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Theorem uspgrn2crct 29051
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.) (Proof shortened by AV, 31-Oct-2021.)
Assertion
Ref Expression
uspgrn2crct ((𝐺 ∈ USPGraph ∧ 𝐹(Circuitsβ€˜πΊ)𝑃) β†’ (β™―β€˜πΉ) β‰  2)
Proof of Theorem uspgrn2crct
Dummy variables π‘₯ π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crctprop 29038 . . 3 (𝐹(Circuitsβ€˜πΊ)𝑃 β†’ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
2 istrl 28942 . . . . . . 7 (𝐹(Trailsβ€˜πΊ)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹))
3 uspgrupgr 28425 . . . . . . . . 9 (𝐺 ∈ USPGraph β†’ 𝐺 ∈ UPGraph)
4 eqid 2732 . . . . . . . . . . . . 13 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
5 eqid 2732 . . . . . . . . . . . . 13 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
64, 5upgriswlk 28887 . . . . . . . . . . . 12 (𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word 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€˜(π‘˜ + 1))})))
7 preq2 4737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘ƒβ€˜2) = (π‘ƒβ€˜0) β†’ {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} = {(π‘ƒβ€˜1), (π‘ƒβ€˜0)})
8 prcom 4735 . . . . . . . . . . . . . . . . . . . . . . . . . 26 {(π‘ƒβ€˜1), (π‘ƒβ€˜0)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}
97, 8eqtrdi 2788 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((π‘ƒβ€˜2) = (π‘ƒβ€˜0) β†’ {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
109eqcoms 2740 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘ƒβ€˜0) = (π‘ƒβ€˜2) β†’ {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
1110eqeq2d 2743 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘ƒβ€˜0) = (π‘ƒβ€˜2) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} ↔ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
1211anbi2d 629 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘ƒβ€˜0) = (π‘ƒβ€˜2) 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)}) ↔ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})))
1312ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 ((((π‘ƒβ€˜0) = (π‘ƒβ€˜2) ∧ ((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word 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)}) ↔ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})))
14 eqtr3 2758 . . . . . . . . . . . . . . . . . . . . . 22 ((((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}) β†’ ((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)))
154, 5uspgrf 28403 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐺 ∈ USPGraph β†’ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
1615adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph) β†’ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
1716adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) β†’ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
1817adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)) β†’ (iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
19 df-f1 6545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ) ↔ (𝐹:(0..^(β™―β€˜πΉ))⟢dom (iEdgβ€˜πΊ) ∧ Fun ◑𝐹))
2019simplbi2 501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹:(0..^(β™―β€˜πΉ))⟢dom (iEdgβ€˜πΊ) β†’ (Fun ◑𝐹 β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ)))
21 wrdf 14465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐹 ∈ Word dom (iEdgβ€˜πΊ) β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom (iEdgβ€˜πΊ))
2220, 21syl11 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun ◑𝐹 β†’ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ)))
2322adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph) β†’ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ)))
2423adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) β†’ (𝐹 ∈ Word dom (iEdgβ€˜πΊ) β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ)))
2524imp 407 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)) β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ))
26 2nn 12281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 ∈ β„•
27 lbfzo0 13668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0 ∈ (0..^2) ↔ 2 ∈ β„•)
2826, 27mpbir 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ (0..^2)
29 1nn0 12484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 ∈ β„•0
