Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1wlkiswwlks1 Structured version   Visualization version   GIF version

Theorem 1wlkiswwlks1 41064
 Description: The sequence of vertices in a walk is a walk as word in a pseudograph. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.)
Assertion
Ref Expression
1wlkiswwlks1 (𝐺 ∈ UPGraph → (𝐹(1Walks‘𝐺)𝑃𝑃 ∈ (WWalkS‘𝐺)))

Proof of Theorem 1wlkiswwlks1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 1wlkn0 40825 . . 3 (𝐹(1Walks‘𝐺)𝑃𝑃 ≠ ∅)
2 wlkv 40815 . . . 4 (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
3 simpr 476 . . . . . . . . 9 (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph ) → 𝐺 ∈ UPGraph )
4 simpl2 1058 . . . . . . . . 9 (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph ) → 𝐹 ∈ V)
5 simpl3 1059 . . . . . . . . 9 (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph ) → 𝑃 ∈ V)
6 eqid 2610 . . . . . . . . . 10 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2610 . . . . . . . . . 10 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7upgriswlk 40849 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
93, 4, 5, 8syl3anc 1318 . . . . . . . 8 (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph ) → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
10 simpr 476 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → 𝑃 ≠ ∅)
11 ffz0iswrd 13187 . . . . . . . . . . . . . 14 (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → 𝑃 ∈ Word (Vtx‘𝐺))
12113ad2ant2 1076 . . . . . . . . . . . . 13 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → 𝑃 ∈ Word (Vtx‘𝐺))
1312ad2antlr 759 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → 𝑃 ∈ Word (Vtx‘𝐺))
14 upgruhgr 25768 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
157uhgrfun 25732 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
16 funfn 5833 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1716biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . 24 (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1814, 15, 173syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1918ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
20 wrdsymbcl 13173 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (𝐹𝑖) ∈ dom (iEdg‘𝐺))
2120ad4ant14 1285 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (𝐹𝑖) ∈ dom (iEdg‘𝐺))
22 fnfvelrn 6264 . . . . . . . . . . . . . . . . . . . . . 22 (((iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ (𝐹𝑖) ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ ran (iEdg‘𝐺))
2319, 21, 22syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ (0..^(#‘𝐹))) → ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ ran (iEdg‘𝐺))
24 edgaval 25794 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
2524ad2antlr 759 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (Edg‘𝐺) = ran (iEdg‘𝐺))
2623, 25eleqtrrd 2691 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ (0..^(#‘𝐹))) → ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ (Edg‘𝐺))
27 eleq1 2676 . . . . . . . . . . . . . . . . . . . . 21 ({(𝑃𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘(𝐹𝑖)) → ({(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ (Edg‘𝐺)))
2827eqcoms 2618 . . . . . . . . . . . . . . . . . . . 20 (((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → ({(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ((iEdg‘𝐺)‘(𝐹𝑖)) ∈ (Edg‘𝐺)))
2926, 28syl5ibrcom 236 . . . . . . . . . . . . . . . . . . 19 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ (0..^(#‘𝐹))) → (((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
3029ralimdva 2945 . . . . . . . . . . . . . . . . . 18 (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) → (∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → ∀𝑖 ∈ (0..^(#‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
3130ex 449 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) → (𝐺 ∈ UPGraph → (∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → ∀𝑖 ∈ (0..^(#‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
3231com23 84 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^(#‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
33323impia 1253 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^(#‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
3433impcom 445 . . . . . . . . . . . . . 14 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) → ∀𝑖 ∈ (0..^(#‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
