 + 𝑆) ∈ ℕ₀)
21	20	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → (𝑥 + 𝑆) ∈ ℕ₀)
22		crctcsh.n	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑁 = (#‘𝐹)
23		elfz3nn0 12303	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
24	22, 23	syl5eqelr 2693	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (0...𝑁) → (#‘𝐹) ∈ ℕ₀)
25	24	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → (#‘𝐹) ∈ ℕ₀)
26		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℤ)
27	26	zred 11358	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℝ)
28	27	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑥 ∈ ℝ)
29		elfzoelz 12339	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℤ)
30	29	zred 11358	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℝ)
31	30	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑆 ∈ ℝ)
32		elfzel2 12211	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
33	32	zred 11358	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (0...𝑁) → 𝑁 ∈ ℝ)
34	33	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
35		leaddsub 10383	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑥 + 𝑆) ≤ 𝑁 ↔ 𝑥 ≤ (𝑁 − 𝑆)))
36	28, 31, 34, 35	syl3anc 1318	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) ≤ 𝑁 ↔ 𝑥 ≤ (𝑁 − 𝑆)))
37	36	biimpar 501	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → (𝑥 + 𝑆) ≤ 𝑁)
38	37, 22	syl6breq 4624	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → (𝑥 + 𝑆) ≤ (#‘𝐹))
39	21, 25, 38	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → ((𝑥 + 𝑆) ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ₀ ∧ (𝑥 + 𝑆) ≤ (#‘𝐹)))
40	15, 39	sylanl1 680	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → ((𝑥 + 𝑆) ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ₀ ∧ (𝑥 + 𝑆) ≤ (#‘𝐹)))
41		elfz2nn0 12300	. . . . . . . . . . . . . 14 ⊢ ((𝑥 + 𝑆) ∈ (0...(#‘𝐹)) ↔ ((𝑥 + 𝑆) ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ₀ ∧ (𝑥 + 𝑆) ≤ (#‘𝐹)))
42	40, 41	sylibr 223	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (0...𝑁)) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → (𝑥 + 𝑆) ∈ (0...(#‘𝐹)))
43	42	adantll 746	. . . . . . . . . . . 12 ⊢ (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑 ∧ 𝑥 ∈ (0...𝑁))) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → (𝑥 + 𝑆) ∈ (0...(#‘𝐹)))
44	14, 43	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑 ∧ 𝑥 ∈ (0...𝑁))) ∧ 𝑥 ≤ (𝑁 − 𝑆)) → (𝑃‘(𝑥 + 𝑆)) ∈ 𝑉)
45		simpll 786	. . . . . . . . . . . 12 ⊢ (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑 ∧ 𝑥 ∈ (0...𝑁))) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → 𝑃:(0...(#‘𝐹))⟶𝑉)
46		elfzoel2 12338	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
47		zaddcl 11294	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑥 + 𝑆) ∈ ℤ)
48	47	adantrr 749	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑥 + 𝑆) ∈ ℤ)
49		simprr 792	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℤ)
50	48, 49	zsubcld 11363	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℤ)
51	50	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℤ)
52		zsubcl 11296	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑁 − 𝑆) ∈ ℤ)
53	52	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 − 𝑆) ∈ ℤ)
54	53	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 − 𝑆) ∈ ℝ)
55		zre 11258	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
56		ltnle 9996	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑁 − 𝑆) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑁 − 𝑆) < 𝑥 ↔ ¬ 𝑥 ≤ (𝑁 − 𝑆)))
57	54, 55, 56	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁 − 𝑆) < 𝑥 ↔ ¬ 𝑥 ≤ (𝑁 − 𝑆)))
58		zre 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
59	58	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
60		zre 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑆 ∈ ℤ → 𝑆 ∈ ℝ)
61	60	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑆 ∈ ℝ)
62	55	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑥 ∈ ℝ)
63		ltsubadd 10377	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑁 − 𝑆) < 𝑥 ↔ 𝑁 < (𝑥 + 𝑆)))
64	59, 61, 62, 63	syl2an23an 1379	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁 − 𝑆) < 𝑥 ↔ 𝑁 < (𝑥 + 𝑆)))
65	59	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 𝑁 ∈ ℝ)
66	48	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑥 + 𝑆) ∈ ℝ)
67	65, 66	posdifd 10493	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 < (𝑥 + 𝑆) ↔ 0 < ((𝑥 + 𝑆) − 𝑁)))
68		0red 9920	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → 0 ∈ ℝ)
69	50	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℝ)
70		ltle 10005	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0 ∈ ℝ ∧ ((𝑥 + 𝑆) − 𝑁) ∈ ℝ) → (0 < ((𝑥 + 𝑆) − 𝑁) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
