Theorem upgr2wlkdc 16147

Description: Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.)

Hypotheses

Ref	Expression
upgr2wlk.v	⊢ 𝑉 = (Vtx‘𝐺)
upgr2wlk.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
upgr2wlkdc	⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

Distinct variable groups:   𝑘,𝐹   𝑃,𝑘   𝑘,𝐺   𝑘,𝐼   𝑘,𝑉

Proof of Theorem upgr2wlkdc

Step	Hyp	Ref	Expression
1		simprl 529	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝐹(Walks‘𝐺)𝑃)
2		upgr2wlk.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
3		upgr2wlk.i	. . . . . . . . . 10 ⊢ 𝐼 = (iEdg‘𝐺)
4	2, 3	upgriswlkdc 16132	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
5	4	adantr 276	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
6	1, 5	mpbid 147	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
7	6	simp1d 1033	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝐹 ∈ Word dom 𝐼)
8		wrdf 11095	. . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
9	7, 8	syl 14	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
10		simprr 531	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝐹 ≈ 2o)
11		hash2en 11083	. . . . . . . . 9 ⊢ (𝐹 ≈ 2o ↔ (𝐹 ∈ Fin ∧ (♯‘𝐹) = 2))
12	11	simprbi 275	. . . . . . . 8 ⊢ (𝐹 ≈ 2o → (♯‘𝐹) = 2)
13	10, 12	syl 14	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (♯‘𝐹) = 2)
14	13	oveq2d 6026	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (0..^(♯‘𝐹)) = (0..^2))
15	14	feq2d 5464	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ↔ 𝐹:(0..^2)⟶dom 𝐼))
16	9, 15	mpbid 147	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝐹:(0..^2)⟶dom 𝐼)
17	6	simp2d 1034	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
18	13	oveq2d 6026	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (0...(♯‘𝐹)) = (0...2))
19	18	feq2d 5464	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
20	17, 19	mpbid 147	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝑃:(0...2)⟶𝑉)
21	6	simp3d 1035	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
22		simpl 109	. . . . . . 7 ⊢ ((DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
23	22	ralimi 2593	. . . . . 6 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∀𝑘 ∈ (0..^(♯‘𝐹))DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
24	21, 23	syl 14	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → ∀𝑘 ∈ (0..^(♯‘𝐹))DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
25		oveq2 6018	. . . . . . 7 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
26		fzo0to2pr 10441	. . . . . . 7 ⊢ (0..^2) = {0, 1}
27	25, 26	eqtrdi 2278	. . . . . 6 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
28	13, 27	syl 14	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (0..^(♯‘𝐹)) = {0, 1})
29	24, 28	raleqtrdv 2736	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
30	16, 20, 29	3jca 1201	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))))
31		simpr 110	. . . . . 6 ⊢ ((DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
32	31	ralimi 2593	. . . . 5 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
33	21, 32	syl 14	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
34	28	raleqdv 2734	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
35		2wlklem 16146	. . . . 5 ⊢ (∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
36	34, 35	bitrdi 196	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
37	33, 36	mpbid 147	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
38	30, 37	jca 306	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
39		simprl1 1066	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → 𝐹:(0..^2)⟶dom 𝐼)
40		2nn0 9402	. . . . . 6 ⊢ 2 ∈ ℕ0
41		iswrdinn0 11094	. . . . . 6 ⊢ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 2 ∈ ℕ0) → 𝐹 ∈ Word dom 𝐼)
42	39, 40, 41	sylancl 413	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → 𝐹 ∈ Word dom 𝐼)
43		simprl2 1067	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → 𝑃:(0...2)⟶𝑉)
44		fnfzo0hash 11075	. . . . . . . . 9 ⊢ ((2 ∈ ℕ0 ∧ 𝐹:(0..^2)⟶dom 𝐼) → (♯‘𝐹) = 2)
45	40, 39, 44	sylancr 414	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → (♯‘𝐹) = 2)
