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Theorem cyclnspth 26159
 Description: A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Assertion
Ref Expression
cyclnspth (𝐹 ≠ ∅ → (𝐹(𝑉 Cycles 𝐸)𝑃 → ¬ 𝐹(𝑉 SPaths 𝐸)𝑃))

Proof of Theorem cyclnspth
Dummy variables 𝑒 𝑓 𝑝 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cycl 26041 . . . 4 Cycles = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
21brovmpt2ex 7236 . . 3 (𝐹(𝑉 Cycles 𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
3 iscycl 26153 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ↔ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
4 pthistrl 26102 . . . . . . . . . . . 12 (𝐹(𝑉 Paths 𝐸)𝑃𝐹(𝑉 Trails 𝐸)𝑃)
5 trliswlk 26069 . . . . . . . . . . . 12 (𝐹(𝑉 Trails 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)
6 2mwlk 26049 . . . . . . . . . . . 12 (𝐹(𝑉 Walks 𝐸)𝑃 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉))
7 lennncl 13180 . . . . . . . . . . . . . . . 16 ((𝐹 ∈ Word dom 𝐸𝐹 ≠ ∅) → (#‘𝐹) ∈ ℕ)
8 df-f1 5809 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑃:(0...(#‘𝐹))–1-1𝑉 ↔ (𝑃:(0...(#‘𝐹))⟶𝑉 ∧ Fun 𝑃))
9 nnne0 10930 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) ∈ ℕ → (#‘𝐹) ≠ 0)
109necomd 2837 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝐹) ∈ ℕ → 0 ≠ (#‘𝐹))
1110neneqd 2787 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝐹) ∈ ℕ → ¬ 0 = (#‘𝐹))
1211adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((#‘𝐹) ∈ ℕ ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ¬ 0 = (#‘𝐹))
13 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((#‘𝐹) ∈ ℕ ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → 𝑃:(0...(#‘𝐹))–1-1𝑉)
14 nnnn0 11176 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) ∈ ℕ → (#‘𝐹) ∈ ℕ0)
15 0elfz 12305 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) ∈ ℕ0 → 0 ∈ (0...(#‘𝐹)))
16 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℕ0)
17 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℝ)
1817leidd 10473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ≤ (#‘𝐹))
19 elfz2nn0 12300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) ∈ (0...(#‘𝐹)) ↔ ((#‘𝐹) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0 ∧ (#‘𝐹) ≤ (#‘𝐹)))
2016, 16, 18, 19syl3anbrc 1239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ (0...(#‘𝐹)))
2115, 20jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) ∈ ℕ0 → (0 ∈ (0...(#‘𝐹)) ∧ (#‘𝐹) ∈ (0...(#‘𝐹))))
2214, 21syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝐹) ∈ ℕ → (0 ∈ (0...(#‘𝐹)) ∧ (#‘𝐹) ∈ (0...(#‘𝐹))))
2322adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((#‘𝐹) ∈ ℕ ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (0 ∈ (0...(#‘𝐹)) ∧ (#‘𝐹) ∈ (0...(#‘𝐹))))
24 f1fveq 6420 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (0 ∈ (0...(#‘𝐹)) ∧ (#‘𝐹) ∈ (0...(#‘𝐹)))) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ 0 = (#‘𝐹)))
2513, 23, 24syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((#‘𝐹) ∈ ℕ ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ 0 = (#‘𝐹)))
2612, 25mtbird 314 . . . . . . . . . . . . . . . . . . . . . . . 24 (((#‘𝐹) ∈ ℕ ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ¬ (𝑃‘0) = (𝑃‘(#‘𝐹)))
2726expcom 450 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((#‘𝐹) ∈ ℕ → ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))))
288, 27sylbir 224 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃:(0...(#‘𝐹))⟶𝑉 ∧ Fun 𝑃) → ((#‘𝐹) ∈ ℕ → ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))))
2928expcom 450 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝑃 → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((#‘𝐹) ∈ ℕ → ¬ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
