Theorem usgr2trlncl 40966

Description: In a simple graph, any trail of length 2 does not start and end at the same vertex. (Contributed by AV, 5-Jun-2021.)

Assertion

Ref	Expression
usgr2trlncl	⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))

Proof of Theorem usgr2trlncl

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trlis1wlk 40905	. . . 4 ⊢ (𝐹(TrailS‘𝐺)𝑃 → 𝐹(1Walks‘𝐺)𝑃)
2		wlkv 40815	. . . 4 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
3	1, 2	syl 17	. . 3 ⊢ (𝐹(TrailS‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
4		usgrupgr 40412	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → 𝐺 ∈ UPGraph )
6	5	adantl 481	. . . . . . 7 ⊢ (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2)) → 𝐺 ∈ UPGraph )
7		simpl2 1058	. . . . . . 7 ⊢ (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2)) → 𝐹 ∈ V)
8		simpl3 1059	. . . . . . 7 ⊢ (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2)) → 𝑃 ∈ V)
9		eqid 2610	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10		eqid 2610	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
11	9, 10	upgrf1istrl 40911	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
12	6, 7, 8, 11	syl3anc 1318	. . . . . 6 ⊢ (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2)) → (𝐹(TrailS‘𝐺)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
13		eqidd 2611	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → 𝐹 = 𝐹)
14		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
15		fzo0to2pr 12420	. . . . . . . . . . . . . . 15 ⊢ (0..^2) = {0, 1}
16	14, 15	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
17		eqidd 2611	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
18	13, 16, 17	f1eq123d 6044	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ 𝐹:{0, 1}–1-1→dom (iEdg‘𝐺)))
19	16	raleqdv 3121	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 2 → (∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
20		c0ex 9913	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
21		1ex 9914	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
22		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝐹‘𝑖) = (𝐹‘0))
23	22	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → ((iEdg‘𝐺)‘(𝐹‘𝑖)) = ((iEdg‘𝐺)‘(𝐹‘0)))
24		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0))
25		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
26		0p1e1 11009	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 + 1) = 1
27	25, 26	syl6eq 2660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 0 → (𝑖 + 1) = 1)
28	27	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝑃‘(𝑖 + 1)) = (𝑃‘1))
29	24, 28	preq12d 4220	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘0), (𝑃‘1)})
30	23, 29	eqeq12d 2625	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → (((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
31		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 1 → (𝐹‘𝑖) = (𝐹‘1))
32	31	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 1 → ((iEdg‘𝐺)‘(𝐹‘𝑖)) = ((iEdg‘𝐺)‘(𝐹‘1)))
33		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
34		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 1 → (𝑖 + 1) = (1 + 1))
35		1p1e2 11011	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 + 1) = 2
36	34, 35	syl6eq 2660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 1 → (𝑖 + 1) = 2)
37	36	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 1 → (𝑃‘(𝑖 + 1)) = (𝑃‘2))
38	33, 37	preq12d 4220	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 1 → {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘1), (𝑃‘2)})
39	32, 38	eqeq12d 2625	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 1 → (((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
40	20, 21, 30, 39	ralpr 4185	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
41	19, 40	syl6bb 275	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 2 → (∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
42	18, 41	anbi12d 743	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 2 → ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ↔ (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
43	42	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ↔ (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
44	20, 21	pm3.2i 470	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ V ∧ 1 ∈ V)
45		0ne1 10965	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 1
46		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ {0, 1} = {0, 1}
47	46	f12dfv 6429	. . . . . . . . . . . . . . 15 ⊢ (((0 ∈ V ∧ 1 ∈ V) ∧ 0 ≠ 1) → (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ↔ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1))))
48	44, 45, 47	mp2an 704	. . . . . . . . . . . . . 14 ⊢ (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ↔ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)))
49		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
50	10, 49	usgrf1oedg 40434	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
51		f1of1 6049	. . . . . . . . . . . . . . . 16 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
52		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 𝐹:{0, 1}⟶dom (iEdg‘𝐺))
53	20	prid1 4241	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ {0, 1}
54	53	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 0 ∈ {0, 1})
