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Theorem upgrewlkle2 30180

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 2761	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	ewlkprop 30177	. . 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 6896	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V
4		hashin 14549	. . . . . . . . . . 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 1212	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → 𝐺 ∈ UPGraph)
7		upgruhgr 29673	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
8	1	uhgrfun 29637	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	7, 8	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10	9	funfnd 6569	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
11	10	3ad2ant3 1153	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12	11	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
13		elfzofz 13803	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1..^(♯‘𝐹)) → 𝑘 ∈ (1...(♯‘𝐹)))
14		fz1fzo0m1 13838	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...(♯‘𝐹)) → (𝑘 − 1) ∈ (0..^(♯‘𝐹)))
15	13, 14	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1..^(♯‘𝐹)) → (𝑘 − 1) ∈ (0..^(♯‘𝐹)))
16		wrdsymbcl 14665	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (𝑘 − 1) ∈ (0..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
17	15, 16	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
18	17	3ad2antl2 1205	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
19		eqid 2761	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
20	19, 1	upgrle 29661	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(♯‘𝐹))) → (♯‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
22	3	inex1 5277	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V
23		hashxrcl 14494	. . . . . . . . . . . . . . 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 14494	. . . . . . . . . . . . . . 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 12410	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
28	27	rexri 11360	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ^*
29	24, 26, 28	3pm3.2i 1358	. . . . . . . . . . . . 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 13280	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 707	. . . . . . . . . 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 21	. . . . . . . . 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 12677	. . . . . . . . . . . . . 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 13280	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 422	. . . . . . . . . . . 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 487	. . . . . . . . . . 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 1151	. . . . . . . . . 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 486	. . . . . . . . 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 16	. . . . . . . 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 3175	. . . . . . 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 1137	. . . . . 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 92	. . . . 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 1128	. . . 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 14671	. . . . . 6 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (♯‘𝐹) ∈ ℕ₀)
50		1zzd 12720	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ ℤ)
51		nn0z 12710	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℤ)
52		fzon 13808	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘𝐹) ∈ ℤ) → ((♯‘𝐹) ≤ 1 ↔ (1..^(♯‘𝐹)) = ∅))
53	50, 51, 52	syl2anc 596	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ((♯‘𝐹) ≤ 1 ↔ (1..^(♯‘𝐹)) = ∅))
54		nn0re 12608	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (♯‘𝐹) ∈ ℝ)
55		1red 11302	. . . . . . . . . . . 12 ⊢ ((♯‘𝐹) ∈ ℕ₀ → 1 ∈ ℝ)
56	54, 55	lenltd 11449	. . . . . . . . . . 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 2971	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (1 < (♯‘𝐹) → (1..^(♯‘𝐹)) ≠ ∅))
60		rspn0 4304	. . . . . . . 8 ⊢ ((1..^(♯‘𝐹)) ≠ ∅ → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2))
61	59, 60	syl6com 38	. . . . . . 7 ⊢ (1 < (♯‘𝐹) → ((♯‘𝐹) ∈ ℕ₀ → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2)))
62	61	com3l 90	. . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2 → (1 < (♯‘𝐹) → 𝑆 ≤ 2)))
63	49, 62	syl 18	. . . . 5 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ 2 → (1 < (♯‘𝐹) → 𝑆 ≤ 2)))
64	63	3ad2ant2 1152	. . . 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 48	. . 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 18	. 2 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → (𝐺 ∈ UPGraph → (1 < (♯‘𝐹) → 𝑆 ≤ 2)))
67	66	3imp21 1131	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (♯‘𝐹)) → 𝑆 ≤ 2)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 Vcvv 3451 ∩ cin 3898 ∅c0 4279 class class class wbr 5103 dom cdm 5651 Fun wfun 6531 Fn wfn 6532 ‘cfv 6537 (class class class)co 7418 0cc0 11193 1c1 11194 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 − cmin 11534 2c2 12390 ℕ₀cn0 12599 ℕ₀^*cxnn0 12672 ℤcz 12686 ...cfz 13632 ..^cfzo 13781 ♯chash 14467 Word cword 14651 Vtxcvtx 29567 iEdgciedg 29568 UHGraphcuhgr 29627 UPGraphcupgr 29651 EdgWalks cewlks 30169

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-oadd 8473 df-er 8710 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-n0 12600 df-xnn0 12673 df-z 12687 df-uz 12959 df-fz 13633 df-fzo 13782 df-hash 14468 df-word 14652 df-uhgr 29629 df-upgr 29653 df-ewlks 30172

This theorem is used by: (None)

W3C validator