uspgrn2crct - Mathbox for Alexander van der Vekens

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Theorem uspgrn2crct 41011

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.)

**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 40998	. . 3 ⊢ (𝐹(CircuitS‘𝐺)𝑃 → (𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))
2		trlis1wlk 40905	. . . . . . 7 ⊢ (𝐹(TrailS‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃)
3		wlkv 40815	. . . . . . . 8 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
4		isTrl 40904	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹)))
5		uspgrupgr 40406	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
6		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	upgriswlk 40849	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
9		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘1), (𝑃‘0)})
10		prcom 4211	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ {(𝑃‘1), (𝑃‘0)} = {(𝑃‘0), (𝑃‘1)}
11	9, 10	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃‘2) = (𝑃‘0) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
12	11	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘1), (𝑃‘2)} = {(𝑃‘0), (𝑃‘1)})
13	12	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑃‘0) = (𝑃‘2) → (((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}))
14	13	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((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)})))
15	14	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((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)})))
16		eqtr3 2631	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘0), (𝑃‘1)}) → ((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)))
17	6, 7	uspgrf 40384	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
18	17	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph ) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
19	18	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph )) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
20	19	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((#‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
21		wrdf 13165	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(#‘𝐹))⟶dom (iEdg‘𝐺))
22		df-f1 5809	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ (𝐹:(0..^(#‘𝐹))⟶dom (iEdg‘𝐺) ∧ Fun ^◡𝐹))
23	22	simplbi2 653	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝐹:(0..^(#‘𝐹))⟶dom (iEdg‘𝐺) → (Fun ^◡𝐹 → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
24	21, 23	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (Fun ^◡𝐹 → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
25	24	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (Fun ^◡𝐹 → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
26	25	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph ) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
27	26	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph )) → (𝐹 ∈ Word dom (iEdg‘𝐺) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺)))
28	27	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((#‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → 𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺))
29		2nn 11062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 2 ∈ ℕ
30		lbfzo0 12375	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (0 ∈ (0..^2) ↔ 2 ∈ ℕ)
31	29, 30	mpbir 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 0 ∈ (0..^2)
32		1nn0 11185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 1 ∈ ℕ₀
33		1lt2 11071	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 1 < 2
34		elfzo0 12376	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ₀ ∧ 2 ∈ ℕ ∧ 1 < 2))
35	32, 29, 33, 34	mpbir3an 1237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 1 ∈ (0..^2)
36	31, 35	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
37		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
38	37	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((#‘𝐹) = 2 → (0 ∈ (0..^(#‘𝐹)) ↔ 0 ∈ (0..^2)))
39	37	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((#‘𝐹) = 2 → (1 ∈ (0..^(#‘𝐹)) ↔ 1 ∈ (0..^2)))
40	38, 39	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝐹) = 2 → ((0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))) ↔ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))))
41	36, 40	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘𝐹) = 2 → (0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))))
42	41	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((#‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (0 ∈ (0..^(#‘𝐹)) ∧ 1 ∈ (0..^(#‘𝐹))))
43		f1cofveqaeq 40323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((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))
44	20, 28, 42, 43	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((#‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → 0 = 1))
45		0ne1 10965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 0 ≠ 1
46		eqneqall 2793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (0 = 1 → (0 ≠ 1 → (𝑃‘0) ≠ (𝑃‘2)))
47	44, 45, 46	syl6mpi 65	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((#‘𝐹) = 2 ∧ (Fun ^◡𝐹 ∧ 𝐺 ∈ USPGraph )) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝑃‘0) ≠ (𝑃‘2)))
48	47	adantll 746	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((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)))
49	16, 48	syl5 33	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((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)))
50	15, 49	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((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	50	expimpd 627	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((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)))
52	51	ex 449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((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))))
53		2a1 28	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((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))))
54	52, 53	pm2.61ine 2865	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((#‘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)))
55		fzo0to2pr 12420	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0..^2) = {0, 1}
56	37, 55	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
57	56	raleqdv 3121	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘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))}))
58		2wlklem 26094	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∀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)}))
59	57, 58	syl6bb 275	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘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)})))
60	59	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘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)}))))
61		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝐹) = 2 → (𝑃‘(#‘𝐹)) = (𝑃‘2))
62	61	neeq2d 2842	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝐹) = 2 → ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) ≠ (𝑃‘2)))
63	60, 62	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘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))))
64	63	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((#‘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))))
65	54, 64	mpbird 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘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))))
66	65	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘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)))))
67	66	com13 86	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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)))))
68	67	expd 451	. . . . . . . . . . . . . . . . . 18 ⊢ ((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))))))
69	68	3adant2 1073	. . . . . . . . . . . . . . . . 17 ⊢ ((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))))))
70	8, 69	syl6bi 242	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun ^◡𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))))
71	70	3expib 1260	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun ^◡𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))))
72	5, 71	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USPGraph → ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun ^◡𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))))
73	72	com25 97	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (𝐹(1Walks‘𝐺)𝑃 → (Fun ^◡𝐹 → ((𝐹 ∈ V ∧ 𝑃 ∈ V) → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))))
74	73	pm2.43i 50	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (𝐹(1Walks‘𝐺)𝑃 → (Fun ^◡𝐹 → ((𝐹 ∈ V ∧ 𝑃 ∈ V) → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))))
75	74	com14 94	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun ^◡𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))))
76	75	3adant1 1072	. . . . . . . . . 10 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(1Walks‘𝐺)𝑃 → (Fun ^◡𝐹 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))))
77	76	impd 446	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → ((𝐹(1Walks‘𝐺)𝑃 ∧ Fun ^◡𝐹) → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))
78	4, 77	sylbid 229	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(TrailS‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))
79	3, 78	syl 17	. . . . . . 7 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐹(TrailS‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))))
80	2, 79	mpcom 37	. . . . . 6 ⊢ (𝐹(TrailS‘𝐺)𝑃 → (𝐺 ∈ USPGraph → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))))
81	80	imp 444	. . . . 5 ⊢ ((𝐹(TrailS‘𝐺)𝑃 ∧ 𝐺 ∈ USPGraph ) → ((#‘𝐹) = 2 → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))))
82	81	necon2d 2805	. . . 4 ⊢ ((𝐹(TrailS‘𝐺)𝑃 ∧ 𝐺 ∈ USPGraph ) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (#‘𝐹) ≠ 2))
83	82	impancom 455	. . 3 ⊢ ((𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝐺 ∈ USPGraph → (#‘𝐹) ≠ 2))
84	1, 83	syl 17	. 2 ⊢ (𝐹(CircuitS‘𝐺)𝑃 → (𝐺 ∈ USPGraph → (#‘𝐹) ≠ 2))
85	84	impcom 445	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹(CircuitS‘𝐺)𝑃) → (#‘𝐹) ≠ 2)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 class class class wbr 4583 ^◡ccnv 5037 dom cdm 5038 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Vtxcvtx 25673 iEdgciedg 25674 UPGraph cupgr 25747 USPGraph cuspgr 40378 1Walksc1wlks 40796 TrailSctrls 40899 CircuitSccrcts 40990

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-uspgr 40380 df-1wlks 40800 df-wlks 40801 df-trls 40901 df-crcts 40992

This theorem is referenced by: usgrn2cycl 41012

W3C validator