71	68, 69, 70	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (0 < ((𝑥 + 𝑆) − 𝑁) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
72	67, 71	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑁 < (𝑥 + 𝑆) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
73	64, 72	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑁 − 𝑆) < 𝑥 → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
74	57, 73	sylbird 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (¬ 𝑥 ≤ (𝑁 − 𝑆) → 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
75	74	imp 444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → 0 ≤ ((𝑥 + 𝑆) − 𝑁))
76	51, 75	jca 553	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
77	76	exp31 628	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℤ → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑥 ≤ (𝑁 − 𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
78	26, 77	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (0...𝑁) → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑥 ≤ (𝑁 − 𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
79	78	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (0...𝑁) → (¬ 𝑥 ≤ (𝑁 − 𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
80	29, 46, 79	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (0..^𝑁) → (𝑥 ∈ (0...𝑁) → (¬ 𝑥 ≤ (𝑁 − 𝑆) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))))
81	80	imp31 447	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
82		elnn0z 11267	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 + 𝑆) − 𝑁) ∈ ℕ₀ ↔ (((𝑥 + 𝑆) − 𝑁) ∈ ℤ ∧ 0 ≤ ((𝑥 + 𝑆) − 𝑁)))
83	81, 82	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ ℕ₀)
84	24	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → (#‘𝐹) ∈ ℕ₀)
85		elfzo0 12376	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ (0..^𝑁) ↔ (𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
86		elfz2nn0 12300	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (0...𝑁) ↔ (𝑥 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑥 ≤ 𝑁))
87		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑆 ∈ ℕ₀ → 𝑆 ∈ ℝ)
88	87	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆 ∈ ℝ)
89		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ∈ ℝ)
90	89	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ ℝ)
91	88, 90	anim12ci 589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑥 ≤ 𝑁)) → (𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ))
92		nnre 10904	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
93	92, 92	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
94	93	3ad2ant2 1076	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
95	94	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑥 ≤ 𝑁)) → (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
96	91, 95	jca 553	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑥 ≤ 𝑁)) → ((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
97		simpr3 1062	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑥 ≤ 𝑁)) → 𝑥 ≤ 𝑁)
98		ltle 10005	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁 → 𝑆 ≤ 𝑁))
99	87, 92, 98	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ) → (𝑆 < 𝑁 → 𝑆 ≤ 𝑁))
100	99	3impia 1253	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆 ≤ 𝑁)
101	100	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑥 ≤ 𝑁)) → 𝑆 ≤ 𝑁)
102	96, 97, 101	jca32 556	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑆 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ (𝑥 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀ ∧ 𝑥 ≤ 𝑁)) → (((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) ∧ (𝑥 ≤ 𝑁 ∧ 𝑆 ≤ 𝑁)))
103	85, 86, 102	syl2anb 495	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) ∧ (𝑥 ≤ 𝑁 ∧ 𝑆 ≤ 𝑁)))
104		le2add 10389	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) → ((𝑥 ≤ 𝑁 ∧ 𝑆 ≤ 𝑁) → (𝑥 + 𝑆) ≤ (𝑁 + 𝑁)))
105	104	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)) ∧ (𝑥 ≤ 𝑁 ∧ 𝑆 ≤ 𝑁)) → (𝑥 + 𝑆) ≤ (𝑁 + 𝑁))
106	103, 105	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 + 𝑆) ≤ (𝑁 + 𝑁))
107	66, 65, 65	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ ℤ ∧ (𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
108	107	ex 449	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℤ → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
109	26, 108	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (0...𝑁) → ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
110	109	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (0...𝑁) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
111	29, 46, 110	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ (0..^𝑁) → (𝑥 ∈ (0...𝑁) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
112	111	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ))