35 lencl 13179 . . . . . . . . . . . . . . . . . . . 20 (𝐹 ∈ Word dom (iEdg‘𝐺) → (#‘𝐹) ∈ ℕ0)
36 ffz0hash 13088 . . . . . . . . . . . . . . . . . . . . . 22 (((#‘𝐹) ∈ ℕ0𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) → (#‘𝑃) = ((#‘𝐹) + 1))
3736ex 449 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → (#‘𝑃) = ((#‘𝐹) + 1)))
38 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑃) = ((#‘𝐹) + 1) → ((#‘𝑃) − 1) = (((#‘𝐹) + 1) − 1))
39 nn0cn 11179 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℂ)
40 pncan1 10333 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝐹) ∈ ℂ → (((#‘𝐹) + 1) − 1) = (#‘𝐹))
4139, 40syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝐹) ∈ ℕ0 → (((#‘𝐹) + 1) − 1) = (#‘𝐹))
4238, 41sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . . . 22 (((#‘𝐹) ∈ ℕ0 ∧ (#‘𝑃) = ((#‘𝐹) + 1)) → ((#‘𝑃) − 1) = (#‘𝐹))
4342ex 449 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝐹) ∈ ℕ0 → ((#‘𝑃) = ((#‘𝐹) + 1) → ((#‘𝑃) − 1) = (#‘𝐹)))
4437, 43syld 46 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐹) ∈ ℕ0 → (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((#‘𝑃) − 1) = (#‘𝐹)))
4535, 44syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐹 ∈ Word dom (iEdg‘𝐺) → (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) → ((#‘𝑃) − 1) = (#‘𝐹)))
4645imp 444 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) → ((#‘𝑃) − 1) = (#‘𝐹))
4746oveq2d 6565 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) → (0..^((#‘𝑃) − 1)) = (0..^(#‘𝐹)))
4847raleqdv 3121 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(#‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
49483adant3 1074 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(#‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5049adantl 481 . . . . . . . . . . . . . 14 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(#‘𝐹)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5134, 50mpbird 246 . . . . . . . . . . . . 13 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) → ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
5251adantr 480 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))
53 eqid 2610 . . . . . . . . . . . . 13 (Edg‘𝐺) = (Edg‘𝐺)
546, 53iswwlks 41039 . . . . . . . . . . . 12 (𝑃 ∈ (WWalkS‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5510, 13, 52, 54syl3anbrc 1239 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ 𝑃 ≠ ∅) → 𝑃 ∈ (WWalkS‘𝐺))
5655ex 449 . . . . . . . . . 10 ((𝐺 ∈ UPGraph ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalkS‘𝐺)))
5756ex 449 . . . . . . . . 9 (𝐺 ∈ UPGraph → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalkS‘𝐺))))
5857adantl 481 . . . . . . . 8 (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph ) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalkS‘𝐺))))
599, 58sylbid 229 . . . . . . 7 (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph ) → (𝐹(1Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalkS‘𝐺))))
6059ex 449 . . . . . 6 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐺 ∈ UPGraph → (𝐹(1Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalkS‘𝐺)))))
6160com23 84 . . . . 5 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → (𝑃 ≠ ∅ → 𝑃 ∈ (WWalkS‘𝐺)))))
6261com34 89 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ → (𝐺 ∈ UPGraph → 𝑃 ∈ (WWalkS‘𝐺)))))
632, 62mpcom 37 . . 3 (𝐹(1Walks‘𝐺)𝑃 → (𝑃 ≠ ∅ → (𝐺 ∈ UPGraph → 𝑃 ∈ (WWalkS‘𝐺))))
641, 63mpd 15 . 2 (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → 𝑃 ∈ (WWalkS‘𝐺)))
6564com12 32 1 (𝐺 ∈ UPGraph → (𝐹(1Walks‘𝐺)𝑃𝑃 ∈ (WWalkS‘𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722   UPGraph cupgr 25747  Edgcedga 25792  1Walksc1wlks 40796  WWalkScwwlks 41028 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-ifp 1007  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-rmo 2904  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-2o 7448  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-cda 8873  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-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-1wlks 40800  df-wlks 40801  df-wwlks 41033 This theorem is referenced by:  1wlklnwwlkln1  41065  1wlkiswwlks  41073  1wlkiswwlkupgr  41075  elwspths2spth  41171
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