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Theorem usgr2trlncl 29006
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 28431 . . . . 5 (𝐺 ∈ USGraph β†’ 𝐺 ∈ UPGraph)
2 eqid 2732 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 eqid 2732 . . . . . 6 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
42, 3upgrf1istrl 28949 . . . . 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 2733 . . . . . . . . . . . 12 ((β™―β€˜πΉ) = 2 β†’ 𝐹 = 𝐹)
7 oveq2 7413 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = (0..^2))
8 fzo0to2pr 13713 . . . . . . . . . . . . 13 (0..^2) = {0, 1}
97, 8eqtrdi 2788 . . . . . . . . . . . 12 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = {0, 1})
10 eqidd 2733 . . . . . . . . . . . 12 ((β™―β€˜πΉ) = 2 β†’ dom (iEdgβ€˜πΊ) = dom (iEdgβ€˜πΊ))
116, 9, 10f1eq123d 6822 . . . . . . . . . . 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 3325 . . . . . . . . . . . 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 28913 . . . . . . . . . . . 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 631 . . . . . . . . . 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 482 . . . . . . . . 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 11204 . . . . . . . . . . . . . 14 0 ∈ V
18 1ex 11206 . . . . . . . . . . . . . 14 1 ∈ V
1917, 18pm3.2i 471 . . . . . . . . . . . . 13 (0 ∈ V ∧ 1 ∈ V)
20 0ne1 12279 . . . . . . . . . . . . 13 0 β‰  1
21 eqid 2732 . . . . . . . . . . . . . 14 {0, 1} = {0, 1}
2221f12dfv 7267 . . . . . . . . . . . . 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 690 . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . 14 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
253, 24usgrf1oedg 28453 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph β†’ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1-ontoβ†’(Edgβ€˜πΊ))
26 f1of1 6829 . . . . . . . . . . . . . 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 7084 . . . . . . . . . . . . . . . . . . . . . 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 7084 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) β†’ (πΉβ€˜1) ∈ dom (iEdgβ€˜πΊ))
3430, 33jca 512 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:{0, 1}⟢dom (iEdgβ€˜πΊ) β†’ ((πΉβ€˜0) ∈ dom (iEdgβ€˜πΊ) ∧ (πΉβ€˜1) ∈ dom (iEdgβ€˜πΊ)))
3534anim1ci 616 . . . . . . . . . . . . . . . . . . . 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 7252 . . . . . . . . . . . . . . . . . . . 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 2961 . . . . . . . . . . . . . . . . . 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 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€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
40 simpr 485 . . . . . . . . . . . . . . . . . . . . . 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 3002 . . . . . . . . . . . . . . . . . . . . 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 2788 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘ƒβ€˜0) = (π‘ƒβ€˜2) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})
4544necon3i 2973 . . . . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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 411 . . . . . . . . . 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 481 . . . . . . . . 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 1131 . . . . . 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 415 . . . . 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 407 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  Vcvv 3474  {cpr 4629   class class class wbr 5147  dom cdm 5675  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405  0cc0 11106  1c1 11107   + caddc 11109  2c2 12263  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Vtxcvtx 28245  iEdgciedg 28246  Edgcedg 28296  UPGraphcupgr 28329  USGraphcusgr 28398  Trailsctrls 28936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-edg 28297  df-uhgr 28307  df-upgr 28331  df-uspgr 28399  df-usgr 28400  df-wlks 28845  df-trls 28938
This theorem is referenced by:  usgr2trlspth  29007  usgr2trlncrct  29049
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