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Theorem usgr2trlncl 29284
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.) (Proof shortened by AV, 31-Oct-2021.)
Assertion
Ref Expression
usgr2trlncl ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(Trailsβ€˜πΊ)𝑃 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
Proof of Theorem usgr2trlncl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 usgrupgr 28709 . . . . 5 (𝐺 ∈ USGraph β†’ 𝐺 ∈ UPGraph)
2 eqid 2730 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 eqid 2730 . . . . . 6 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
42, 3upgrf1istrl 29227 . . . . 5 (𝐺 ∈ UPGraph β†’ (𝐹(Trailsβ€˜πΊ)𝑃 ↔ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
51, 4syl 17 . . . 4 (𝐺 ∈ USGraph β†’ (𝐹(Trailsβ€˜πΊ)𝑃 ↔ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
6 eqidd 2731 . . . . . . . . . . . 12 ((β™―β€˜πΉ) = 2 β†’ 𝐹 = 𝐹)
7 oveq2 7419 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = (0..^2))
8 fzo0to2pr 13721 . . . . . . . . . . . . 13 (0..^2) = {0, 1}
97, 8eqtrdi 2786 . . . . . . . . . . . 12 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = {0, 1})
10 eqidd 2731 . . . . . . . . . . . 12 ((β™―β€˜πΉ) = 2 β†’ dom (iEdgβ€˜πΊ) = dom (iEdgβ€˜πΊ))
116, 9, 10f1eq123d 6824 . . . . . . . . . . 11 ((β™―β€˜πΉ) = 2 β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ↔ 𝐹:{0, 1}–1-1β†’dom (iEdgβ€˜πΊ)))
129raleqdv 3323 . . . . . . . . . . . 12 ((β™―β€˜πΉ) = 2 β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ βˆ€π‘– ∈ {0, 1} ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
13 2wlklem 29191 . . . . . . . . . . . 12 (βˆ€π‘– ∈ {0, 1} ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
1412, 13bitrdi 286 . . . . . . . . . . 11 ((β™―β€˜πΉ) = 2 β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})))
1511, 14anbi12d 629 . . . . . . . . . 10 ((β™―β€˜πΉ) = 2 β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ↔ (𝐹:{0, 1}–1-1β†’dom (iEdgβ€˜πΊ) ∧ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
1615adantl 480 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ↔ (𝐹:{0, 1}–1-1β†’dom (iEdgβ€˜πΊ) ∧ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))))
17 c0ex 11212 . . . . . . . . . . . . . 14 0 ∈ V
18 1ex 11214 . . . . . . . . . . . . . 14 1 ∈ V
1917, 18pm3.2i 469 . . . . . . . . . . . . 13 (0 ∈ V ∧ 1 ∈ V)
20 0ne1 12287 . . . . . . . . . . . . 13 0 β‰  1
21 eqid 2730 . . . . . . . . . . . . . 14 {0, 1} = {0, 1}
2221f12dfv 7273 . . . . . . . . . . . . 13 (((0 ∈ V ∧ 1 ∈ V) ∧ 0 β‰  1) β†’ (𝐹:{0, 1}–1-1β†’dom (iEdgβ€˜πΊ) ↔ (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜0) β‰  (πΉβ€˜1))))
2319, 20, 22mp2an 688 . . . . . . . . . . . 12 (𝐹:{0, 1}–1-1β†’dom (iEdgβ€˜πΊ) ↔ (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜0) β‰  (πΉβ€˜1)))
24 eqid 2730 . . . . . . . . . . . . . 14 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
253, 24usgrf1oedg 28731 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph β†’ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1-ontoβ†’(Edgβ€˜πΊ))
26 f1of1 6831 . . . . . . . . . . . . . 14 ((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1-ontoβ†’(Edgβ€˜πΊ) β†’ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ))
27 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) β†’ 𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ))
2817prid1 4765 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ {0, 1}
2928a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) β†’ 0 ∈ {0, 1})
3027, 29ffvelcdmd 7086 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) β†’ (πΉβ€˜0) ∈ dom (iEdgβ€˜πΊ))
3118prid2 4766 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ {0, 1}
3231a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) β†’ 1 ∈ {0, 1})
3327, 32ffvelcdmd 7086 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) β†’ (πΉβ€˜1) ∈ dom (iEdgβ€˜πΊ))
3430, 33jca 510 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) β†’ ((πΉβ€˜0) ∈ dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜1) ∈ dom (iEdgβ€˜πΊ)))
3534anim1ci 614 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ)) β†’ ((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ) ∧ ((πΉβ€˜0) ∈ dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜1) ∈ dom (iEdgβ€˜πΊ))))
36 f1veqaeq 7258 . . . . . . . . . . . . . . . . . . . 20 (((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ) ∧ ((πΉβ€˜0) ∈ dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜1) ∈ dom (iEdgβ€˜πΊ))) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) β†’ (πΉβ€˜0) = (πΉβ€˜1)))
3735, 36syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ)) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) β†’ (πΉβ€˜0) = (πΉβ€˜1)))
3837necon3d 2959 . . . . . . . . . . . . . . . . . 18 ((𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ)) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜1) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) β‰  ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1))))
39 simpl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
