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Theorem upgrewlkle2 27390

Description: In a pseudograph, there is no s-walk of edges of length greater than 1 with s>2. (Contributed by AV, 4-Jan-2021.)

**Assertion**
Ref	Expression
upgrewlkle2	⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (♯‘𝐹)) → 𝑆 ≤ 2)

Proof of Theorem upgrewlkle2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2823	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	ewlkprop 27387	. . 3 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
3		fvex 6685	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V
4		hashin 13775	. . . . . . . . . . 11 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V → (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))))
5	3, 4	ax-mp 5	. . . . . . . . . 10 ⊢ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))))
6		simpl3 1189	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → 𝐺 ∈ UPGraph)
7		upgruhgr 26889	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8	1	uhgrfun 26853	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	7, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10	9	funfnd 6388	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
11	10	3ad2ant3 1131	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12	11	adantr 483	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
13		elfzofz 13056	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1..^(♯‘𝐹)) → 𝑘 ∈ (1...(♯‘𝐹)))
14		fz1fzo0m1 13088	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...(♯‘𝐹)) → (𝑘 − 1) ∈ (0..^(♯‘𝐹)))
15	13, 14	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1..^(♯‘𝐹)) → (𝑘 − 1) ∈ (0..^(♯‘𝐹)))
16		wrdsymbcl 13878	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (𝑘 − 1) ∈ (0..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
17	15, 16	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
18	17	3ad2antl2 1182	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
19		eqid 2823	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
20	19, 1	upgrle 26877	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺)) → (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
21	6, 12, 18, 20	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
22	3	inex1 5223	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V
23		hashxrcl 13721	. . . . . . . . . . . . . . 15 ⊢ ((((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V → (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^*)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^*
25		hashxrcl 13721	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V → (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ^*)
26	3, 25	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ^*
27		2re 11714	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
28	27	rexri 10701	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ^*
29	24, 26, 28	3pm3.2i 1335	. . . . . . . . . . . . 13 ⊢ ((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^* ∧ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ^* ∧ 2 ∈ ℝ^*)
30	29	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → ((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^ ∧ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ^* ∧ 2 ∈ ℝ^*))
31		xrletr 12554	. . . . . . . . . . . 12 ⊢ (((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^* ∧ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ^* ∧ 2 ∈ ℝ^*) → (((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∧ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2) → (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2))
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∧ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2) → (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2))
33	21, 32	mpan2d 692	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → ((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) → (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2))
34	5, 33	mpi 20	. . . . . . . . 9 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2)
35		xnn0xr 11975	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℕ₀^* → 𝑆 ∈ ℝ^*)
36	24	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℕ₀^* → (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^*)
37	28	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℕ₀^* → 2 ∈ ℝ^*)
38		xrletr 12554	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ ℝ^* ∧ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^* ∧ 2 ∈ ℝ^*) → ((𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∧ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2) → 𝑆 ≤ 2))
39	35, 36, 37, 38	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ ℕ₀^* → ((𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∧ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2) → 𝑆 ≤ 2))
40	39	expcomd 419	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℕ₀^* → ((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2 → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝑆 ≤ 2)))
41	40	adantl 484	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) → ((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2 → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝑆 ≤ 2)))
42	41	3ad2ant1 1129	. . . . . . . . . 10 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → ((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2 → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝑆 ≤ 2)))
43	42	adantr 483	. . . . . . . . 9 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → ((♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ≤ 2 → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝑆 ≤ 2)))
44	34, 43	mpd 15	. . . . . . . 8 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝑆 ≤ 2))
45	44	ralimdva 3179	. . . . . . 7 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2))
46	45	3exp 1115	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) → (𝐹 ∈ Word dom (iEdg‘𝐺) → (𝐺 ∈ UPGraph → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2))))
47	46	com34 91	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) → (𝐹 ∈ Word dom (iEdg‘𝐺) → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → (𝐺 ∈ UPGraph → ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2))))
48	47	3imp 1107	. . . 4 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → (𝐺 ∈ UPGraph → ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2))
49		lencl 13885	. . . . . 6 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (♯‘𝐹) ∈ ℕ₀)
50		1zzd 12016	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ ℤ)
51		nn0z 12008	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℤ)
52		fzon 13061	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘𝐹) ∈ ℤ) → ((♯‘𝐹) ≤ 1 ↔ (1..^(♯‘𝐹)) = ∅))
53	50, 51, 52	syl2anc 586	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ≤ 1 ↔ (1..^(♯‘𝐹)) = ∅))
54		nn0re 11909	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℝ)
55		1red 10644	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ ℝ)
56	54, 55	lenltd 10788	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ≤ 1 ↔ ¬ 1 < (♯‘𝐹)))
57	53, 56	bitr3d 283	. . . . . . . . . 10 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((1..^(♯‘𝐹)) = ∅ ↔ ¬ 1 < (♯‘𝐹)))
58	57	biimpd 231	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((1..^(♯‘𝐹)) = ∅ → ¬ 1 < (♯‘𝐹)))
59	58	necon2ad 3033	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (1 < (♯‘𝐹) → (1..^(♯‘𝐹)) ≠ ∅))
60		rspn0 4315	. . . . . . . 8 ⊢ ((1..^(♯‘𝐹)) ≠ ∅ → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2))
61	59, 60	syl6com 37	. . . . . . 7 ⊢ (1 < (♯‘𝐹) → ((♯‘𝐹) ∈ ℕ₀ → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2)))
62	61	com3l 89	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2 → (1 < (♯‘𝐹) → 𝑆 ≤ 2)))
63	49, 62	syl 17	. . . . 5 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2 → (1 < (♯‘𝐹) → 𝑆 ≤ 2)))
64	63	3ad2ant2 1130	. . . 4 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2 → (1 < (♯‘𝐹) → 𝑆 ≤ 2)))
65	48, 64	syld 47	. . 3 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → (𝐺 ∈ UPGraph → (1 < (♯‘𝐹) → 𝑆 ≤ 2)))
66	2, 65	syl 17	. 2 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → (𝐺 ∈ UPGraph → (1 < (♯‘𝐹) → 𝑆 ≤ 2)))
67	66	3imp21 1110	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (♯‘𝐹)) → 𝑆 ≤ 2)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 Vcvv 3496 ∩ cin 3937 ∅c0 4293 class class class wbr 5068 dom cdm 5557 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 − cmin 10872 2c2 11695 ℕ₀cn0 11900 ℕ₀^*cxnn0 11970 ℤcz 11984 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 Vtxcvtx 26783 iEdgciedg 26784 UHGraphcuhgr 26843 UPGraphcupgr 26867 EdgWalks cewlks 27379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-uhgr 26845 df-upgr 26869 df-ewlks 27382

This theorem is referenced by: (None)

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