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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  cr 9879  0cc0 9880  1c1 9881  *cxr 10017   < clt 10018  cle 10019  cmin 10210  2c2 11014  0cn0 11236  0*cxnn0 11307  cz 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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