Theorem upgrwlkupwlk 46276

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.

Step	Hyp	Ref	Expression
1		wlkv 28729	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2		eqid 2731	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2731	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	iswlk 28727	. . . . . . . 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-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
5		simpr1 1194	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → 𝐹 ∈ Word dom (iEdg‘𝐺))
6		simpr2 1195	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
7		df-ifp 1062	. . . . . . . . . . . . . . . . 17 ⊢ (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ↔ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) ∨ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
8		dfsn2 4632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘𝑘)}
9		preq2 4728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘), (𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
10	8, 9	eqtrid 2783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → {(𝑃‘𝑘)} = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
11	10	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} ↔ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
12	11	biimpa 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
13	12	a1d 25	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) → ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
14		simpr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → 𝐺 ∈ UPGraph)
15		simpl 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → 𝐹 ∈ Word dom (iEdg‘𝐺))
16		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
17	3, 16	upgredginwlk 28753	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (𝑘 ∈ (0..^(♯‘𝐹)) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)))
18	14, 15, 17	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → (𝑘 ∈ (0..^(♯‘𝐹)) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)))
19	18	imp 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺))
20		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → 𝐺 ∈ UPGraph)
21	20	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → 𝐺 ∈ UPGraph)
22	21	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) → 𝐺 ∈ UPGraph)
23	22	adantr 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)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝐺 ∈ UPGraph)
24		simplr 767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺))
25		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))
26		df-ne 2940	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
27		fvexd 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘𝑘) ∈ V)
28		fvexd 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘(𝑘 + 1)) ∈ V)
29		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
30	27, 28, 29	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
31	26, 30	sylbir 234	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
32	31	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
33	32	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
34	2, 16	upgredgpr 28262	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ UPGraph ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∧ ((𝑃‘𝑘) ∈ V ∧ (𝑃‘(𝑘 + 1)) ∈ V ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
35	23, 24, 25, 33, 34	syl31anc 1373	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
36	35	eqcomd 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺)) ∧ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
37	36	exp31 420	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((iEdg‘𝐺)‘(𝐹‘𝑘)) ∈ (Edg‘𝐺) → ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
38	19, 37	mpd 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
39	38	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
40	13, 39	jaoi 855	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) ∨ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
41	40	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}) ∨ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
42	7, 41	biimtrid 241	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
43	42	ralimdva 3166	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
44	43	ex 413	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
45	44	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
46	45	3impia 1117	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
47	46	impcom 408	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
48	5, 6, 47	3jca 1128	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ 𝐺 ∈ UPGraph) ∧ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
49	48	exp31 420	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐺 ∈ UPGraph → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
50	49	com23 86	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → (𝐺 ∈ UPGraph → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
51	4, 50	sylbid 239	. . . . . . 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)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
52	51	impd 411	. . . . . 6 ⊢ ((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)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
53	52	impcom 408	. . . . 5 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐺 ∈ UPGraph) ∧ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
54	2, 3	isupwlk 46272	. . . . . 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)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
55	54	adantl 482	. . . . 5 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐺 ∈ 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)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
56	53, 55	mpbird 256	. . . 4 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐺 ∈ UPGraph) ∧ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) → 𝐹(UPWalks‘𝐺)𝑃)
57	56	exp31 420	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → 𝐹(UPWalks‘𝐺)𝑃)))
58	1, 57	mpid 44	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ UPGraph → 𝐹(UPWalks‘𝐺)𝑃))
59	58	impcom 408	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845  if-wif 1061   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  Vcvv 3470   ⊆ wss 3941  {csn 4619  {cpr 4621   class class class wbr 5138  dom cdm 5666  ⟶wf 6525  ‘cfv 6529  (class class class)co 7390  0cc0 11089  1c1 11090   + caddc 11092  ...cfz 13463  ..^cfzo 13606  ♯chash 14269  Word cword 14443  Vtxcvtx 28116  iEdgciedg 28117  Edgcedg 28167  UPGraphcupgr 28200  Walkscwlks 28713  UPWalkscupwlks 46269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705  ax-cnex 11145  ax-resscn 11146  ax-1cn 11147  ax-icn 11148  ax-addcl 11149  ax-addrcl 11150  ax-mulcl 11151  ax-mulrcl 11152  ax-mulcom 11153  ax-addass 11154  ax-mulass 11155  ax-distr 11156  ax-i2m1 11157  ax-1ne0 11158  ax-1rid 11159  ax-rnegex 11160  ax-rrecex 11161  ax-cnre 11162  ax-pre-lttri 11163  ax-pre-lttrn 11164  ax-pre-ltadd 11165  ax-pre-mulgt0 11166

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6286  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-riota 7346  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7836  df-1st 7954  df-2nd 7955  df-frecs 8245  df-wrecs 8276  df-recs 8350  df-rdg 8389  df-1o 8445  df-2o 8446  df-oadd 8449  df-er 8683  df-map 8802  df-en 8920  df-dom 8921  df-sdom 8922  df-fin 8923  df-dju 9875  df-card 9913  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11425  df-neg 11426  df-nn 12192  df-2 12254  df-n0 12452  df-xnn0 12524  df-z 12538  df-uz 12802  df-fz 13464  df-fzo 13607  df-hash 14270  df-word 14444  df-edg 28168  df-uhgr 28178  df-upgr 28202  df-wlks 28716  df-upwlks 46270

This theorem is referenced by:  upgrwlkupwlkb  46277