uspgrn2crct - Metamath Proof Explorer

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Theorem uspgrn2crct 29496

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 29483	. . 3 ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2		istrl 29387	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹))
3		uspgrupgr 28870	. . . . . . . . 9 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
4		eqid 2731	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2731	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	upgriswlk 29332	. . . . . . . . . . . 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 4738	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘1), (𝑃‘0)})
8		prcom 4736	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {(𝑃‘1), (𝑃‘0)} = {(𝑃‘0), (𝑃‘1)}
9	7, 8	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
10	9	eqcoms 2739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
11	10	eqeq2d 2742	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘0) = (𝑃‘2) → (((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
12	11	anbi2d 628	. . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . 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 2757	. . . . . . . . . . . . . . . . . . . . . 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 28848	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
16	15	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
17	16	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
18	17	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 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 6548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ (𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺) ∧ Fun ^◡𝐹))
20	19	simplbi2 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom (iEdg‘𝐺) → (Fun ^◡𝐹 → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
21		wrdf 14476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
24	23	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺)))
25	24	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → 𝐹:(0..^(♯‘𝐹))–1-1→dom (iEdg‘𝐺))
26		2nn 12292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 2 ∈ ℕ
27		lbfzo0 13679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
28	26, 27	mpbir 230	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ (0..^2)
29		1nn0 12495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 ∈ ℕ₀
30		1lt2 12390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 < 2
31		elfzo0 13680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ ∧ 1 < 2))
32	29, 26, 30, 31	mpbir3an 1340	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ (0..^2)
33	28, 32	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
34		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
35	34	eleq2d 2818	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝐹) = 2 → (0 ∈ (0..^(♯‘𝐹)) ↔ 0 ∈ (0..^2)))
36	34	eleq2d 2818	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝐹) = 2 → (1 ∈ (0..^(♯‘𝐹)) ↔ 1 ∈ (0..^2)))
37	35, 36	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝐹) = 2 → ((0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))) ↔ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))))
38	33, 37	mpbiri 258	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝐹) = 2 → (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))))
39	38	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))))
40		f1cofveqaeq 7260	. . . . . . . . . . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
42		0ne1 12290	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ≠ 1
43		eqneqall 2950	. . . . . . . . . . . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . 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 3024	. . . . . . . . . . . . . . . . 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 13724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0..^2) = {0, 1}
53	34, 52	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
54	53	raleqdv 3324	. . . . . . . . . . . . . . . . . . . . 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 29358	. . . . . . . . . . . . . . . . . . . . 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	bitrdi 287	. . . . . . . . . . . . . . . . . . . 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 628	. . . . . . . . . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = (𝑃‘2))
59	58	neeq2d 3000	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘(♯‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘2)))
60	57, 59	imbi12d 344	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 1130	. . . . . . . . . . . 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 253	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 → (Fun ^◡𝐹 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))))
68	67	impd 410	. . . . . . . . . 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 216	. . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))))
73	72	imp 406	. . . . 5 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ 𝐺 ∈ USPGraph) → ((♯‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))))
74	73	necon2d 2962	. . . 4 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ 𝐺 ∈ USPGraph) → ((𝑃‘0) = (𝑃‘(♯‘𝐹)) → (♯‘𝐹) ≠ 2))
75	74	impancom 451	. . 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 407	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 {crab 3431 ∖ cdif 3945 ∅c0 4322 𝒫 cpw 4602 {csn 4628 {cpr 4630 class class class wbr 5148 ^◡ccnv 5675 dom cdm 5676 Fun wfun 6537 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7412 0cc0 11116 1c1 11117 + caddc 11119 < clt 11255 ≤ cle 11256 ℕcn 12219 2c2 12274 ℕ₀cn0 12479 ...cfz 13491 ..^cfzo 13634 ♯chash 14297 Word cword 14471 Vtxcvtx 28690 iEdgciedg 28691 UPGraphcupgr 28774 USPGraphcuspgr 28842 Walkscwlks 29287 Trailsctrls 29381 Circuitsccrcts 29475

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-oadd 8476 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-hash 14298 df-word 14472 df-edg 28742 df-uhgr 28752 df-upgr 28776 df-uspgr 28844 df-wlks 29290 df-trls 29383 df-crcts 29477

This theorem is referenced by: usgrn2cycl 29497

W3C validator