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Theorem dlwwlknonclwlknonf1olem1OLD 27735
Description: Obsolete version of dlwwlknondlwlknonf1olem1 27734 as of 12-Oct-2022. (Contributed by AV, 29-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
dlwwlknonclwlknonf1olem1OLD (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))

Proof of Theorem dlwwlknonclwlknonf1olem1OLD
StepHypRef Expression
1 clwlkwlk 27029 . . . . 5 (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
2 wlkcpr 26878 . . . . 5 (𝑐 ∈ (Walks‘𝐺) ↔ (1st𝑐)(Walks‘𝐺)(2nd𝑐))
31, 2sylib 210 . . . 4 (𝑐 ∈ (ClWalks‘𝐺) → (1st𝑐)(Walks‘𝐺)(2nd𝑐))
4 eqid 2799 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
54wlkpwrd 26867 . . . 4 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
63, 5syl 17 . . 3 (𝑐 ∈ (ClWalks‘𝐺) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
763ad2ant2 1165 . 2 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
8 eluzge2nn0 11971 . . . . . . 7 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ0)
983ad2ant3 1166 . . . . . 6 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → 𝑁 ∈ ℕ0)
10 eleq1 2866 . . . . . . 7 ((♯‘(1st𝑐)) = 𝑁 → ((♯‘(1st𝑐)) ∈ ℕ0𝑁 ∈ ℕ0))
11103ad2ant1 1164 . . . . . 6 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → ((♯‘(1st𝑐)) ∈ ℕ0𝑁 ∈ ℕ0))
129, 11mpbird 249 . . . . 5 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (♯‘(1st𝑐)) ∈ ℕ0)
13 nn0fz0 12692 . . . . 5 ((♯‘(1st𝑐)) ∈ ℕ0 ↔ (♯‘(1st𝑐)) ∈ (0...(♯‘(1st𝑐))))
1412, 13sylib 210 . . . 4 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (♯‘(1st𝑐)) ∈ (0...(♯‘(1st𝑐))))
15 fzelp1 12647 . . . 4 ((♯‘(1st𝑐)) ∈ (0...(♯‘(1st𝑐))) → (♯‘(1st𝑐)) ∈ (0...((♯‘(1st𝑐)) + 1)))
1614, 15syl 17 . . 3 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (♯‘(1st𝑐)) ∈ (0...((♯‘(1st𝑐)) + 1)))
17 wlklenvp1 26868 . . . . . . . 8 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(2nd𝑐)) = ((♯‘(1st𝑐)) + 1))
1817eqcomd 2805 . . . . . . 7 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) + 1) = (♯‘(2nd𝑐)))
193, 18syl 17 . . . . . 6 (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1st𝑐)) + 1) = (♯‘(2nd𝑐)))
2019oveq2d 6894 . . . . 5 (𝑐 ∈ (ClWalks‘𝐺) → (0...((♯‘(1st𝑐)) + 1)) = (0...(♯‘(2nd𝑐))))
2120eleq2d 2864 . . . 4 (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1st𝑐)) ∈ (0...((♯‘(1st𝑐)) + 1)) ↔ (♯‘(1st𝑐)) ∈ (0...(♯‘(2nd𝑐)))))
22213ad2ant2 1165 . . 3 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → ((♯‘(1st𝑐)) ∈ (0...((♯‘(1st𝑐)) + 1)) ↔ (♯‘(1st𝑐)) ∈ (0...(♯‘(2nd𝑐)))))
2316, 22mpbid 224 . 2 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (♯‘(1st𝑐)) ∈ (0...(♯‘(2nd𝑐))))
24 2nn 11386 . . . . . . 7 2 ∈ ℕ
2524a1i 11 . . . . . 6 (𝑁 ∈ (ℤ‘2) → 2 ∈ ℕ)
26 eluz2nn 11970 . . . . . 6 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ)
27 eluzle 11943 . . . . . 6 (𝑁 ∈ (ℤ‘2) → 2 ≤ 𝑁)
28 elfz1b 12663 . . . . . 6 (2 ∈ (1...𝑁) ↔ (2 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 2 ≤ 𝑁))
2925, 26, 27, 28syl3anbrc 1444 . . . . 5 (𝑁 ∈ (ℤ‘2) → 2 ∈ (1...𝑁))
30 ubmelfzo 12788 . . . . 5 (2 ∈ (1...𝑁) → (𝑁 − 2) ∈ (0..^𝑁))
3129, 30syl 17 . . . 4 (𝑁 ∈ (ℤ‘2) → (𝑁 − 2) ∈ (0..^𝑁))
32313ad2ant3 1166 . . 3 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (𝑁 − 2) ∈ (0..^𝑁))
33 oveq2 6886 . . . . 5 ((♯‘(1st𝑐)) = 𝑁 → (0..^(♯‘(1st𝑐))) = (0..^𝑁))
3433eleq2d 2864 . . . 4 ((♯‘(1st𝑐)) = 𝑁 → ((𝑁 − 2) ∈ (0..^(♯‘(1st𝑐))) ↔ (𝑁 − 2) ∈ (0..^𝑁)))
35343ad2ant1 1164 . . 3 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → ((𝑁 − 2) ∈ (0..^(♯‘(1st𝑐))) ↔ (𝑁 − 2) ∈ (0..^𝑁)))
3632, 35mpbird 249 . 2 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (𝑁 − 2) ∈ (0..^(♯‘(1st𝑐))))
37 swrd0fvOLD 13692 . 2 (((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (0...(♯‘(2nd𝑐))) ∧ (𝑁 − 2) ∈ (0..^(♯‘(1st𝑐)))) → (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))
387, 23, 36, 37syl3anc 1491 1 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1108   = wceq 1653  wcel 2157  cop 4374   class class class wbr 4843  cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  0cc0 10224  1c1 10225   + caddc 10227  cle 10364  cmin 10556  cn 11312  2c2 11368  0cn0 11580  cuz 11930  ...cfz 12580  ..^cfzo 12720  chash 13370  Word cword 13534   substr csubstr 13664  Vtxcvtx 26231  Walkscwlks 26846  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-2 11376  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-substr 13665  df-wlks 26849  df-clwlks 27025
This theorem is referenced by:  dlwwlknondlwlknonf1oOLD  27737
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