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Theorem dlwwlknondlwlknonf1olem1 28147
Description: Lemma 1 for dlwwlknondlwlknonf1o 28148. (Contributed by AV, 29-May-2022.) (Revised by AV, 1-Nov-2022.)
Assertion
Ref Expression
dlwwlknondlwlknonf1olem1 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))

Proof of Theorem dlwwlknondlwlknonf1olem1
StepHypRef Expression
1 clwlkwlk 27562 . . . . 5 (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
2 wlkcpr 27416 . . . . 5 (𝑐 ∈ (Walks‘𝐺) ↔ (1st𝑐)(Walks‘𝐺)(2nd𝑐))
31, 2sylib 221 . . . 4 (𝑐 ∈ (ClWalks‘𝐺) → (1st𝑐)(Walks‘𝐺)(2nd𝑐))
4 eqid 2822 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
54wlkpwrd 27405 . . . 4 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
63, 5syl 17 . . 3 (𝑐 ∈ (ClWalks‘𝐺) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
763ad2ant2 1131 . 2 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
8 eluzge2nn0 12275 . . . . . . 7 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ0)
983ad2ant3 1132 . . . . . 6 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → 𝑁 ∈ ℕ0)
10 eleq1 2901 . . . . . . 7 ((♯‘(1st𝑐)) = 𝑁 → ((♯‘(1st𝑐)) ∈ ℕ0𝑁 ∈ ℕ0))
11103ad2ant1 1130 . . . . . 6 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → ((♯‘(1st𝑐)) ∈ ℕ0𝑁 ∈ ℕ0))
129, 11mpbird 260 . . . . 5 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (♯‘(1st𝑐)) ∈ ℕ0)
13 nn0fz0 13000 . . . . 5 ((♯‘(1st𝑐)) ∈ ℕ0 ↔ (♯‘(1st𝑐)) ∈ (0...(♯‘(1st𝑐))))
1412, 13sylib 221 . . . 4 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (♯‘(1st𝑐)) ∈ (0...(♯‘(1st𝑐))))
15 fzelp1 12954 . . . 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 27406 . . . . . . . 8 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(2nd𝑐)) = ((♯‘(1st𝑐)) + 1))
1817eqcomd 2828 . . . . . . 7 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) + 1) = (♯‘(2nd𝑐)))
193, 18syl 17 . . . . . 6 (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1st𝑐)) + 1) = (♯‘(2nd𝑐)))
2019oveq2d 7156 . . . . 5 (𝑐 ∈ (ClWalks‘𝐺) → (0...((♯‘(1st𝑐)) + 1)) = (0...(♯‘(2nd𝑐))))
2120eleq2d 2899 . . . 4 (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1st𝑐)) ∈ (0...((♯‘(1st𝑐)) + 1)) ↔ (♯‘(1st𝑐)) ∈ (0...(♯‘(2nd𝑐)))))
22213ad2ant2 1131 . . 3 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → ((♯‘(1st𝑐)) ∈ (0...((♯‘(1st𝑐)) + 1)) ↔ (♯‘(1st𝑐)) ∈ (0...(♯‘(2nd𝑐)))))
2316, 22mpbid 235 . 2 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (♯‘(1st𝑐)) ∈ (0...(♯‘(2nd𝑐))))
24 2nn 11698 . . . . . . 7 2 ∈ ℕ
2524a1i 11 . . . . . 6 (𝑁 ∈ (ℤ‘2) → 2 ∈ ℕ)
26 eluz2nn 12272 . . . . . 6 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ)
27 eluzle 12244 . . . . . 6 (𝑁 ∈ (ℤ‘2) → 2 ≤ 𝑁)
28 elfz1b 12971 . . . . . 6 (2 ∈ (1...𝑁) ↔ (2 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 2 ≤ 𝑁))
2925, 26, 27, 28syl3anbrc 1340 . . . . 5 (𝑁 ∈ (ℤ‘2) → 2 ∈ (1...𝑁))
30 ubmelfzo 13097 . . . . 5 (2 ∈ (1...𝑁) → (𝑁 − 2) ∈ (0..^𝑁))
3129, 30syl 17 . . . 4 (𝑁 ∈ (ℤ‘2) → (𝑁 − 2) ∈ (0..^𝑁))
32313ad2ant3 1132 . . 3 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (𝑁 − 2) ∈ (0..^𝑁))
33 oveq2 7148 . . . . 5 ((♯‘(1st𝑐)) = 𝑁 → (0..^(♯‘(1st𝑐))) = (0..^𝑁))
3433eleq2d 2899 . . . 4 ((♯‘(1st𝑐)) = 𝑁 → ((𝑁 − 2) ∈ (0..^(♯‘(1st𝑐))) ↔ (𝑁 − 2) ∈ (0..^𝑁)))
35343ad2ant1 1130 . . 3 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → ((𝑁 − 2) ∈ (0..^(♯‘(1st𝑐))) ↔ (𝑁 − 2) ∈ (0..^𝑁)))
3632, 35mpbird 260 . 2 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (𝑁 − 2) ∈ (0..^(♯‘(1st𝑐))))
37 pfxfv 14035 . 2 (((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (0...(♯‘(2nd𝑐))) ∧ (𝑁 − 2) ∈ (0..^(♯‘(1st𝑐)))) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))
387, 23, 36, 37syl3anc 1368 1 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2114   class class class wbr 5042  cfv 6334  (class class class)co 7140  1st c1st 7673  2nd c2nd 7674  0cc0 10526  1c1 10527   + caddc 10529  cle 10665  cmin 10859  cn 11625  2c2 11680  0cn0 11885  cuz 12231  ...cfz 12885  ..^cfzo 13028  chash 13686  Word cword 13857   prefix cpfx 14023  Vtxcvtx 26787  Walkscwlks 27384  ClWalkscclwlks 27557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-substr 13994  df-pfx 14024  df-wlks 27387  df-clwlks 27558
This theorem is referenced by:  dlwwlknondlwlknonf1o  28148
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