3029com13 86 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝐹) ∈ ℕ → (𝑃:(0...(#‘𝐹))⟶𝑉 → (Fun 𝑃 → ¬ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
3130imp 444 . . . . . . . . . . . . . . . . . . 19 (((#‘𝐹) ∈ ℕ ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → (Fun 𝑃 → ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))))
3231con2d 128 . . . . . . . . . . . . . . . . . 18 (((#‘𝐹) ∈ ℕ ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → ¬ Fun 𝑃))
3332ex 449 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) ∈ ℕ → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → ¬ Fun 𝑃)))
3433com23 84 . . . . . . . . . . . . . . . 16 ((#‘𝐹) ∈ ℕ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑃:(0...(#‘𝐹))⟶𝑉 → ¬ Fun 𝑃)))
357, 34syl 17 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom 𝐸𝐹 ≠ ∅) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑃:(0...(#‘𝐹))⟶𝑉 → ¬ Fun 𝑃)))
3635ex 449 . . . . . . . . . . . . . 14 (𝐹 ∈ Word dom 𝐸 → (𝐹 ≠ ∅ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑃:(0...(#‘𝐹))⟶𝑉 → ¬ Fun 𝑃))))
3736com24 93 . . . . . . . . . . . . 13 (𝐹 ∈ Word dom 𝐸 → (𝑃:(0...(#‘𝐹))⟶𝑉 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝐹 ≠ ∅ → ¬ Fun 𝑃))))
3837imp 444 . . . . . . . . . . . 12 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝐹 ≠ ∅ → ¬ Fun 𝑃)))
394, 5, 6, 384syl 19 . . . . . . . . . . 11 (𝐹(𝑉 Paths 𝐸)𝑃 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝐹 ≠ ∅ → ¬ Fun 𝑃)))
4039imp 444 . . . . . . . . . 10 ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝐹 ≠ ∅ → ¬ Fun 𝑃))
4140adantl 481 . . . . . . . . 9 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) → (𝐹 ≠ ∅ → ¬ Fun 𝑃))
4241imp 444 . . . . . . . 8 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) ∧ 𝐹 ≠ ∅) → ¬ Fun 𝑃)
4342intnand 953 . . . . . . 7 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) ∧ 𝐹 ≠ ∅) → ¬ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃))
44 isspth 26099 . . . . . . . . 9 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 SPaths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)))
4544adantr 480 . . . . . . . 8 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) → (𝐹(𝑉 SPaths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)))
4645adantr 480 . . . . . . 7 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) ∧ 𝐹 ≠ ∅) → (𝐹(𝑉 SPaths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)))
4743, 46mtbird 314 . . . . . 6 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) ∧ 𝐹 ≠ ∅) → ¬ 𝐹(𝑉 SPaths 𝐸)𝑃)
4847ex 449 . . . . 5 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) → (𝐹 ≠ ∅ → ¬ 𝐹(𝑉 SPaths 𝐸)𝑃))
4948ex 449 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝐹 ≠ ∅ → ¬ 𝐹(𝑉 SPaths 𝐸)𝑃)))
503, 49sylbid 229 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝐹 ≠ ∅ → ¬ 𝐹(𝑉 SPaths 𝐸)𝑃)))
512, 50mpcom 37 . 2 (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝐹 ≠ ∅ → ¬ 𝐹(𝑉 SPaths 𝐸)𝑃))
5251com12 32 1 (𝐹 ≠ ∅ → (𝐹(𝑉 Cycles 𝐸)𝑃 → ¬ 𝐹(𝑉 SPaths 𝐸)𝑃))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815   ≤ cle 9954  ℕcn 10897  ℕ0cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146   Walks cwalk 26026   Trails ctrail 26027   Paths cpath 26028   SPaths cspath 26029   Cycles ccycl 26035 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-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-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-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-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-cycl 26041 This theorem is referenced by: (None)
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