40 simpr 483 . . . . . . . . . . . . . . . . . . . . . 22 ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})
4139, 40neeq12d 3000 . . . . . . . . . . . . . . . . . . . . 21 ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) β‰  ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) ↔ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} β‰  {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
42 preq1 4736 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘ƒβ€˜0) = (π‘ƒβ€˜2) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)})
43 prcom 4735 . . . . . . . . . . . . . . . . . . . . . . 23 {(π‘ƒβ€˜2), (π‘ƒβ€˜1)} = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}
4442, 43eqtrdi 2786 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘ƒβ€˜0) = (π‘ƒβ€˜2) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})
4544necon3i 2971 . . . . . . . . . . . . . . . . . . . . 21 ({(π‘ƒβ€˜0), (π‘ƒβ€˜1)} β‰  {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))
4641, 45syl6bi 252 . . . . . . . . . . . . . . . . . . . 20 ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) β‰  ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
4746com12 32 . . . . . . . . . . . . . . . . . . 19 (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) β‰  ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) β†’ ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
4847a1d 25 . . . . . . . . . . . . . . . . . 18 (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) β‰  ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) β†’ (𝐺 ∈ USGraph β†’ ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))))
4938, 48syl6 35 . . . . . . . . . . . . . . . . 17 ((𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ)) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜1) β†’ (𝐺 ∈ USGraph β†’ ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))))
5049expcom 412 . . . . . . . . . . . . . . . 16 ((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ) β†’ (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) β†’ ((πΉβ€˜0) β‰  (πΉβ€˜1) β†’ (𝐺 ∈ USGraph β†’ ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))))))
5150impd 409 . . . . . . . . . . . . . . 15 ((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ) β†’ ((𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜0) β‰  (πΉβ€˜1)) β†’ (𝐺 ∈ USGraph β†’ ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))))
5251com23 86 . . . . . . . . . . . . . 14 ((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’(Edgβ€˜πΊ) β†’ (𝐺 ∈ USGraph β†’ ((𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜0) β‰  (πΉβ€˜1)) β†’ ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))))
5326, 52syl 17 . . . . . . . . . . . . 13 ((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1-ontoβ†’(Edgβ€˜πΊ) β†’ (𝐺 ∈ USGraph β†’ ((𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜0) β‰  (πΉβ€˜1)) β†’ ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))))
5425, 53mpcom 38 . . . . . . . . . . . 12 (𝐺 ∈ USGraph β†’ ((𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜0) β‰  (πΉβ€˜1)) β†’ ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))))
5523, 54biimtrid 241 . . . . . . . . . . 11 (𝐺 ∈ USGraph β†’ (𝐹:{0, 1}–1-1β†’dom (iEdgβ€˜πΊ) β†’ ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))))
5655impd 409 . . . . . . . . . 10 (𝐺 ∈ USGraph β†’ ((𝐹:{0, 1}–1-1β†’dom (iEdgβ€˜πΊ) ∧ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
5756adantr 479 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ ((𝐹:{0, 1}–1-1β†’dom (iEdgβ€˜πΊ) ∧ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
5816, 57sylbid 239 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
5958com12 32 . . . . . . 7 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
60593adant2 1129 . . . . . 6 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
6160expdcom 413 . . . . 5 (𝐺 ∈ USGraph β†’ ((β™―β€˜πΉ) = 2 β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))))
6261com23 86 . . . 4 (𝐺 ∈ USGraph β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))))
635, 62sylbid 239 . . 3 (𝐺 ∈ USGraph β†’ (𝐹(Trailsβ€˜πΊ)𝑃 β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))))
6463com23 86 . 2 (𝐺 ∈ USGraph β†’ ((β™―β€˜πΉ) = 2 β†’ (𝐹(Trailsβ€˜πΊ)𝑃 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))))
6564imp 405 1 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(Trailsβ€˜πΊ)𝑃 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  Vcvv 3472  {cpr 4629   class class class wbr 5147  dom cdm 5675  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115  2c2 12271  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Vtxcvtx 28523  iEdgciedg 28524  Edgcedg 28574  UPGraphcupgr 28607  USGraphcusgr 28676  Trailsctrls 29214
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-edg 28575  df-uhgr 28585  df-upgr 28609  df-uspgr 28677  df-usgr 28678  df-wlks 29123  df-trls 29216
This theorem is referenced by:  usgr2trlspth  29285  usgr2trlncrct  29327
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