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Theorem clwlksf1clwwlklem3OLD 27248
 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
clwlksfclwwlk.1 𝐴 = (1st𝑐)
clwlksfclwwlk.2 𝐵 = (2nd𝑐)
clwlksfclwwlk.c 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlk.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 clwlksfclwwlk.1 . . 3 𝐴 = (1st𝑐)
2 clwlksfclwwlk.2 . . 3 𝐵 = (2nd𝑐)
3 clwlksfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
4 clwlksfclwwlk.f . . 3 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
51, 2, 3, 4clwlksf1clwwlklem0OLD 27245 . 2 (𝑊𝐶 → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))) ∧ (♯‘(1st𝑊)) = 𝑁))
6 lencl 13520 . . . . 5 ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → (♯‘(1st𝑊)) ∈ ℕ0)
7 nn0z 11602 . . . . . . . . 9 ((♯‘(1st𝑊)) ∈ ℕ0 → (♯‘(1st𝑊)) ∈ ℤ)
8 fzval3 12745 . . . . . . . . 9 ((♯‘(1st𝑊)) ∈ ℤ → (0...(♯‘(1st𝑊))) = (0..^((♯‘(1st𝑊)) + 1)))
97, 8syl 17 . . . . . . . 8 ((♯‘(1st𝑊)) ∈ ℕ0 → (0...(♯‘(1st𝑊))) = (0..^((♯‘(1st𝑊)) + 1)))
109feq2d 6171 . . . . . . 7 ((♯‘(1st𝑊)) ∈ ℕ0 → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0..^((♯‘(1st𝑊)) + 1))⟶(Vtx‘𝐺)))
1110biimpa 462 . . . . . 6 (((♯‘(1st𝑊)) ∈ ℕ0 ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → (2nd𝑊):(0..^((♯‘(1st𝑊)) + 1))⟶(Vtx‘𝐺))
12 iswrdi 13505 . . . . . 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 569 . . . 4 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
15143adant3 1126 . . 3 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
1615adantr 466 . 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 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  {crab 3065  ⟨cop 4322   ↦ cmpt 4863  dom cdm 5249  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  0cc0 10138  1c1 10139   + caddc 10141  ℕ0cn0 11494  ℤcz 11579  ...cfz 12533  ..^cfzo 12673  ♯chash 13321  Word cword 13487   substr csubstr 13491  Vtxcvtx 26095  iEdgciedg 26096  ClWalkscclwlks 26901 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-wlks 26730  df-clwlks 26902 This theorem is referenced by:  clwlksf1clwwlklemOLD  27249
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