113		lesubadd 10379	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 + 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((𝑥 + 𝑆) − 𝑁) ≤ 𝑁 ↔ (𝑥 + 𝑆) ≤ (𝑁 + 𝑁)))
114	112, 113	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → (((𝑥 + 𝑆) − 𝑁) ≤ 𝑁 ↔ (𝑥 + 𝑆) ≤ (𝑁 + 𝑁)))
115	106, 114	mpbird 246	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) − 𝑁) ≤ 𝑁)
116	115	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → ((𝑥 + 𝑆) − 𝑁) ≤ 𝑁)
117	116, 22	syl6breq 4624	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹))
118	83, 84, 117	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ (0..^𝑁) ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ₀ ∧ ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹)))
119	15, 118	sylanl1 680	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → (((𝑥 + 𝑆) − 𝑁) ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ₀ ∧ ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹)))
120		elfz2nn0 12300	. . . . . . . . . . . . . 14 ⊢ (((𝑥 + 𝑆) − 𝑁) ∈ (0...(#‘𝐹)) ↔ (((𝑥 + 𝑆) − 𝑁) ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ₀ ∧ ((𝑥 + 𝑆) − 𝑁) ≤ (#‘𝐹)))
121	119, 120	sylibr 223	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (0...𝑁)) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ (0...(#‘𝐹)))
122	121	adantll 746	. . . . . . . . . . . 12 ⊢ (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑 ∧ 𝑥 ∈ (0...𝑁))) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → ((𝑥 + 𝑆) − 𝑁) ∈ (0...(#‘𝐹)))
123	45, 122	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ (((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑 ∧ 𝑥 ∈ (0...𝑁))) ∧ ¬ 𝑥 ≤ (𝑁 − 𝑆)) → (𝑃‘((𝑥 + 𝑆) − 𝑁)) ∈ 𝑉)
124	44, 123	ifclda 4070	. . . . . . . . . 10 ⊢ ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ (𝜑 ∧ 𝑥 ∈ (0...𝑁))) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)
125	124	exp32 629	. . . . . . . . 9 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 → (𝜑 → (𝑥 ∈ (0...𝑁) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)))
126	13, 125	syl 17	. . . . . . . 8 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝜑 → (𝑥 ∈ (0...𝑁) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)))
127	11, 126	mpcom 37	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0...𝑁) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉))
128	127	imp 444	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0...𝑁)) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) ∈ 𝑉)
129		crctcsh.q	. . . . . 6 ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
130	128, 129	fmptd 6292	. . . . 5 ⊢ (𝜑 → 𝑄:(0...𝑁)⟶𝑉)
131	12, 4, 2, 22, 15, 1	crctcshlem2 41021	. . . . . . 7 ⊢ (𝜑 → (#‘𝐻) = 𝑁)
132	131	oveq2d 6565	. . . . . 6 ⊢ (𝜑 → (0...(#‘𝐻)) = (0...𝑁))
133	132	feq2d 5944	. . . . 5 ⊢ (𝜑 → (𝑄:(0...(#‘𝐻))⟶𝑉 ↔ 𝑄:(0...𝑁)⟶𝑉))
134	130, 133	mpbird 246	. . . 4 ⊢ (𝜑 → 𝑄:(0...(#‘𝐻))⟶𝑉)
135	134	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝑄:(0...(#‘𝐻))⟶𝑉)
136		wlkv 40815	. . . . . . . . 9 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
137	136	simp1d 1066	. . . . . . . 8 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝐺 ∈ V)
138	12, 4	is1wlkg 40816	. . . . . . . 8 ⊢ (𝐺 ∈ V → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))))))
139	137, 138	syl 17	. . . . . . 7 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))))))
140	139	ibi 255	. . . . . 6 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))))
141	2, 3, 140	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))))
142	141	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))))
143	22	eqcomi 2619	. . . . . . . . . 10 ⊢ (#‘𝐹) = 𝑁
144	143	oveq2i 6560	. . . . . . . . 9 ⊢ (0..^(#‘𝐹)) = (0..^𝑁)
145	144	raleqi 3119	. . . . . . . 8 ⊢ (∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))) ↔ ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))))
146		fzo1fzo0n0 12386	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ (0..^𝑁) ∧ 𝑆 ≠ 0))
147	146	simplbi2 653	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (0..^𝑁) → (𝑆 ≠ 0 → 𝑆 ∈ (1..^𝑁)))
148	15, 147	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ≠ 0 → 𝑆 ∈ (1..^𝑁)))
149	148	imp 444	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝑆 ∈ (1..^𝑁))
150	149	ad2antlr 759	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))) → 𝑆 ∈ (1..^𝑁))
151		simplll 794	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))) → 𝐹 ∈ Word dom 𝐼)
152		1wlkslem1 40809	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → (if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))) ↔ if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
153	152	cbvralv 3147	. . . . . . . . . . . . 13 ⊢ (∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))) ↔ ∀𝑘 ∈ (0..^𝑁)if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
