uspgrn2crct - Metamath Proof Explorer

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Theorem uspgrn2crct 27157

Description: In a simple pseudograph there are no circuits with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 3-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)

**Assertion**
Ref	Expression
uspgrn2crct	⊢ ((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)

Proof of Theorem uspgrn2crct Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crctprop 27144	. . 3 ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2		istrl 27047	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹))
3		uspgrupgr 26525	. . . . . . . . 9 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
4		eqid 2777	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2777	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	upgriswlk 26988	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
7		preq2 4500	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘1), (𝑃‘0)})
8		prcom 4498	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {(𝑃‘1), (𝑃‘0)} = {(𝑃‘0), (𝑃‘1)}
9	7, 8	syl6eq 2829	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
10	9	eqcoms 2785	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
11	10	eqeq2d 2787	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘0) = (𝑃‘2) → (((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
12	11	anbi2d 622	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘2) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)})))
13	12	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)})))
14		eqtr3 2800	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → ((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)))
15	4, 5	uspgrf 26503	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
16	15	adantl 475	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
17	16	adantl 475	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18	17	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
19		df-f1 6140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ (𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺) ∧ Fun ^◡𝐹))
20	19	simplbi2 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺) → (Fun ^◡𝐹 → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
21		wrdf 13604	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺))
22	20, 21	syl11 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun ^◡𝐹 → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
23	22	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
24	23	adantl 475	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
25	24	imp 397	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺))
26		2nn 11448	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 2 ∈ ℕ
27		lbfzo0 12827	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
28	26, 27	mpbir 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ (0..^2)
29		1nn0 11660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 ∈ ℕ₀
30		1lt2 11553	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 < 2
31		elfzo0 12828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ ∧ 1 < 2))
32	29, 26, 30, 31	mpbir3an 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ (0..^2)
33	28, 32	pm3.2i 464	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
34		oveq2 6930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
35	34	eleq2d 2844	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝐹) = 2 → (0 ∈ (0..^(♯‘𝐹)) ↔ 0 ∈ (0..^2)))
36	34	eleq2d 2844	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝐹) = 2 → (1 ∈ (0..^(♯‘𝐹)) ↔ 1 ∈ (0..^2)))
37	35, 36	anbi12d 624	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝐹) = 2 → ((0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))) ↔ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))))
38	33, 37	mpbiri 250	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝐹) = 2 → (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))))
39	38	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))))
40		f1cofveqaeq 6787	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)) ∧ (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹)))) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
41	18, 25, 39, 40	syl21anc 828	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
42		0ne1 11446	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ≠ 1
43		eqneqall 2979	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 = 1 → (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘2)))
44	41, 42, 43	syl6mpi 67	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝑃‘0) ≠ (𝑃‘2)))
45	44	adantll 704	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝑃‘0) ≠ (𝑃‘2)))
46	14, 45	syl5 34	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → (𝑃‘0) ≠ (𝑃‘2)))
47	13, 46	sylbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))
48	47	expimpd 447	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃‘0) = (𝑃‘2) ∧ ((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph))) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2)))
49	48	ex 403	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃‘0) = (𝑃‘2) → (((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2))))
50		2a1 28	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃‘0) ≠ (𝑃‘2) → (((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2))))
51	49, 50	pm2.61ine 3052	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2)))
52		fzo0to2pr 12872	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0..^2) = {0, 1}
53	34, 52	syl6eq 2829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
54	53	raleqdv 3339	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝐹) = 2 → (∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
55		2wlklem 27014	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑘 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
56	54, 55	syl6bb 279	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) = 2 → (∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
57	56	anbi2d 622	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐹) = 2 → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
58		fveq2 6446	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = (𝑃‘2))
59	58	neeq2d 3028	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘2)))
60	57, 59	imbi12d 336	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐹) = 2 → (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) ↔ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2))))
61	60	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) → (((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) ↔ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2))))
62	51, 61	mpbird 249	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))
63	62	ex 403	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐹) = 2 → ((Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph) → ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
64	63	com13 88	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
65	64	expd 406	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (Fun ^◡𝐹 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))))
66	65	3adant2 1122	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (Fun ^◡𝐹 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))))
67	6, 66	syl6bi 245	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → (Fun ^◡𝐹 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))))
68	67	impd 400	. . . . . . . . . 10 ⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹) → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))))
69	68	com23 86	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))))
70	3, 69	mpcom 38	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
71	70	com12 32	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹) → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
72	2, 71	sylbi 209	. . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
73	72	imp 397	. . . . 5 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ 𝐺 ∈ USPGraph) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))
74	73	necon2d 2991	. . . 4 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ 𝐺 ∈ USPGraph) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (♯‘𝐹) ≠ 2))
75	74	impancom 445	. . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝐺 ∈ USPGraph → (♯‘𝐹) ≠ 2))
76	1, 75	syl 17	. 2 ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐺 ∈ USPGraph → (♯‘𝐹) ≠ 2))
77	76	impcom 398	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∀wral 3089 {crab 3093 ∖ cdif 3788 ∅c0 4140 𝒫 cpw 4378 {csn 4397 {cpr 4399 class class class wbr 4886 ^◡ccnv 5354 dom cdm 5355 Fun wfun 6129 ⟶wf 6131 –1-1→wf1 6132 ‘cfv 6135 (class class class)co 6922 0cc0 10272 1c1 10273 + caddc 10275 < clt 10411 ≤ cle 10412 ℕcn 11374 2c2 11430 ℕ₀cn0 11642 ...cfz 12643 ..^cfzo 12784 ♯chash 13435 Word cword 13599 Vtxcvtx 26344 iEdgciedg 26345 UPGraphcupgr 26428 USPGraphcuspgr 26497 Walkscwlks 26944 Trailsctrls 27041 Circuitsccrcts 27136

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ifp 1047 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-edg 26396 df-uhgr 26406 df-upgr 26430 df-uspgr 26499 df-wlks 26947 df-trls 27043 df-crcts 27138

This theorem is referenced by: usgrn2cycl 27158

W3C validator