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Theorem numclwwlkovgelim 26616
 Description: Properties of an element of the value of operation 𝐺. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
numclwwlk.g 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
Assertion
Ref Expression
numclwwlkovgelim ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐺𝑁) → ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))))
Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑤,𝐶   𝑤,𝑁   𝐶,𝑛,𝑣,𝑤   𝑣,𝑁   𝑛,𝑋,𝑣,𝑤   𝑣,𝑉   𝑤,𝐸   𝑤,𝑉   𝑤,𝐹   𝑤,𝑃
Allowed substitution hints:   𝑃(𝑣,𝑛)   𝐸(𝑣)   𝐹(𝑣,𝑛)   𝐺(𝑤,𝑣,𝑛)

Proof of Theorem numclwwlkovgelim
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 numclwwlk.c . . . 4 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
2 numclwwlk.f . . . 4 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
3 numclwwlk.g . . . 4 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
41, 2, 3numclwwlkovgel 26615 . . 3 ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐺𝑁) ↔ (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))
543adant1 1072 . 2 ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐺𝑁) ↔ (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))
6 usgrav 25867 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
7 eluzge2nn0 11603 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ0)
86, 7anim12i 588 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
9 df-3an 1033 . . . . . . . . . . . 12 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
108, 9sylibr 223 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
11 isclwwlkn 26297 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑃 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑃) = 𝑁)))
1210, 11syl 17 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑃 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑃) = 𝑁)))
13 isclwwlk 26296 . . . . . . . . . . . . 13 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)))
146, 13syl 17 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)))
1514adantr 480 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)))
1615anbi1d 737 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2)) → ((𝑃 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑃) = 𝑁) ↔ ((𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) ∧ (#‘𝑃) = 𝑁)))
1712, 16bitrd 267 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ ((𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) ∧ (#‘𝑃) = 𝑁)))
18 simp1 1054 . . . . . . . . . . 11 ((𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → 𝑃 ∈ Word 𝑉)
1918a1i 11 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2)) → ((𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → 𝑃 ∈ Word 𝑉))
2019anim1d 586 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2)) → (((𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) ∧ (#‘𝑃) = 𝑁) → (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁)))
2117, 20sylbid 229 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁)))
22213adant2 1073 . . . . . . 7 ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁)))
2322com12 32 . . . . . 6 (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁)))
24233ad2ant1 1075 . . . . 5 ((𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)) → ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁)))
2524impcom 445 . . . 4 (((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))) → (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁))
26 3simpc 1053 . . . . 5 ((𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)) → ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))
2726adantl 481 . . . 4 (((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))) → ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))
2825, 27jca 553 . . 3 (((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))) → ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))
2928ex 449 . 2 ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) → ((𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)) → ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))))
305, 29sylbid 229 1 ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐺𝑁) → ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  2c2 10947  ℕ0cn0 11169  ℤ≥cuz 11563  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   USGrph cusg 25859   ClWWalks cclwwlk 26276   ClWWalksN cclwwlkn 26277 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-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-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  numclwlk1lem2f1  26621
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