154	153	biimpi 205	. . . . . . . . . . . 12 ⊢ (∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
155	154	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))) → ∀𝑘 ∈ (0..^𝑁)if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
156		crctprop 40998	. . . . . . . . . . . . . 14 ⊢ (𝐹(CircuitS‘𝐺)𝑃 → (𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))
157	143	fveq2i 6106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃‘(#‘𝐹)) = (𝑃‘𝑁)
158	157	eqeq2i 2622	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃‘𝑁))
159	158	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑃‘0) = (𝑃‘𝑁))
160	159	eqcomd 2616	. . . . . . . . . . . . . . 15 ⊢ ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑃‘𝑁) = (𝑃‘0))
161	160	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑃‘𝑁) = (𝑃‘0))
162	2, 156, 161	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃‘𝑁) = (𝑃‘0))
163	162	ad2antrl 760	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) → (𝑃‘𝑁) = (𝑃‘0))
164	163	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))) → (𝑃‘𝑁) = (𝑃‘0))
165	150, 129, 1, 22, 151, 155, 164	crctcsh1wlkn0lem7 41019	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))) → ∀𝑗 ∈ (0..^𝑁)if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))
166	131	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0..^(#‘𝐻)) = (0..^𝑁))
167	166	raleqdv 3121	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))) ↔ ∀𝑗 ∈ (0..^𝑁)if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗)))))
168	167	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) → (∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))) ↔ ∀𝑗 ∈ (0..^𝑁)if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗)))))
169	168	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))) → (∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))) ↔ ∀𝑗 ∈ (0..^𝑁)if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗)))))
170	165, 169	mpbird 246	. . . . . . . . 9 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) ∧ ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))
171	170	ex 449	. . . . . . . 8 ⊢ (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) → (∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗)))))
172	145, 171	syl5bi 231	. . . . . . 7 ⊢ (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (𝜑 ∧ 𝑆 ≠ 0)) → (∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗)))))
173	172	ex 449	. . . . . 6 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → ((𝜑 ∧ 𝑆 ≠ 0) → (∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))))
174	173	com23 84	. . . . 5 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → (∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))) → ((𝜑 ∧ 𝑆 ≠ 0) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))))
175	174	3impia 1253	. . . 4 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖)))) → ((𝜑 ∧ 𝑆 ≠ 0) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗)))))
176	142, 175	mpcom 37	. . 3 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))
177	10, 135, 176	3jca 1235	. 2 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝐻 ∈ Word dom 𝐼 ∧ 𝑄:(0...(#‘𝐻))⟶𝑉 ∧ ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗)))))
178	12, 4, 2, 22, 15, 1, 129	crctcshlem3 41022	. . . 4 ⊢ (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
179	178	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
180	12, 4	is1wlk 40813	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V) → (𝐻(1Walks‘𝐺)𝑄 ↔ (𝐻 ∈ Word dom 𝐼 ∧ 𝑄:(0...(#‘𝐻))⟶𝑉 ∧ ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))))
181	179, 180	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝐻(1Walks‘𝐺)𝑄 ↔ (𝐻 ∈ Word dom 𝐼 ∧ 𝑄:(0...(#‘𝐻))⟶𝑉 ∧ ∀𝑗 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))))
182	177, 181	mpbird 246	1 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝐻(1Walks‘𝐺)𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 if-wif 1006 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ifcif 4036 {csn 4125 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 cyclShift ccsh 13385 Vtxcvtx 25673 iEdgciedg 25674 1Walksc1wlks 40796 TrailSctrls 40899 CircuitSccrcts 40990

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 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 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-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-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 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-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 df-csh 13386 df-1wlks 40800 df-trls 40901 df-crcts 40992

This theorem is referenced by: crctcsh1wlk 41025

W3C validator