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Theorem upgrewlkle2 26558
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.
StepHypRef Expression
1 eqid 2651 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
21ewlkprop 26555 . . 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)‘(𝐹𝑘))))))
3 fvex 6239 . . . . . . . . . . 11 ((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V
4 hashin 13237 . . . . . . . . . . 11 (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V → (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))))
53, 4ax-mp 5 . . . . . . . . . 10 (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))))
6 simpl3 1086 . . . . . . . . . . . 12 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → 𝐺 ∈ UPGraph)
7 upgruhgr 26042 . . . . . . . . . . . . . . . 16 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
81uhgrfun 26006 . . . . . . . . . . . . . . . 16 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
97, 8syl 17 . . . . . . . . . . . . . . 15 (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10 funfn 5956 . . . . . . . . . . . . . . 15 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
119, 10sylib 208 . . . . . . . . . . . . . 14 (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12113ad2ant3 1104 . . . . . . . . . . . . 13 (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1312adantr 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 12524 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1..^(#‘𝐹)) → 𝑘 ∈ (1...(#‘𝐹)))
16 fz1fzo0m1 12555 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...(#‘𝐹)) → (𝑘 − 1) ∈ (0..^(#‘𝐹)))
1715, 16syl 17 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (1..^(#‘𝐹)) → (𝑘 − 1) ∈ (0..^(#‘𝐹)))
1817adantl 481 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝑘 − 1) ∈ (0..^(#‘𝐹)))
1914, 18jca 553 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (𝑘 − 1) ∈ (0..^(#‘𝐹))))
20 wrdsymbcl 13350 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (𝑘 − 1) ∈ (0..^(#‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
2119, 20syl 17 . . . . . . . . . . . . 13 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
22213ad2antl2 1244 . . . . . . . . . . . 12 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
23 eqid 2651 . . . . . . . . . . . . 13 (Vtx‘𝐺) = (Vtx‘𝐺)
2423, 1upgrle 26030 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺)) → (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
256, 13, 22, 24syl3anc 1366 . . . . . . . . . . 11 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
263inex1 4832 . . . . . . . . . . . . . . 15 (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘))) ∈ V
27 hashxrcl 13186 . . . . . . . . . . . . . . 15 ((((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘))) ∈ V → (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∈ ℝ*)
2826, 27ax-mp 5 . . . . . . . . . . . . . 14 (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∈ ℝ*
29 hashxrcl 13186 . . . . . . . . . . . . . . 15 (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V → (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ*)
303, 29ax-mp 5 . . . . . . . . . . . . . 14 (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ*
31 2re 11128 . . . . . . . . . . . . . . 15 2 ∈ ℝ
3231rexri 10135 . . . . . . . . . . . . . 14 2 ∈ ℝ*
3328, 30, 323pm3.2i 1259 . . . . . . . . . . . . 13 ((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∈ ℝ* ∧ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ* ∧ 2 ∈ ℝ*)
3433a1i 11 . . . . . . . . . . . 12 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → ((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∈ ℝ* ∧ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ* ∧ 2 ∈ ℝ*))
35 xrletr 12027 . . . . . . . . . . . 12 (((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∈ ℝ* ∧ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∈ ℝ* ∧ 2 ∈ ℝ*) → (((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∧ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2) → (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2))
3634, 35syl 17 . . . . . . . . . . 11 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ∧ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2) → (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2))
3725, 36mpan2d 710 . . . . . . . . . 10 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → ((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) → (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2))
385, 37mpi 20 . . . . . . . . 9 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2)
39 xnn0xr 11406 . . . . . . . . . . . . . 14 (𝑆 ∈ ℕ0*𝑆 ∈ ℝ*)
4028a1i 11 . . . . . . . . . . . . . 14 (𝑆 ∈ ℕ0* → (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∈ ℝ*)
4132a1i 11 . . . . . . . . . . . . . 14 (𝑆 ∈ ℕ0* → 2 ∈ ℝ*)
42 xrletr 12027 . . . . . . . . . . . . . 14 ((𝑆 ∈ ℝ* ∧ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∈ ℝ* ∧ 2 ∈ ℝ*) → ((𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∧ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2) → 𝑆 ≤ 2))
4339, 40, 41, 42syl3anc 1366 . . . . . . . . . . . . 13 (𝑆 ∈ ℕ0* → ((𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∧ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2) → 𝑆 ≤ 2))
