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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  upgrwlkupwlk Structured version   Visualization version   GIF version

Theorem upgrwlkupwlk 42046
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 26564 . . 3 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2 eqid 2651 . . . . . . . . 9 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2651 . . . . . . . . 9 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3iswlk 26562 . . . . . . . 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 1087 . . . . . . . . . . 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 1088 . . . . . . . . . . 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 1033 . . . . . . . . . . . . . . . . 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 4223 . . . . . . . . . . . . . . . . . . . . . . 23 {(𝑃𝑘)} = {(𝑃𝑘), (𝑃𝑘)}
9 preq2 4301 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘), (𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
108, 9syl5eq 2697 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃𝑘)} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
1110eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘)} ↔ ((iEdg‘𝐺)‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
1211biimpa 500 . . . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → 𝐺 ∈ UPGraph)
15 simpl 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) → 𝐹 ∈ Word dom (iEdg‘𝐺))
16 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . 24 (Edg‘𝐺) = (Edg‘𝐺)
173, 16upgredginwlk 26588 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (𝑘 ∈ (0..^(#‘𝐹)) → ((iEdg‘𝐺)‘(𝐹𝑘)) ∈ (Edg‘𝐺)))
1814, 15, 17syl2anr 494 . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → 𝐺 ∈ UPGraph)
2120adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)))
27 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ∈ V)
28 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘(𝑘 + 1)) ∈ V)
29 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
3027, 28, 293jca 1261 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹𝑘))) → ((𝑃𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
3332adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 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 26082 . . . . . . . . . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . . . . . . . . . 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 2657 . . . . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . . . . . . 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 2991 . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . 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 1280 . . . . . . . . . . . 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 445 . . . . . . . . . . 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 1261 . . . . . . . . . 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 629 . . . . . . . . 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 446 . . . . . 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 445 . . . . 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 42042 . . . . . 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 481 . . . . 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 629 . . 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 445 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  wo 382  wa 383  if-wif 1032  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  wss 3607  {csn 4210  {cpr 4212   class class class wbr 4685  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  UPGraphcupgr 26020  Walkscwlks 26548  UPWalkscupwlks 42039
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-ifp 1033  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-rmo 2949  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-2o 7606  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-cda 9028  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-edg 25985  df-uhgr 25998  df-upgr 26022  df-wlks 26551  df-upwlks 42040
This theorem is referenced by:  upgrwlkupwlkb  42047
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