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Theorem clwlksf1clwwlklem3OLD 27408
 Description: Obsolete as of 24-May-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
clwlksfclwwlkOLD.1 𝐴 = (1st𝑐)
clwlksfclwwlkOLD.2 𝐵 = (2nd𝑐)
clwlksfclwwlkOLD.c 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlkOLD.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
Assertion
Ref Expression
clwlksf1clwwlklem3OLD (𝑊𝐶 → (2nd𝑊) ∈ Word (Vtx‘𝐺))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝑊,𝑐   𝐶,𝑐   𝐹,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlksf1clwwlklem3OLD
StepHypRef Expression
1 clwlksfclwwlkOLD.1 . . 3 𝐴 = (1st𝑐)
2 clwlksfclwwlkOLD.2 . . 3 𝐵 = (2nd𝑐)
3 clwlksfclwwlkOLD.c . . 3 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
4 clwlksfclwwlkOLD.f . . 3 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
51, 2, 3, 4clwlksf1clwwlklem0OLD 27405 . 2 (𝑊𝐶 → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))) ∧ (♯‘(1st𝑊)) = 𝑁))
6 lencl 13553 . . . . 5 ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → (♯‘(1st𝑊)) ∈ ℕ0)
7 nn0z 11690 . . . . . . . . 9 ((♯‘(1st𝑊)) ∈ ℕ0 → (♯‘(1st𝑊)) ∈ ℤ)
8 fzval3 12792 . . . . . . . . 9 ((♯‘(1st𝑊)) ∈ ℤ → (0...(♯‘(1st𝑊))) = (0..^((♯‘(1st𝑊)) + 1)))
97, 8syl 17 . . . . . . . 8 ((♯‘(1st𝑊)) ∈ ℕ0 → (0...(♯‘(1st𝑊))) = (0..^((♯‘(1st𝑊)) + 1)))
109feq2d 6242 . . . . . . 7 ((♯‘(1st𝑊)) ∈ ℕ0 → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0..^((♯‘(1st𝑊)) + 1))⟶(Vtx‘𝐺)))
1110biimpa 469 . . . . . 6 (((♯‘(1st𝑊)) ∈ ℕ0 ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → (2nd𝑊):(0..^((♯‘(1st𝑊)) + 1))⟶(Vtx‘𝐺))
12 iswrdi 13538 . . . . . 6 ((2nd𝑊):(0..^((♯‘(1st𝑊)) + 1))⟶(Vtx‘𝐺) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
1311, 12syl 17 . . . . 5 (((♯‘(1st𝑊)) ∈ ℕ0 ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
146, 13sylan 576 . . . 4 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
15143adant3 1163 . . 3 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
1615adantr 473 . 2 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))) ∧ (♯‘(1st𝑊)) = 𝑁) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
175, 16syl 17 1 (𝑊𝐶 → (2nd𝑊) ∈ Word (Vtx‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  {crab 3093  ⟨cop 4374   ↦ cmpt 4922  dom cdm 5312  ⟶wf 6097  ‘cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  0cc0 10224  1c1 10225   + caddc 10227  ℕ0cn0 11580  ℤcz 11666  ...cfz 12580  ..^cfzo 12720  ♯chash 13370  Word cword 13534   substr csubstr 13664  Vtxcvtx 26231  iEdgciedg 26232  ClWalkscclwlks 27024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ifp 1087  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-wlks 26849  df-clwlks 27025 This theorem is referenced by:  clwlksf1clwwlklemOLD  27409
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