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Theorem upgrewlkle2 27548

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 2738	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	ewlkprop 27545	. . 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 6687	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V
4		hashin 13864	. . . . . . . . . . 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 1194	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → 𝐺 ∈ UPGraph)
7		upgruhgr 27047	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8	1	uhgrfun 27011	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	7, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10	9	funfnd 6370	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
11	10	3ad2ant3 1136	. . . . . . . . . . . . 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 13144	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1..^(♯‘𝐹)) → 𝑘 ∈ (1...(♯‘𝐹)))
14		fz1fzo0m1 13176	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...(♯‘𝐹)) → (𝑘 − 1) ∈ (0..^(♯‘𝐹)))
15	13, 14	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1..^(♯‘𝐹)) → (𝑘 − 1) ∈ (0..^(♯‘𝐹)))
16		wrdsymbcl 13968	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (𝑘 − 1) ∈ (0..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
17	15, 16	sylan2 596	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
18	17	3ad2antl2 1187	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
19		eqid 2738	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
20	19, 1	upgrle 27035	. . . . . . . . . . . 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 1372	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
22	3	inex1 5185	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V
23		hashxrcl 13810	. . . . . . . . . . . . . . 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 13810	. . . . . . . . . . . . . . 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 11790	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
28	27	rexri 10777	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ^*
29	24, 26, 28	3pm3.2i 1340	. . . . . . . . . . . . 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 12634	. . . . . . . . . . . 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 694	. . . . . . . . . 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 12053	. . . . . . . . . . . . . 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 12634	. . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . 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 1134	. . . . . . . . . 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 3091	. . . . . . 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 1120	. . . . . 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 1112	. . . 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 13974	. . . . . 6 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (♯‘𝐹) ∈ ℕ₀)
50		1zzd 12094	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ ℤ)
51		nn0z 12086	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℤ)
52		fzon 13149	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘𝐹) ∈ ℤ) → ((♯‘𝐹) ≤ 1 ↔ (1..^(♯‘𝐹)) = ∅))
53	50, 51, 52	syl2anc 587	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ≤ 1 ↔ (1..^(♯‘𝐹)) = ∅))
54		nn0re 11985	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℝ)
55		1red 10720	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ ℝ)
56	54, 55	lenltd 10864	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ≤ 1 ↔ ¬ 1 < (♯‘𝐹)))
57	53, 56	bitr3d 284	. . . . . . . . . 10 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((1..^(♯‘𝐹)) = ∅ ↔ ¬ 1 < (♯‘𝐹)))
58	57	biimpd 232	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((1..^(♯‘𝐹)) = ∅ → ¬ 1 < (♯‘𝐹)))
59	58	necon2ad 2949	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (1 < (♯‘𝐹) → (1..^(♯‘𝐹)) ≠ ∅))
60		rspn0 4241	. . . . . . . 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 1135	. . . 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 1115	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (♯‘𝐹)) → 𝑆 ≤ 2)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 Vcvv 3398 ∩ cin 3842 ∅c0 4211 class class class wbr 5030 dom cdm 5525 Fun wfun 6333 Fn wfn 6334 ‘cfv 6339 (class class class)co 7170 0cc0 10615 1c1 10616 ℝ^*cxr 10752 < clt 10753 ≤ cle 10754 − cmin 10948 2c2 11771 ℕ₀cn0 11976 ℕ₀^*cxnn0 12048 ℤcz 12062 ...cfz 12981 ..^cfzo 13124 ♯chash 13782 Word cword 13955 Vtxcvtx 26941 iEdgciedg 26942 UHGraphcuhgr 27001 UPGraphcupgr 27025 EdgWalks cewlks 27537

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-oadd 8135 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-n0 11977 df-xnn0 12049 df-z 12063 df-uz 12325 df-fz 12982 df-fzo 13125 df-hash 13783 df-word 13956 df-uhgr 27003 df-upgr 27027 df-ewlks 27540

This theorem is referenced by: (None)

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