upgrewlkle2 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > upgrewlkle2		Structured version Visualization version GIF version

Theorem upgrewlkle2 40808

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 2610	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	ewlkprop 40805	. . 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 6113	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V
4		hashin 13060	. . . . . . . . . . 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 1059	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → 𝐺 ∈ UPGraph )
7		upgruhgr 25768	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
8	1	uhgrfun 25732	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	7, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10		funfn 5833	. . . . . . . . . . . . . . 15 ⊢ (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
11	9, 10	sylib 207	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12	11	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
13	12	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14		simpl 472	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → 𝐹 ∈ Word dom (iEdg‘𝐺))
15		elfzofz 12354	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1..^(#‘𝐹)) → 𝑘 ∈ (1...(#‘𝐹)))
16		fz1fzo0m1 12383	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...(#‘𝐹)) → (𝑘 − 1) ∈ (0..^(#‘𝐹)))
17	15, 16	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1..^(#‘𝐹)) → (𝑘 − 1) ∈ (0..^(#‘𝐹)))
18	17	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝑘 − 1) ∈ (0..^(#‘𝐹)))
19	14, 18	jca 553	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (𝑘 − 1) ∈ (0..^(#‘𝐹))))
20		wrdsymbcl 13173	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (𝑘 − 1) ∈ (0..^(#‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
21	19, 20	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
22	21	3ad2antl2 1217	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
23		eqid 2610	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24	23, 1	upgrle 25757	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺)) → (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
25	6, 13, 22, 24	syl3anc 1318	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
26	3	inex1 4727	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V
27		hashxrcl 13010	. . . . . . . . . . . . . . 15 ⊢ ((((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V → (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^*)
28	26, 27	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^*
29		hashxrcl 13010	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V → (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ^*)
30	3, 29	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ^*
31		2re 10967	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
32	31	rexri 9976	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ^*
33	28, 30, 32	3pm3.2i 1232	. . . . . . . . . . . . 13 ⊢ ((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^* ∧ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ^* ∧ 2 ∈ ℝ^*)
34	33	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^*))
35		xrletr 11865	. . . . . . . . . . . 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))
36	34, 35	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))
37	25, 36	mpan2d 706	. . . . . . . . . 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))
38	5, 37	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)
39		xnn0xr 11245	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℕ₀^* → 𝑆 ∈ ℝ^*)
40	28	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℕ₀^* → (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^*)
41	32	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℕ₀^* → 2 ∈ ℝ^*)
42		xrletr 11865	. . . . . . . . . . . . . 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))
43	39, 40, 41, 42	syl3anc 1318	. . . . . . . . . . . . 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))
44	43	expcomd 453	. . . . . . . . . . . 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)))
45	44	adantl 481	. . . . . . . . . . 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)))
46	45	3ad2ant1 1075	. . . . . . . . . 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)))
47	46	adantr 480	. . . . . . . . 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)))
48	38, 47	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))
49	48	ralimdva 2945	. . . . . . 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))
50	49	3exp 1256	. . . . . 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))))
51	50	com34 89	. . . . 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))))
52	51	3imp 1249	. . . 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))
53		lencl 13179	. . . . . 6 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (#‘𝐹) ∈ ℕ₀)
54		1zzd 11285	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝐹) ∈ ℕ₀ → 1 ∈ ℤ)
55		nn0z 11277	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝐹) ∈ ℕ₀ → (#‘𝐹) ∈ ℤ)
56	54, 55	jca 553	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) ∈ ℕ₀ → (1 ∈ ℤ ∧ (#‘𝐹) ∈ ℤ))
57		fzon 12358	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℤ ∧ (#‘𝐹) ∈ ℤ) → ((#‘𝐹) ≤ 1 ↔ (1..^(#‘𝐹)) = ∅))
58	56, 57	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) ∈ ℕ₀ → ((#‘𝐹) ≤ 1 ↔ (1..^(#‘𝐹)) = ∅))
59	58	bicomd 212	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) ∈ ℕ₀ → ((1..^(#‘𝐹)) = ∅ ↔ (#‘𝐹) ≤ 1))
60		nn0re 11178	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) ∈ ℕ₀ → (#‘𝐹) ∈ ℝ)
61		1red 9934	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) ∈ ℕ₀ → 1 ∈ ℝ)
62	60, 61	jca 553	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) ∈ ℕ₀ → ((#‘𝐹) ∈ ℝ ∧ 1 ∈ ℝ))
63		lenlt 9995	. . . . . . . . . . . . . . . 16 ⊢ (((#‘𝐹) ∈ ℝ ∧ 1 ∈ ℝ) → ((#‘𝐹) ≤ 1 ↔ ¬ 1 < (#‘𝐹)))
64	62, 63	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) ∈ ℕ₀ → ((#‘𝐹) ≤ 1 ↔ ¬ 1 < (#‘𝐹)))
65	59, 64	bitrd 267	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) ∈ ℕ₀ → ((1..^(#‘𝐹)) = ∅ ↔ ¬ 1 < (#‘𝐹)))
66	65	biimpd 218	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) ∈ ℕ₀ → ((1..^(#‘𝐹)) = ∅ → ¬ 1 < (#‘𝐹)))
67	66	necon2ad 2797	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) ∈ ℕ₀ → (1 < (#‘𝐹) → (1..^(#‘𝐹)) ≠ ∅))
68	67	impcom 445	. . . . . . . . . . 11 ⊢ ((1 < (#‘𝐹) ∧ (#‘𝐹) ∈ ℕ₀) → (1..^(#‘𝐹)) ≠ ∅)
69		rspn0 3892	. . . . . . . . . . 11 ⊢ ((1..^(#‘𝐹)) ≠ ∅ → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2))
70	68, 69	syl 17	. . . . . . . . . 10 ⊢ ((1 < (#‘𝐹) ∧ (#‘𝐹) ∈ ℕ₀) → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2))
71	70	ex 449	. . . . . . . . 9 ⊢ (1 < (#‘𝐹) → ((#‘𝐹) ∈ ℕ₀ → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2)))
72	71	com23 84	. . . . . . . 8 ⊢ (1 < (#‘𝐹) → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → ((#‘𝐹) ∈ ℕ₀ → 𝑆 ≤ 2)))
73	72	com13 86	. . . . . . 7 ⊢ ((#‘𝐹) ∈ ℕ₀ → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → (1 < (#‘𝐹) → 𝑆 ≤ 2)))
74	73	a1i 11	. . . . . 6 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → ((#‘𝐹) ∈ ℕ₀ → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → (1 < (#‘𝐹) → 𝑆 ≤ 2))))
75	53, 74	mpd 15	. . . . 5 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → (1 < (#‘𝐹) → 𝑆 ≤ 2)))
76	75	3ad2ant2 1076	. . . 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)))
77	52, 76	syld 46	. . 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)))
78	2, 77	syl 17	. 2 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → (𝐺 ∈ UPGraph → (1 < (#‘𝐹) → 𝑆 ≤ 2)))
79	78	3imp21 1269	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (#‘𝐹)) → 𝑆 ≤ 2)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∩ cin 3539 ∅c0 3874 class class class wbr 4583 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 2c2 10947 ℕ₀cn0 11169 ℕ₀^*cxnn0 11240 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 UPGraph cupgr 25747 EdgWalks cewlks 40795

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

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-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-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-uhgr 25724 df-upgr 25749 df-ewlks 40799

This theorem is referenced by: (None)

W3C validator