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Theorem clwlkf1clwwlklem3 26375
 Description: Lemma 3 for clwlkf1clwwlklem 26376. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
Hypotheses
Ref Expression
clwlkfclwwlk.1 𝐴 = (1st𝑐)
clwlkfclwwlk.2 𝐵 = (2nd𝑐)
clwlkfclwwlk.c 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
clwlkfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
Assertion
Ref Expression
clwlkf1clwwlklem3 (𝑊𝐶 → (2nd𝑊) ∈ Word 𝑉)
Distinct variable groups:   𝐸,𝑐   𝑁,𝑐   𝑉,𝑐   𝑊,𝑐   𝐶,𝑐   𝐹,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlkf1clwwlklem3
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.1 . . . . . 6 𝐴 = (1st𝑐)
2 fveq2 6103 . . . . . 6 (𝑐 = 𝑊 → (1st𝑐) = (1st𝑊))
31, 2syl5eq 2656 . . . . 5 (𝑐 = 𝑊𝐴 = (1st𝑊))
43fveq2d 6107 . . . 4 (𝑐 = 𝑊 → (#‘𝐴) = (#‘(1st𝑊)))
54eqeq1d 2612 . . 3 (𝑐 = 𝑊 → ((#‘𝐴) = 𝑁 ↔ (#‘(1st𝑊)) = 𝑁))
6 clwlkfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
75, 6elrab2 3333 . 2 (𝑊𝐶 ↔ (𝑊 ∈ (𝑉 ClWalks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁))
8 eqid 2610 . . . . 5 (1st𝑊) = (1st𝑊)
9 eqid 2610 . . . . 5 (2nd𝑊) = (2nd𝑊)
108, 9clwlkcompim 26292 . . . 4 (𝑊 ∈ (𝑉 ClWalks 𝐸) → (((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑖)) = {((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊))))))
11 lencl 13179 . . . . . 6 ((1st𝑊) ∈ Word dom 𝐸 → (#‘(1st𝑊)) ∈ ℕ0)
12 nn0z 11277 . . . . . . . . . 10 ((#‘(1st𝑊)) ∈ ℕ0 → (#‘(1st𝑊)) ∈ ℤ)
13 fzval3 12404 . . . . . . . . . 10 ((#‘(1st𝑊)) ∈ ℤ → (0...(#‘(1st𝑊))) = (0..^((#‘(1st𝑊)) + 1)))
1412, 13syl 17 . . . . . . . . 9 ((#‘(1st𝑊)) ∈ ℕ0 → (0...(#‘(1st𝑊))) = (0..^((#‘(1st𝑊)) + 1)))
1514feq2d 5944 . . . . . . . 8 ((#‘(1st𝑊)) ∈ ℕ0 → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉 ↔ (2nd𝑊):(0..^((#‘(1st𝑊)) + 1))⟶𝑉))
1615biimpa 500 . . . . . . 7 (((#‘(1st𝑊)) ∈ ℕ0 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) → (2nd𝑊):(0..^((#‘(1st𝑊)) + 1))⟶𝑉)
17 iswrdi 13164 . . . . . . 7 ((2nd𝑊):(0..^((#‘(1st𝑊)) + 1))⟶𝑉 → (2nd𝑊) ∈ Word 𝑉)
1816, 17syl 17 . . . . . 6 (((#‘(1st𝑊)) ∈ ℕ0 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) → (2nd𝑊) ∈ Word 𝑉)
1911, 18sylan 487 . . . . 5 (((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) → (2nd𝑊) ∈ Word 𝑉)
2019adantr 480 . . . 4 ((((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑖)) = {((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊))))) → (2nd𝑊) ∈ Word 𝑉)
2110, 20syl 17 . . 3 (𝑊 ∈ (𝑉 ClWalks 𝐸) → (2nd𝑊) ∈ Word 𝑉)
2221adantr 480 . 2 ((𝑊 ∈ (𝑉 ClWalks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁) → (2nd𝑊) ∈ Word 𝑉)
237, 22sylbi 206 1 (𝑊𝐶 → (2nd𝑊) ∈ Word 𝑉)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  {cpr 4127  ⟨cop 4131   ↦ cmpt 4643  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   substr csubstr 13150   ClWalks cclwlk 26275 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-clwlk 26278 This theorem is referenced by:  clwlkf1clwwlklem  26376
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