55	52, 54	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘0) ∈ dom (iEdg‘𝐺))
56	21	prid2 4242	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 1 ∈ {0, 1}
57	56	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 1 ∈ {0, 1})
58	52, 57	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘1) ∈ dom (iEdg‘𝐺))
59	55, 58	jca 553	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺)))
60	59	anim2i 591	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) ∧ 𝐹:{0, 1}⟶dom (iEdg‘𝐺)) → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) ∧ ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺))))
61	60	ancoms 468	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)) → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) ∧ ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺))))
62		f1veqaeq 6418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) ∧ ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺))) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝐹‘0) = (𝐹‘1)))
63	61, 62	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)) → (((iEdg‘𝐺)‘(𝐹‘0)) = ((iEdg‘𝐺)‘(𝐹‘1)) → (𝐹‘0) = (𝐹‘1)))
64	63	necon3d 2803	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)) → ((𝐹‘0) ≠ (𝐹‘1) → ((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1))))
65		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)})
66		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})
67	65, 66	neeq12d 2843	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1)) ↔ {(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)}))
68		preq1 4212	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘0), (𝑃‘1)} = {(𝑃‘2), (𝑃‘1)})
69		prcom 4211	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {(𝑃‘2), (𝑃‘1)} = {(𝑃‘1), (𝑃‘2)}
70	68, 69	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃‘0) = (𝑃‘2) → {(𝑃‘0), (𝑃‘1)} = {(𝑃‘1), (𝑃‘2)})
71	70	necon3i 2814	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)} → (𝑃‘0) ≠ (𝑃‘2))
72	67, 71	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1)) → (𝑃‘0) ≠ (𝑃‘2)))
73	72	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))
74	73	a1d 25	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((iEdg‘𝐺)‘(𝐹‘0)) ≠ ((iEdg‘𝐺)‘(𝐹‘1)) → (𝐺 ∈ USGraph → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2))))
75	64, 74	syl6 34	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)) → ((𝐹‘0) ≠ (𝐹‘1) → (𝐺 ∈ USGraph → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))))
76	75	expcom 450	. . . . . . . . . . . . . . . . . 18 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) → (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → ((𝐹‘0) ≠ (𝐹‘1) → (𝐺 ∈ USGraph → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2))))))
77	76	impd 446	. . . . . . . . . . . . . . . . 17 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) → ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)) → (𝐺 ∈ USGraph → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))))
78	77	com23 84	. . . . . . . . . . . . . . . 16 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺) → (𝐺 ∈ USGraph → ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))))
79	51, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (𝐺 ∈ USGraph → ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2)))))
80	50, 79	mpcom 37	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ USGraph → ((𝐹:{0, 1}⟶dom (iEdg‘𝐺) ∧ (𝐹‘0) ≠ (𝐹‘1)) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2))))
81	48, 80	syl5bi 231	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) → ((((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) → (𝑃‘0) ≠ (𝑃‘2))))
82	81	impd 446	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → ((𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2)))
83	82	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝐹:{0, 1}–1-1→dom (iEdg‘𝐺) ∧ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) → (𝑃‘0) ≠ (𝑃‘2)))
84	43, 83	sylbid 229	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃‘0) ≠ (𝑃‘2)))
85	84	com12 32	. . . . . . . . 9 ⊢ ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2)))
86	85	3adant2 1073	. . . . . . . 8 ⊢ ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2)))
87	86	com12 32	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃‘0) ≠ (𝑃‘2)))
88	87	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2)) → ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃‘0) ≠ (𝑃‘2)))
89	12, 88	sylbid 229	. . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2)) → (𝐹(TrailS‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))
90	89	ex 449	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2))))
91	90	com23 84	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(TrailS‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2))))
92	3, 91	mpcom 37	. 2 ⊢ (𝐹(TrailS‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2)))
93	92	com12 32	1 ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  {cpr 4127   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ...cfz 12197  ..^cfzo 12334  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747  Edgcedga 25792   USGraph cusgr 40379  1Walksc1wlks 40796  TrailSctrls 40899

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-usgr 40381  df-1wlks 40800  df-wlks 40801  df-trls 40901

This theorem is referenced by:  usgr2trlspth  40967  usgr2trlncrct  41009