Theorem upgr2wlkdc 16231

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 531	. . . . . . . 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 16214	. . . . . . . . 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 1035	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝐹 ∈ Word dom 𝐼)
8		wrdf 11120	. . . . . 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 533	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝐹 ≈ 2o)
11		hash2en 11108	. . . . . . . . 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 6034	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (0..^(♯‘𝐹)) = (0..^2))
15	14	feq2d 5470	. . . . 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 1036	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
18	13	oveq2d 6034	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → (0...(♯‘𝐹)) = (0...2))
19	18	feq2d 5470	. . . . 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 1037	. . . . . 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 2595	. . . . . 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 6026	. . . . . . 7 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
26		fzo0to2pr 10464	. . . . . . 7 ⊢ (0..^2) = {0, 1}
27	25, 26	eqtrdi 2280	. . . . . 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 2738	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹 ≈ 2o)) → ∀𝑘 ∈ {0, 1}DECID (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
30	16, 20, 29	3jca 1203	. . 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 2595	. . . . 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 2736	. . . . 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 16230	. . . . 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 1068	. . . . . 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 9419	. . . . . 6 ⊢ 2 ∈ ℕ0
41		iswrdinn0 11119	. . . . . 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 1069	. . . . . 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 11100	. . . . . . . . 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 6034	. . . . . . 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 5470	. . . . . 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 1070	. . . . . . . 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 2740	. . . . . . 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 533	. . . . . . . . 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 2740	. . . . . . 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 2659	. . . . . 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 1203	. . . 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 9490	. . . . . . . . 9 ⊢ 0 ∈ ℤ
62		2z 9507	. . . . . . . . 9 ⊢ 2 ∈ ℤ
63		fzofig 10695	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 2 ∈ ℤ) → (0..^2) ∈ Fin)
64	61, 62, 63	mp2an 426	. . . . . . . 8 ⊢ (0..^2) ∈ Fin
65		fex 5883	. . . . . . . 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 5485	. . . . . . 7 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → Fun 𝐹)
68		fundmeng 6982	. . . . . . 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 6957	. . . . 5 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → 𝐹 ≈ dom 𝐹)
71		fdm 5488	. . . . . . 7 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → dom 𝐹 = (0..^2))
72	71, 26	eqtrdi 2280	. . . . . 6 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → dom 𝐹 = {0, 1})
73		1z 9505	. . . . . . 7 ⊢ 1 ∈ ℤ
74		0ne1 9210	. . . . . . 7 ⊢ 0 ≠ 1
75		pr2nelem 7396	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 0 ≠ 1) → {0, 1} ≈ 2o)
76	61, 73, 74, 75	mp3an 1373	. . . . . 6 ⊢ {0, 1} ≈ 2o
77	72, 76	eqbrtrdi 4127	. . . . 5 ⊢ (𝐹:(0..^2)⟶dom 𝐼 → dom 𝐹 ≈ 2o)
78		entr 6958	. . . . 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 600	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 841   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  ∀wral 2510  Vcvv 2802  {cpr 3670   class class class wbr 4088  dom cdm 4725  Fun wfun 5320  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018  2_oc2o 6576   ≈ cen 6907  Fincfn 6909  0cc0 8032  1c1 8033   + caddc 8035  2c2 9194  ℕ₀cn0 9402  ℤcz 9479  ...cfz 10243  ..^cfzo 10377  ♯chash 11038  Word cword 11114  Vtxcvtx 15866  iEdgciedg 15867  UPGraphcupgr 15945  Walkscwlks 16171

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-map 6819  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11115  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-edg 15912  df-uhgrm 15923  df-upgren 15947  df-wlks 16172

This theorem is referenced by: (None)