30 1lt2 12379 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 < 2
31 elfzo0 13669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ (0..^2) ↔ (1 ∈ β„•0 ∧ 2 ∈ β„• ∧ 1 < 2))
3229, 26, 30, 31mpbir3an 1341 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ (0..^2)
3328, 32pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
34 oveq2 7413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = (0..^2))
3534eleq2d 2819 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜πΉ) = 2 β†’ (0 ∈ (0..^(β™―β€˜πΉ)) ↔ 0 ∈ (0..^2)))
3634eleq2d 2819 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜πΉ) = 2 β†’ (1 ∈ (0..^(β™―β€˜πΉ)) ↔ 1 ∈ (0..^2)))
3735, 36anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((β™―β€˜πΉ) = 2 β†’ ((0 ∈ (0..^(β™―β€˜πΉ)) ∧ 1 ∈ (0..^(β™―β€˜πΉ))) ↔ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))))
3833, 37mpbiri 257 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β™―β€˜πΉ) = 2 β†’ (0 ∈ (0..^(β™―β€˜πΉ)) ∧ 1 ∈ (0..^(β™―β€˜πΉ))))
3938ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)) β†’ (0 ∈ (0..^(β™―β€˜πΉ)) ∧ 1 ∈ (0..^(β™―β€˜πΉ))))
40 f1cofveqaeq 7253 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((iEdgβ€˜πΊ):dom (iEdgβ€˜πΊ)–1-1β†’{π‘₯ ∈ (𝒫 (Vtxβ€˜πΊ) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2} ∧ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom (iEdgβ€˜πΊ)) ∧ (0 ∈ (0..^(β™―β€˜πΉ)) ∧ 1 ∈ (0..^(β™―β€˜πΉ)))) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) β†’ 0 = 1))
4118, 25, 39, 40syl21anc 836 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) β†’ 0 = 1))
42 0ne1 12279 . . . . . . . . . . . . . . . . . . . . . . . 24 0 β‰  1
43 eqneqall 2951 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = 1 β†’ (0 β‰  1 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
4441, 42, 43syl6mpi 67 . . . . . . . . . . . . . . . . . . . . . . 23 ((((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
4544adantll 712 . . . . . . . . . . . . . . . . . . . . . 22 ((((π‘ƒβ€˜0) = (π‘ƒβ€˜2) ∧ ((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word dom (iEdgβ€˜πΊ)) β†’ (((iEdgβ€˜πΊ)β€˜(πΉβ€˜0)) = ((iEdgβ€˜πΊ)β€˜(πΉβ€˜1)) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
4614, 45syl5 34 . . . . . . . . . . . . . . . . . . . . 21 ((((π‘ƒβ€˜0) = (π‘ƒβ€˜2) ∧ ((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word 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€˜0), (π‘ƒβ€˜1)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
4713, 46sylbid 239 . . . . . . . . . . . . . . . . . . . 20 ((((π‘ƒβ€˜0) = (π‘ƒβ€˜2) ∧ ((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph))) ∧ 𝐹 ∈ Word 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)))
4847expimpd 454 . . . . . . . . . . . . . . . . . . 19 (((π‘ƒβ€˜0) = (π‘ƒβ€˜2) ∧ ((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph))) β†’ ((𝐹 ∈ Word 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)))
4948ex 413 . . . . . . . . . . . . . . . . . 18 ((π‘ƒβ€˜0) = (π‘ƒβ€˜2) β†’ (((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) β†’ ((𝐹 ∈ Word 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))))
50 2a1 28 . . . . . . . . . . . . . . . . . 18 ((π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2) β†’ (((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) β†’ ((𝐹 ∈ Word 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))))
5149, 50pm2.61ine 3025 . . . . . . . . . . . . . . . . 17 (((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) β†’ ((𝐹 ∈ Word 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)))
52 fzo0to2pr 13713 . . . . . . . . . . . . . . . . . . . . . . 23 (0..^2) = {0, 1}
5334, 52eqtrdi 2788 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = {0, 1})
5453raleqdv 3325 . . . . . . . . . . . . . . . . . . . . 21 ((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€˜(π‘˜ + 1))} ↔ βˆ€π‘˜ ∈ {0, 1} ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}))
55 2wlklem 28913 . . . . . . . . . . . . . . . . . . . . 21 (βˆ€π‘˜ ∈ {0, 1} ((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 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)}))
5654, 55bitrdi 286 . . . . . . . . . . . . . . . . . . . 20 ((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€˜(π‘˜ + 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)})))
