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Theorem upgrewlkle2 26372
 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 2621 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
21ewlkprop 26369 . . 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 6158 . . . . . . . . . . 11 ((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V
4 hashin 13139 . . . . . . . . . . 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 1064 . . . . . . . . . . . 12 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → 𝐺 ∈ UPGraph )
7 upgruhgr 25892 . . . . . . . . . . . . . . . 16 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
81uhgrfun 25857 . . . . . . . . . . . . . . . 16 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
97, 8syl 17 . . . . . . . . . . . . . . 15 (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10 funfn 5877 . . . . . . . . . . . . . . 15 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
119, 10sylib 208 . . . . . . . . . . . . . 14 (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12113ad2ant3 1082 . . . . . . . . . . . . 13 (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1312adantr 481 . . . . . . . . . . . 12 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14 simpl 473 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → 𝐹 ∈ Word dom (iEdg‘𝐺))
15 elfzofz 12426 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1..^(#‘𝐹)) → 𝑘 ∈ (1...(#‘𝐹)))
16 fz1fzo0m1 12456 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...(#‘𝐹)) → (𝑘 − 1) ∈ (0..^(#‘𝐹)))
1715, 16syl 17 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (1..^(#‘𝐹)) → (𝑘 − 1) ∈ (0..^(#‘𝐹)))
1817adantl 482 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝑘 − 1) ∈ (0..^(#‘𝐹)))
1914, 18jca 554 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (𝑘 − 1) ∈ (0..^(#‘𝐹))))
20 wrdsymbcl 13257 . . . . . . . . . . . . . 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 1222 . . . . . . . . . . . 12 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (𝐹‘(𝑘 − 1)) ∈ dom (iEdg‘𝐺))
23 eqid 2621 . . . . . . . . . . . . 13 (Vtx‘𝐺) = (Vtx‘𝐺)
2423, 1upgrle 25881 . . . . . . . . . . . 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 1323 . . . . . . . . . . 11 ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑘 ∈ (1..^(#‘𝐹))) → (#‘((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1)))) ≤ 2)
263inex1 4759 . . . . . . . . . . . . . . 15 (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘))) ∈ V
27 hashxrcl 13088 . . . . . . . . . . . . . . 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 13088 . . . . . . . . . . . . . . 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 11034 . . . . . . . . . . . . . . 15 2 ∈ ℝ
3231rexri 10041 . . . . . . . . . . . . . 14 2 ∈ ℝ*
3328, 30, 323pm3.2i 1237 . . . . . . . . . . . . 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 11933 . . . . . . . . . . . 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 709 . . . . . . . . . 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 11312 . . . . . . . . . . . . . 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 11933 . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . 13 (𝑆 ∈ ℕ0* → ((𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ∧ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2) → 𝑆 ≤ 2))
4443expcomd 454 . . . . . . . . . . . 12 (𝑆 ∈ ℕ0* → ((#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) ≤ 2 → (𝑆 ≤ (#‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹𝑘)))) → 𝑆 ≤ 2)))
4544adantl 482 . . . . . . . . . . 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 1080 . . . . . . . . . 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 481 . . . . . . . . 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 2956 . . . . . . 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 1261 . . . . . 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 1254 . . . 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 13263 . . . . . 6 (𝐹 ∈ Word dom (iEdg‘𝐺) → (#‘𝐹) ∈ ℕ0)
54 1zzd 11352 . . . . . . . . . . . . . . . . . 18 ((#‘𝐹) ∈ ℕ0 → 1 ∈ ℤ)
55 nn0z 11344 . . . . . . . . . . . . . . . . . 18 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℤ)
5654, 55jca 554 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) ∈ ℕ0 → (1 ∈ ℤ ∧ (#‘𝐹) ∈ ℤ))
57 fzon 12430 . . . . . . . . . . . . . . . . 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 11245 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℝ)
61 1red 9999 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) ∈ ℕ0 → 1 ∈ ℝ)
6260, 61jca 554 . . . . . . . . . . . . . . . 16 ((#‘𝐹) ∈ ℕ0 → ((#‘𝐹) ∈ ℝ ∧ 1 ∈ ℝ))
63 lenlt 10060 . . . . . . . . . . . . . . . 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 2805 . . . . . . . . . . . 12 ((#‘𝐹) ∈ ℕ0 → (1 < (#‘𝐹) → (1..^(#‘𝐹)) ≠ ∅))
6867impcom 446 . . . . . . . . . . 11 ((1 < (#‘𝐹) ∧ (#‘𝐹) ∈ ℕ0) → (1..^(#‘𝐹)) ≠ ∅)
69 rspn0 3910 . . . . . . . . . . 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 450 . . . . . . . . 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 1081 . . . 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 1274 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  Vcvv 3186   ∩ cin 3554  ∅c0 3891   class class class wbr 4613  dom cdm 5074  Fun wfun 5841   Fn wfn 5842  ‘cfv 5847  (class class class)co 6604  ℝcr 9879  0cc0 9880  1c1 9881  ℝ*cxr 10017   < clt 10018   ≤ cle 10019   − cmin 10210  2c2 11014  ℕ0cn0 11236  ℕ0*cxnn0 11307  ℤcz 11321  ...cfz 12268  ..^cfzo 12406  #chash 13057  Word cword 13230  Vtxcvtx 25774  iEdgciedg 25775   UHGraph cuhgr 25847   UPGraph cupgr 25871   EdgWalks cewlks 26361 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-uhgr 25849  df-upgr 25873  df-ewlks 26364 This theorem is referenced by: (None)
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