4443expcomd 453 . . . . . . . . . . . 12 (𝑆 ∈ ℕ0* → ((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2 → (𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) → 𝑆 ≤ 2)))
4544adantl 481 . . . . . . . . . . 11 ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) → ((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2 → (𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) → 𝑆 ≤ 2)))
46453ad2ant1 1102 . . . . . . . . . 10 (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → ((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2 → (𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) → 𝑆 ≤ 2)))
4746adantr 480 . . . . . . . . 9 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → ((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2 → (𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) → 𝑆 ≤ 2)))
4838, 47mpd 15 . . . . . . . 8 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) → 𝑆 ≤ 2))
4948ralimdva 2991 . . . . . . 7 (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph) → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) → ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2))
50493exp 1283 . . . . . 6 ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) → (𝐹 ∈ Word dom (iEdg‘𝐺) → (𝐺 ∈ UPGraph → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) → ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2))))
5150com34 91 . . . . 5 ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) → (𝐹 ∈ Word dom (iEdg‘𝐺) → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) → (𝐺 ∈ UPGraph → ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2))))
52513imp 1275 . . . 4 (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘))))) → (𝐺 ∈ UPGraph → ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2))
53 lencl 13356 . . . . . 6 (𝐹 ∈ Word dom (iEdg‘𝐺) → (#‘𝐹) ∈ ℕ0)
54 1zzd 11446 . . . . . . . . . . . . . . . . . 18 ((#‘𝐹) ∈ ℕ0 → 1 ∈ ℤ)
55 nn0z 11438 . . . . . . . . . . . . . . . . . 18 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℤ)
5654, 55jca 553 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) ∈ ℕ0 → (1 ∈ ℤ ∧ (#‘𝐹) ∈ ℤ))
57 fzon 12528 . . . . . . . . . . . . . . . . 17 ((1 ∈ ℤ ∧ (#‘𝐹) ∈ ℤ) → ((#‘𝐹) ≤ 1 ↔ (1..^(#‘𝐹)) = ∅))
5856, 57syl 17 . . . . . . . . . . . . . . . 16 ((#‘𝐹) ∈ ℕ0 → ((#‘𝐹) ≤ 1 ↔ (1..^(#‘𝐹)) = ∅))
5958bicomd 213 . . . . . . . . . . . . . . 15 ((#‘𝐹) ∈ ℕ0 → ((1..^(#‘𝐹)) = ∅ ↔ (#‘𝐹) ≤ 1))
60 nn0re 11339 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℝ)
61 1red 10093 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) ∈ ℕ0 → 1 ∈ ℝ)
6260, 61jca 553 . . . . . . . . . . . . . . . 16 ((#‘𝐹) ∈ ℕ0 → ((#‘𝐹) ∈ ℝ ∧ 1 ∈ ℝ))
63 lenlt 10154 . . . . . . . . . . . . . . . 16 (((#‘𝐹) ∈ ℝ ∧ 1 ∈ ℝ) → ((#‘𝐹) ≤ 1 ↔ ¬ 1 < (#‘𝐹)))
6462, 63syl 17 . . . . . . . . . . . . . . 15 ((#‘𝐹) ∈ ℕ0 → ((#‘𝐹) ≤ 1 ↔ ¬ 1 < (#‘𝐹)))
6559, 64bitrd 268 . . . . . . . . . . . . . 14 ((#‘𝐹) ∈ ℕ0 → ((1..^(#‘𝐹)) = ∅ ↔ ¬ 1 < (#‘𝐹)))
6665biimpd 219 . . . . . . . . . . . . 13 ((#‘𝐹) ∈ ℕ0 → ((1..^(#‘𝐹)) = ∅ → ¬ 1 < (#‘𝐹)))
6766necon2ad 2838 . . . . . . . . . . . 12 ((#‘𝐹) ∈ ℕ0 → (1 < (#‘𝐹) → (1..^(#‘𝐹)) ≠ ∅))
6867impcom 445 . . . . . . . . . . 11 ((1 < (#‘𝐹) ∧ (#‘𝐹) ∈ ℕ0) → (1..^(#‘𝐹)) ≠ ∅)
69 rspn0 3967 . . . . . . . . . . 11 ((1..^(#‘𝐹)) ≠ ∅ → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2))
7068, 69syl 17 . . . . . . . . . 10 ((1 < (#‘𝐹) ∧ (#‘𝐹) ∈ ℕ0) → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2))
7170ex 449 . . . . . . . . 9 (1 < (#‘𝐹) → ((#‘𝐹) ∈ ℕ0 → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → 𝑆 ≤ 2)))
7271com23 86 . . . . . . . 8 (1 < (#‘𝐹) → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → ((#‘𝐹) ∈ ℕ0𝑆 ≤ 2)))
7372com13 88 . . . . . . 7 ((#‘𝐹) ∈ ℕ0 → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → (1 < (#‘𝐹) → 𝑆 ≤ 2)))
7473a1i 11 . . . . . 6 (𝐹 ∈ Word dom (iEdg‘𝐺) → ((#‘𝐹) ∈ ℕ0 → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → (1 < (#‘𝐹) → 𝑆 ≤ 2))))
7553, 74mpd 15 . . . . 5 (𝐹 ∈ Word dom (iEdg‘𝐺) → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → (1 < (#‘𝐹) → 𝑆 ≤ 2)))
76753ad2ant2 1103 . . . 4 (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘))))) → (∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ 2 → (1 < (#‘𝐹) → 𝑆 ≤ 2)))
7752, 76syld 47 . . 3 (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘))))) → (𝐺 ∈ UPGraph → (1 < (#‘𝐹) → 𝑆 ≤ 2)))
782, 77syl 17 . 2 (𝐹 ∈ (𝐺 EdgWalks 𝑆) → (𝐺 ∈ UPGraph → (1 < (#‘𝐹) → 𝑆 ≤ 2)))
79783imp21 1298 1 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (#‘𝐹)) → 𝑆 ≤ 2)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cin 3606  c0 3948   class class class wbr 4685  dom cdm 5143  Fun wfun 5920   Fn wfn 5921  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975  *cxr 10111   < clt 10112  cle 10113  cmin 10304  2c2 11108  0cn0 11330  0*cxnn0 11401  cz 11415  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323  Vtxcvtx 25919  iEdgciedg 25920  UHGraphcuhgr 25996  UPGraphcupgr 26020   EdgWalks cewlks 26547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-uhgr 25998  df-upgr 26022  df-ewlks 26550
This theorem is referenced by: (None)
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