46	45	oveq2d 6026	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → (0...(♯‘𝐹)) = (0...2))
47	46	feq2d 5464	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → (𝑃:(0...(♯‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
48	43, 47	mpbird 167	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
49		simprl3 1068	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
50	45, 27	syl 14	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → (0..^(♯‘𝐹)) = {0, 1})
51	49, 50	raleqtrrdv 2738	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → ∀𝑘 ∈ (0..^(♯‘𝐹))DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
52		simprr 531	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
53	52, 35	sylibr 134	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
54	53, 50	raleqtrrdv 2738	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
55	51, 54	jca 306	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → (∀𝑘 ∈ (0..^(♯‘𝐹))DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
56		r19.26 2657	. . . . . 6 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (∀𝑘 ∈ (0..^(♯‘𝐹))DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
57	55, 56	sylibr 134	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
58	42, 48, 57	3jca 1201	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
59	4	adantr 276	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))))
60	58, 59	mpbird 167	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → 𝐹(Walks‘𝐺)𝑃)
61		0z 9473	. . . . . . . . 9 ⊢ 0 ∈ ℤ
62		2z 9490	. . . . . . . . 9 ⊢ 2 ∈ ℤ
63		fzofig 10671	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 2 ∈ ℤ) → (0..^2) ∈ Fin)
64	61, 62, 63	mp2an 426	. . . . . . . 8 ⊢ (0..^2) ∈ Fin
65		fex 5875	. . . . . . . 8 ⊢ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (0..^2) ∈ Fin) → 𝐹 ∈ V)
66	64, 65	mpan2 425	. . . . . . 7 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → 𝐹 ∈ V)
67		ffun 5479	. . . . . . 7 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → Fun 𝐹)
68		fundmeng 6973	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹)
69	66, 67, 68	syl2anc 411	. . . . . 6 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → dom 𝐹 ≈ 𝐹)
70	69	ensymd 6948	. . . . 5 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → 𝐹 ≈ dom 𝐹)
71		fdm 5482	. . . . . . 7 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → dom 𝐹 = (0..^2))
72	71, 26	eqtrdi 2278	. . . . . 6 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → dom 𝐹 = {0, 1})
73		1z 9488	. . . . . . 7 ⊢ 1 ∈ ℤ
74		0ne1 9193	. . . . . . 7 ⊢ 0 ≠ 1
75		pr2nelem 7380	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ≠ 1) → {0, 1} ≈ 2o)
76	61, 73, 74, 75	mp3an 1371	. . . . . 6 ⊢ {0, 1} ≈ 2o
77	72, 76	eqbrtrdi 4122	. . . . 5 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → dom 𝐹 ≈ 2o)
78		entr 6949	. . . . 5 ⊢ ((𝐹 ≈ dom 𝐹 ∧ dom 𝐹 ≈ 2o) → 𝐹 ≈ 2o)
79	70, 77, 78	syl2anc 411	. . . 4 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → 𝐹 ≈ 2o)
80	39, 79	syl 14	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → 𝐹 ≈ 2o)
81	60, 80	jca 306	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o))
82	38, 81	impbida 598	1 ⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1))) ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 839   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∀wral 2508  Vcvv 2799  {cpr 3667   class class class wbr 4083  dom cdm 4720  Fun wfun 5315  ⟶wf 5317  ‘cfv 5321  (class class class)co 6010  2_oc2o 6567   ≈ cen 6898  Fincfn 6900  0cc0 8015  1c1 8016   + caddc 8018  2c2 9177  ℕ₀cn0 9385  ℤcz 9462  ...cfz 10221  ..^cfzo 10355  ♯chash 11014  Word cword 11089  Vtxcvtx 15834  iEdgciedg 15835  UPGraphcupgr 15912  Walkscwlks 16089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-frec 6548  df-1o 6573  df-2o 6574  df-oadd 6577  df-er 6693  df-map 6810  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-z 9463  df-dec 9595  df-uz 9739  df-fz 10222  df-fzo 10356  df-ihash 11015  df-word 11090  df-ndx 13056  df-slot 13057  df-base 13059  df-edgf 15827  df-vtx 15836  df-iedg 15837  df-edg 15880  df-uhgrm 15890  df-upgren 15914  df-wlks 16090

This theorem is referenced by: (None)