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Theorem uspgrn2crct 29329
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 29316 . . 3 (𝐹(Circuitsβ€˜πΊ)𝑃 β†’ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
2 istrl 29220 . . . . . . 7 (𝐹(Trailsβ€˜πΊ)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹))
3 uspgrupgr 28703 . . . . . . . . 9 (𝐺 ∈ USPGraph β†’ 𝐺 ∈ UPGraph)
4 eqid 2730 . . . . . . . . . . . . 13 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
5 eqid 2730 . . . . . . . . . . . . 13 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
64, 5upgriswlk 29165 . . . . . . . . . . . 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 2786 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((π‘ƒβ€˜2) = (π‘ƒβ€˜0) β†’ {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
109eqcoms 2738 . . . . . . . . . . . . . . . . . . . . . . . 24 ((π‘ƒβ€˜0) = (π‘ƒβ€˜2) β†’ {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
1110eqeq2d 2741 . . . . . . . . . . . . . . . . . . . . . . 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 627 . . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . . 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 2756 . . . . . . . . . . . . . . . . . . . . . 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 28681 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . . . 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 6547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . . . . . . . . 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 12289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 ∈ β„•
27 lbfzo0 13676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0 ∈ (0..^2) ↔ 2 ∈ β„•)
2826, 27mpbir 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ (0..^2)
29 1nn0 12492 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 ∈ β„•0
30 1lt2 12387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 < 2
31 elfzo0 13677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ (0..^2) ↔ (1 ∈ β„•0 ∧ 2 ∈ β„• ∧ 1 < 2))
3229, 26, 30, 31mpbir3an 1339 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ (0..^2)
3328, 32pm3.2i 469 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
34 oveq2 7419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = (0..^2))
3534eleq2d 2817 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜πΉ) = 2 β†’ (0 ∈ (0..^(β™―β€˜πΉ)) ↔ 0 ∈ (0..^2)))
3634eleq2d 2817 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((β™―β€˜πΉ) = 2 β†’ (1 ∈ (0..^(β™―β€˜πΉ)) ↔ 1 ∈ (0..^2)))
3735, 36anbi12d 629 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . . . . . . . . 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 7259 . . . . . . . . . . . . . . . . . . . . . . . . 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 834 . . . . . . . . . . . . . . . . . . . . . . . 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 12287 . . . . . . . . . . . . . . . . . . . . . . . 24 0 β‰  1
43 eqneqall 2949 . . . . . . . . . . . . . . . . . . . . . . . 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 710 . . . . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . 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 3023 . . . . . . . . . . . . . . . . 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 13721 . . . . . . . . . . . . . . . . . . . . . . 23 (0..^2) = {0, 1}
5334, 52eqtrdi 2786 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = {0, 1})
5453raleqdv 3323 . . . . . . . . . . . . . . . . . . . . 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 29191 . . . . . . . . . . . . . . . . . . . . 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 627 . . . . . . . . . . . . . . . . . . 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 6890 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜πΉ) = 2 β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜2))
5958neeq2d 2999 . . . . . . . . . . . . . . . . . . 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 343 . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . 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 1129 . . . . . . . . . . . 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 409 . . . . . . . . . 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 405 . . . . 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 2961 . . . 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 450 . . 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 406 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  {crab 3430   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 6536  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   ≀ cle 11253  β„•cn 12216  2c2 12271  β„•0cn0 12476  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Word cword 14468  Vtxcvtx 28523  iEdgciedg 28524  UPGraphcupgr 28607  USPGraphcuspgr 28675  Walkscwlks 29120  Trailsctrls 29214  Circuitsccrcts 29308
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-wlks 29123  df-trls 29216  df-crcts 29310
This theorem is referenced by:  usgrn2cycl  29330
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