uspgrn2crct - Metamath Proof Explorer

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Theorem uspgrn2crct 29738

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 29722	. . 3 ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
2		istrl 29624	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹))
3		uspgrupgr 29105	. . . . . . . . 9 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
4		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2729	. . . . . . . . . . . . 13 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	upgriswlk 29569	. . . . . . . . . . . 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 4698	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘1), (𝑃‘0)})
8		prcom 4696	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ {(𝑃‘1), (𝑃‘0)} = {(𝑃‘0), (𝑃‘1)}
9	7, 8	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
10	9	eqcoms 2737	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
11	10	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘0) = (𝑃‘2) → (((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
12	11	anbi2d 630	. . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . . . . . . . . . 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 29081	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6516	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12259	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 2 ∈ ℕ
27		lbfzo0 13660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
28	26, 27	mpbir 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ (0..^2)
29		1nn0 12458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 ∈ ℕ₀
30		1lt2 12352	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 < 2
31		elfzo0 13661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ ∧ 1 < 2))
32	29, 26, 30, 31	mpbir3an 1342	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ (0..^2)
33	28, 32	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
34		oveq2 7395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = (0..^2))
35	34	eleq2d 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝐹) = 2 → (0 ∈ (0..^(♯‘𝐹)) ↔ 0 ∈ (0..^2)))
36	34	eleq2d 2814	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝐹) = 2 → (1 ∈ (0..^(♯‘𝐹)) ↔ 1 ∈ (0..^2)))
37	35, 36	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (0 ∈ (0..^(♯‘𝐹)) ∧ 1 ∈ (0..^(♯‘𝐹))))
40		f1cofveqaeq 7232	. . . . . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((♯‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
42		0ne1 12257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ≠ 1
43		eqneqall 2936	. . . . . . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . . . . 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 3008	. . . . . . . . . . . . . . . . 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 13711	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0..^2) = {0, 1}
53	34, 52	eqtrdi 2780	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝐹) = 2 → (0..^(♯‘𝐹)) = {0, 1})
54	53	raleqdv 3299	. . . . . . . . . . . . . . . . . . . . 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 29595	. . . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) = 2 → (𝑃‘(♯‘𝐹)) = (𝑃‘2))
59	58	neeq2d 2985	. . . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . 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	biimtrdi 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 217	. . . . . 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 2948	. . . 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 {crab 3405 ∖ cdif 3911 ∅c0 4296 𝒫 cpw 4563 {csn 4589 {cpr 4591 class class class wbr 5107 ^◡ccnv 5637 dom cdm 5638 Fun wfun 6505 ⟶wf 6507 –1-1→wf1 6508 ‘cfv 6511 (class class class)co 7387 0cc0 11068 1c1 11069 + caddc 11071 < clt 11208 ≤ cle 11209 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ...cfz 13468 ..^cfzo 13615 ♯chash 14295 Word cword 14478 Vtxcvtx 28923 iEdgciedg 28924 UPGraphcupgr 29007 USPGraphcuspgr 29075 Walkscwlks 29524 Trailsctrls 29618 Circuitsccrcts 29714

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-hash 14296 df-word 14479 df-edg 28975 df-uhgr 28985 df-upgr 29009 df-uspgr 29077 df-wlks 29527 df-trls 29620 df-crcts 29716

This theorem is referenced by: usgrn2cycl 29739

W3C validator