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

Theorem upgrwlkupwlk 42249
Description: In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.)
Assertion
Ref Expression
upgrwlkupwlk ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃)

Proof of Theorem upgrwlkupwlk
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 wlkv 26743 . . 3 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2 eqid 2771 . . . . . . . . 9 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2771 . . . . . . . . 9 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3iswlk 26741 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))))))
5 simpr1 1233 . . . . . . . . . . 11 ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))))) → 𝐹 ∈ Word dom (iEdg‘𝐺))
6 simpr2 1235 . . . . . . . . . . 11 ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
7 df-ifp 1050 . . . . . . . . . . . . . . . . 17 (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) ↔ (((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}) ∨ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))))
8 dfsn2 4329 . . . . . . . . . . . . . . . . . . . . . . 23 {(𝑃𝑘)} = {(𝑃𝑘), (𝑃𝑘)}
9 preq2 4405 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘), (𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
108, 9syl5eq 2817 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
1110eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)} ↔ ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
1211biimpa 462 . . . . . . . . . . . . . . . . . . . 20 (((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}) → ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
1312a1d 25 . . . . . . . . . . . . . . . . . . 19 (((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}) → ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
14 simpr 471 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → 𝐺 ∈ UPGraph)
15 simpl 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 𝐹 ∈ Word dom (iEdg‘𝐺))
16 eqid 2771 . . . . . . . . . . . . . . . . . . . . . . . 24 (Edg‘𝐺) = (Edg‘𝐺)
173, 16upgredginwlk 26767 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (𝑘 ∈ (0..^(♯‘𝐹)) → ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)))
1814, 15, 17syl2anr 584 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → (𝑘 ∈ (0..^(♯‘𝐹)) → ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)))
1918imp 393 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺))
20 simprr 756 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → 𝐺 ∈ UPGraph)
2120adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → 𝐺 ∈ UPGraph)
2221adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)) → 𝐺 ∈ UPGraph)
2322adantr 466 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → 𝐺 ∈ UPGraph)
24 simplr 752 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺))
25 simprr 756 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))
26 df-ne 2944 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)))
27 fvexd 6344 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ∈ V)
28 fvexd 6344 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘(𝑘 + 1)) ∈ V)
29 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
3027, 28, 293jca 1122 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
3126, 30sylbir 225 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
3231adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
3332adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
342, 16upgredgpr 26259 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐺 ∈ UPGraph ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) ∧ ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))) → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = ((iEdg‘𝐺)‘(𝐹𝑘)))
3523, 24, 25, 33, 34syl31anc 1479 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = ((iEdg‘𝐺)‘(𝐹𝑘)))
3635eqcomd 2777 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
3736exp31 406 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺) → ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
3819, 37mpd 15 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
3938com12 32 . . . . . . . . . . . . . . . . . . 19 ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
4013, 39jaoi 846 . . . . . . . . . . . . . . . . . 18 ((((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}) ∨ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
4140com12 32 . . . . . . . . . . . . . . . . 17 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((((𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}) ∨ (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
427, 41syl5bi 232 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
4342ralimdva 3111 . . . . . . . . . . . . . . 15 (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
4443ex 397 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
4544com23 86 . . . . . . . . . . . . 13 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
46453impia 1109 . . . . . . . . . . . 12 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
4746impcom 394 . . . . . . . . . . 11 ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))))) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
485, 6, 473jca 1122 . . . . . . . . . 10 ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
4948exp31 406 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐺 ∈ UPGraph → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))))
5049com23 86 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘)))) → (𝐺 ∈ UPGraph → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))))
514, 50sylbid 230 . . . . . . 7 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))))
5251impd 396 . . . . . 6 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → ((𝐹(Walks‘𝐺)𝑃𝐺 ∈ UPGraph) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
5352impcom 394 . . . . 5 (((𝐹(Walks‘𝐺)𝑃𝐺 ∈ UPGraph) ∧ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
542, 3isupwlk 42245 . . . . . 6 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
5554adantl 467 . . . . 5 (((𝐹(Walks‘𝐺)𝑃𝐺 ∈ UPGraph) ∧ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
5653, 55mpbird 247 . . . 4 (((𝐹(Walks‘𝐺)𝑃𝐺 ∈ UPGraph) ∧ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) → 𝐹(UPWalks‘𝐺)𝑃)
5756exp31 406 . . 3 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → 𝐹(UPWalks‘𝐺)𝑃)))
581, 57mpid 44 . 2 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → 𝐹(UPWalks‘𝐺)𝑃))
5958impcom 394 1 ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836  if-wif 1049  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  Vcvv 3351  wss 3723  {csn 4316  {cpr 4318   class class class wbr 4786  dom cdm 5249  wf 6027  cfv 6031  (class class class)co 6793  0cc0 10138  1c1 10139   + caddc 10141  ...cfz 12533  ..^cfzo 12673  chash 13321  Word cword 13487  Vtxcvtx 26095  iEdgciedg 26096  Edgcedg 26160  UPGraphcupgr 26196  Walkscwlks 26727  UPWalkscupwlks 42242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-wlks 26730  df-upwlks 42243
This theorem is referenced by:  upgrwlkupwlkb  42250
  Copyright terms: Public domain W3C validator