5756anbi2d 629 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜πΉ) = 2 β†’ ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) ↔ (𝐹 ∈ Word 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)}))))
58 fveq2 6888 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜2))
5958neeq2d 3001 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜πΉ) = 2 β†’ ((π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)) ↔ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
6057, 59imbi12d 344 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜πΉ) = 2 β†’ (((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))) ↔ ((𝐹 ∈ Word 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))))
6160adantr 481 . . . . . . . . . . . . . . . . 17 (((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) β†’ (((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))) ↔ ((𝐹 ∈ Word 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))))
6251, 61mpbird 256 . . . . . . . . . . . . . . . 16 (((β™―β€˜πΉ) = 2 ∧ (Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph)) β†’ ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))))
6362ex 413 . . . . . . . . . . . . . . 15 ((β™―β€˜πΉ) = 2 β†’ ((Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph) β†’ ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))))
6463com13 88 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) β†’ ((Fun ◑𝐹 ∧ 𝐺 ∈ USPGraph) β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))))
6564expd 416 . . . . . . . . . . . . 13 ((𝐹 ∈ Word dom (iEdgβ€˜πΊ) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))((iEdgβ€˜πΊ)β€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}) β†’ (Fun ◑𝐹 β†’ (𝐺 ∈ USPGraph β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))))))
66653adant2 1131 . . . . . . . . . . . 12 ((𝐹 ∈ Word 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€˜(π‘˜ + 1))}) β†’ (Fun ◑𝐹 β†’ (𝐺 ∈ USPGraph β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))))))
676, 66syl6bi 252 . . . . . . . . . . 11 (𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ (Fun ◑𝐹 β†’ (𝐺 ∈ USPGraph β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))))))
6867impd 411 . . . . . . . . . 10 (𝐺 ∈ UPGraph β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹) β†’ (𝐺 ∈ USPGraph β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))))))
6968com23 86 . . . . . . . . 9 (𝐺 ∈ UPGraph β†’ (𝐺 ∈ USPGraph β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹) β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))))))
703, 69mpcom 38 . . . . . . . 8 (𝐺 ∈ USPGraph β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹) β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))))
7170com12 32 . . . . . . 7 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹) β†’ (𝐺 ∈ USPGraph β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))))
722, 71sylbi 216 . . . . . 6 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ (𝐺 ∈ USPGraph β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))))
7372imp 407 . . . . 5 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ 𝐺 ∈ USPGraph) β†’ ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ))))
7473necon2d 2963 . . . 4 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ 𝐺 ∈ USPGraph) β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) β‰  2))
7574impancom 452 . . 3 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ (𝐺 ∈ USPGraph β†’ (β™―β€˜πΉ) β‰  2))
761, 75syl 17 . 2 (𝐹(Circuitsβ€˜πΊ)𝑃 β†’ (𝐺 ∈ USPGraph β†’ (β™―β€˜πΉ) β‰  2))
7776impcom 408 1 ((𝐺 ∈ USPGraph ∧ 𝐹(Circuitsβ€˜πΊ)𝑃) β†’ (β™―β€˜πΉ) β‰  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  {crab 3432   βˆ– cdif 3944  βˆ…c0 4321  π’« cpw 4601  {csn 4627  {cpr 4629   class class class wbr 5147  β—‘ccnv 5674  dom cdm 5675  Fun wfun 6534  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€˜cfv 6540  (class class class)co 7405  0cc0 11106  1c1 11107   + caddc 11109   < clt 11244   ≀ cle 11245  β„•cn 12208  2c2 12263  β„•0cn0 12468  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Word cword 14460  Vtxcvtx 28245  iEdgciedg 28246  UPGraphcupgr 28329  USPGraphcuspgr 28397  Walkscwlks 28842  Trailsctrls 28936  Circuitsccrcts 29030
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-wlks 28845  df-trls 28938  df-crcts 29032
This theorem is referenced by:  usgrn2cycl  29052
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