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Theorem upgrewlkle2 29804

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 2762	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	ewlkprop 29801	. . 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 6880	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V
4		hashin 14424	. . . . . . . . . . 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 1207	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → 𝐺 ∈ UPGraph)
7		upgruhgr 29300	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8	1	uhgrfun 29264	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	7, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10	9	funfnd 6552	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
11	10	3ad2ant3 1148	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12	11	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
13		elfzofz 13681	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1..^(♯‘𝐹)) → 𝑘 ∈ (1...(♯‘𝐹)))
14		fz1fzo0m1 13716	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...(♯‘𝐹)) → (𝑘 − 1) ∈ (0..^(♯‘𝐹)))
15	13, 14	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1..^(♯‘𝐹)) → (𝑘 − 1) ∈ (0..^(♯‘𝐹)))
16		wrdsymbcl 14540	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (𝑘 − 1) ∈ (0..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
17	15, 16	sylan2 602	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
18	17	3ad2antl2 1200	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
19		eqid 2762	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
20	19, 1	upgrle 29288	. . . . . . . . . . . 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 1390	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
22	3	inex1 5273	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V
23		hashxrcl 14370	. . . . . . . . . . . . . . 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 14370	. . . . . . . . . . . . . . 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 12292	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
28	27	rexri 11240	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ^*
29	24, 26, 28	3pm3.2i 1353	. . . . . . . . . . . . 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 13160	. . . . . . . . . . . 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 704	. . . . . . . . . 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 12559	. . . . . . . . . . . . . 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 13160	. . . . . . . . . . . . . 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 1390	. . . . . . . . . . . . 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 420	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 1146	. . . . . . . . . 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 484	. . . . . . . . 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 3174	. . . . . . 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 1132	. . . . . 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 1123	. . . 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 14546	. . . . . 6 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (♯‘𝐹) ∈ ℕ₀)
50		1zzd 12602	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ ℤ)
51		nn0z 12592	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℤ)
52		fzon 13686	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘𝐹) ∈ ℤ) → ((♯‘𝐹) ≤ 1 ↔ (1..^(♯‘𝐹)) = ∅))
53	50, 51, 52	syl2anc 593	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ≤ 1 ↔ (1..^(♯‘𝐹)) = ∅))
54		nn0re 12490	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℝ)
55		1red 11182	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ ℝ)
56	54, 55	lenltd 11329	. . . . . . . . . . 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 2972	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (1 < (♯‘𝐹) → (1..^(♯‘𝐹)) ≠ ∅))
60		rspn0 4309	. . . . . . . 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 1147	. . . 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 1126	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 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∀wral 3076 Vcvv 3454 ∩ cin 3903 ∅c0 4285 class class class wbr 5100 dom cdm 5647 Fun wfun 6515 Fn wfn 6516 ‘cfv 6521 (class class class)co 7396 0cc0 11073 1c1 11074 ℝ^*cxr 11215 < clt 11216 ≤ cle 11217 − cmin 11414 2c2 12272 ℕ₀cn0 12481 ℕ₀^*cxnn0 12554 ℤcz 12568 ...cfz 13512 ..^cfzo 13659 ♯chash 14343 Word cword 14526 Vtxcvtx 29194 iEdgciedg 29195 UHGraphcuhgr 29254 UPGraphcupgr 29278 EdgWalks cewlks 29793

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-oadd 8441 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-n0 12482 df-xnn0 12555 df-z 12569 df-uz 12840 df-fz 13513 df-fzo 13660 df-hash 14344 df-word 14527 df-uhgr 29256 df-upgr 29280 df-ewlks 29796

This theorem is referenced by: (None)

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