Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  usgr2trlncl Structured version   Visualization version   GIF version

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.
StepHypRef Expression
1 trlis1wlk 40905 . . . 4 (𝐹(TrailS‘𝐺)𝑃𝐹(1Walks‘𝐺)𝑃)
2 wlkv 40815 . . . 4 (𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
31, 2syl 17 . . 3 (𝐹(TrailS‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
4 usgrupgr 40412 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
54adantr 480 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → 𝐺 ∈ UPGraph )
65adantl 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)
119, 10upgrf1istrl 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)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
126, 7, 8, 11syl3anc 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)‘(𝐹𝑖)) = {(𝑃𝑖), (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}
1614, 15syl6eq 2660 . . . . . . . . . . . . . 14 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
17 eqidd 2611 . . . . . . . . . . . . . 14 ((#‘𝐹) = 2 → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
1813, 16, 17f1eq123d 6044 . . . . . . . . . . . . 13 ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ↔ 𝐹:{0, 1}–1-1→dom (iEdg‘𝐺)))
1916raleqdv 3121 . . . . . . . . . . . . . 14 ((#‘𝐹) = 2 → (∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
20 c0ex 9913 . . . . . . . . . . . . . . 15 0 ∈ V
21 1ex 9914 . . . . . . . . . . . . . . 15 1 ∈ V
22 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝐹𝑖) = (𝐹‘0))
2322fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → ((iEdg‘𝐺)‘(𝐹𝑖)) = ((iEdg‘𝐺)‘(𝐹‘0)))
24 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑃𝑖) = (𝑃‘0))
25 oveq1 6556 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
26 0p1e1 11009 . . . . . . . . . . . . . . . . . . 19 (0 + 1) = 1
2725, 26syl6eq 2660 . . . . . . . . . . . . . . . . . 18 (𝑖 = 0 → (𝑖 + 1) = 1)
2827fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑃‘(𝑖 + 1)) = (𝑃‘1))
2924, 28preq12d 4220 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → {(𝑃𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘0), (𝑃‘1)})
3023, 29eqeq12d 2625 . . . . . . . . . . . . . . 15 (𝑖 = 0 → (((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ↔ ((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
31 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑖 = 1 → (𝐹𝑖) = (𝐹‘1))
3231fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑖 = 1 → ((iEdg‘𝐺)‘(𝐹𝑖)) = ((iEdg‘𝐺)‘(𝐹‘1)))
33 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑖 = 1 → (𝑃𝑖) = (𝑃‘1))
34 oveq1 6556 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 1 → (𝑖 + 1) = (1 + 1))
35 1p1e2 11011 . . . . . . . . . . . . . . . . . . 19 (1 + 1) = 2
3634, 35syl6eq 2660 . . . . . . . . . . . . . . . . . 18 (𝑖 = 1 → (𝑖 + 1) = 2)
3736fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝑖 = 1 → (𝑃‘(𝑖 + 1)) = (𝑃‘2))
3833, 37preq12d 4220 . . . . . . . . . . . . . . . 16 (𝑖 = 1 → {(𝑃𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘1), (𝑃‘2)})
3932, 38eqeq12d 2625 . . . . . . . . . . . . . . 15 (𝑖 = 1 → (((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ↔ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
4020, 21, 30, 39ralpr 4185 . . . . . . . . . . . . . 14 (∀𝑖 ∈ {0, 1} ((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
4119, 40syl6bb 275 . . . . . . . . . . . . 13 ((#‘𝐹) = 2 → (∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ↔ (((iEdg‘𝐺)‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ ((iEdg‘𝐺)‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
4218, 41anbi12d 743 . . . . . . . . . . . 12 ((#‘𝐹) = 2 → ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (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)}))))
4342adantl 481 . . . . . . . . . . 11 ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (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)}))))
4420, 21pm3.2i 470 . . . . . . . . . . . . . . 15 (0 ∈ V ∧ 1 ∈ V)
45 0ne1 10965 . . . . . . . . . . . . . . 15 0 ≠ 1
46 eqid 2610 . . . . . . . . . . . . . . . 16 {0, 1} = {0, 1}
4746f12dfv 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))))
4844, 45, 47mp2an 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)
5010, 49usgrf1oedg 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))
5320prid1 4241 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ {0, 1}
5453a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 0 ∈ {0, 1})
5552, 54ffvelrnd 6268 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘0) ∈ dom (iEdg‘𝐺))
5621prid2 4242 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 ∈ {0, 1}
5756a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → 1 ∈ {0, 1})
5852, 57ffvelrnd 6268 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → (𝐹‘1) ∈ dom (iEdg‘𝐺))
5955, 58jca 553 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:{0, 1}⟶dom (iEdg‘𝐺) → ((𝐹‘0) ∈ dom (iEdg‘𝐺) ∧ (𝐹‘1) ∈ dom (iEdg‘𝐺)))
6059anim2i 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))))
6160ancoms 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)))
6361, 62syl 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)))
6463necon3d 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)})
6765, 66neeq12d 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)}
7068, 69syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃‘0) = (𝑃‘2) → {(𝑃‘0), (𝑃‘1)} = {(𝑃‘1), (𝑃‘2)})
7170necon3i 2814 . . . . . . . . . . . . . . . . . . . . . . 23 ({(𝑃‘0), (𝑃‘1)} ≠ {(𝑃‘1), (𝑃‘2)} → (𝑃‘0) ≠ (𝑃‘2))
7267, 71syl6bi 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)))
7372com12 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)))
7473a1d 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))))
7564, 74syl6 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)))))
7675expcom 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))))))
7776impd 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)))))
7877com23 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)))))
7951, 78syl 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)))))
8050, 79mpcom 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))))
8148, 80syl5bi 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))))
8281impd 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)))
8382adantr 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)))
8443, 83sylbid 229 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃‘0) ≠ (𝑃‘2)))
8584com12 32 . . . . . . . . 9 ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2)))
86853adant2 1073 . . . . . . . 8 ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2)))
8786com12 32 . . . . . . 7 ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → ((𝐹:(0..^(#‘𝐹))–1-1→dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))((iEdg‘𝐺)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃‘0) ≠ (𝑃‘2)))
8887adantl 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)‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (𝑃‘0) ≠ (𝑃‘2)))
8912, 88sylbid 229 . . . . 5 (((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐺 ∈ USGraph ∧ (#‘𝐹) = 2)) → (𝐹(TrailS‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))
9089ex 449 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2))))
9190com23 84 . . 3 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(TrailS‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2))))
923, 91mpcom 37 . 2 (𝐹(TrailS‘𝐺)𝑃 → ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝑃‘0) ≠ (𝑃‘2)))
9392com12 32 1 ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘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  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
 Copyright terms: Public domain W3C validator