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Theorem clwlknf1oclwwlknlem1 27787
Description: Lemma 1 for clwlknf1oclwwlkn 27790. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)
Assertion
Ref Expression
clwlknf1oclwwlknlem1 ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝐶))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶)))

Proof of Theorem clwlknf1oclwwlknlem1
StepHypRef Expression
1 clwlkwlk 27483 . . 3 (𝐶 ∈ (ClWalks‘𝐺) → 𝐶 ∈ (Walks‘𝐺))
2 wlkcpr 27337 . . . 4 (𝐶 ∈ (Walks‘𝐺) ↔ (1st𝐶)(Walks‘𝐺)(2nd𝐶))
3 eqid 2818 . . . . . . . 8 (Vtx‘𝐺) = (Vtx‘𝐺)
43wlkpwrd 27326 . . . . . . 7 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (2nd𝐶) ∈ Word (Vtx‘𝐺))
5 lencl 13871 . . . . . . . . 9 ((2nd𝐶) ∈ Word (Vtx‘𝐺) → (♯‘(2nd𝐶)) ∈ ℕ0)
64, 5syl 17 . . . . . . . 8 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (♯‘(2nd𝐶)) ∈ ℕ0)
7 wlklenvm1 27330 . . . . . . . . . . 11 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (♯‘(1st𝐶)) = ((♯‘(2nd𝐶)) − 1))
87breq2d 5069 . . . . . . . . . 10 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (1 ≤ (♯‘(1st𝐶)) ↔ 1 ≤ ((♯‘(2nd𝐶)) − 1)))
9 1red 10630 . . . . . . . . . . . . 13 ((♯‘(2nd𝐶)) ∈ ℕ0 → 1 ∈ ℝ)
10 nn0re 11894 . . . . . . . . . . . . 13 ((♯‘(2nd𝐶)) ∈ ℕ0 → (♯‘(2nd𝐶)) ∈ ℝ)
119, 9, 10leaddsub2d 11230 . . . . . . . . . . . 12 ((♯‘(2nd𝐶)) ∈ ℕ0 → ((1 + 1) ≤ (♯‘(2nd𝐶)) ↔ 1 ≤ ((♯‘(2nd𝐶)) − 1)))
12 1p1e2 11750 . . . . . . . . . . . . . 14 (1 + 1) = 2
1312breq1i 5064 . . . . . . . . . . . . 13 ((1 + 1) ≤ (♯‘(2nd𝐶)) ↔ 2 ≤ (♯‘(2nd𝐶)))
1413biimpi 217 . . . . . . . . . . . 12 ((1 + 1) ≤ (♯‘(2nd𝐶)) → 2 ≤ (♯‘(2nd𝐶)))
1511, 14syl6bir 255 . . . . . . . . . . 11 ((♯‘(2nd𝐶)) ∈ ℕ0 → (1 ≤ ((♯‘(2nd𝐶)) − 1) → 2 ≤ (♯‘(2nd𝐶))))
164, 5, 153syl 18 . . . . . . . . . 10 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (1 ≤ ((♯‘(2nd𝐶)) − 1) → 2 ≤ (♯‘(2nd𝐶))))
178, 16sylbid 241 . . . . . . . . 9 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (1 ≤ (♯‘(1st𝐶)) → 2 ≤ (♯‘(2nd𝐶))))
1817imp 407 . . . . . . . 8 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → 2 ≤ (♯‘(2nd𝐶)))
19 ige2m1fz 12985 . . . . . . . 8 (((♯‘(2nd𝐶)) ∈ ℕ0 ∧ 2 ≤ (♯‘(2nd𝐶))) → ((♯‘(2nd𝐶)) − 1) ∈ (0...(♯‘(2nd𝐶))))
206, 18, 19syl2an2r 681 . . . . . . 7 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → ((♯‘(2nd𝐶)) − 1) ∈ (0...(♯‘(2nd𝐶))))
21 pfxlen 14033 . . . . . . 7 (((2nd𝐶) ∈ Word (Vtx‘𝐺) ∧ ((♯‘(2nd𝐶)) − 1) ∈ (0...(♯‘(2nd𝐶)))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = ((♯‘(2nd𝐶)) − 1))
224, 20, 21syl2an2r 681 . . . . . 6 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = ((♯‘(2nd𝐶)) − 1))
237eqcomd 2824 . . . . . . 7 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → ((♯‘(2nd𝐶)) − 1) = (♯‘(1st𝐶)))
2423adantr 481 . . . . . 6 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → ((♯‘(2nd𝐶)) − 1) = (♯‘(1st𝐶)))
2522, 24eqtrd 2853 . . . . 5 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶)))
2625ex 413 . . . 4 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (1 ≤ (♯‘(1st𝐶)) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶))))
272, 26sylbi 218 . . 3 (𝐶 ∈ (Walks‘𝐺) → (1 ≤ (♯‘(1st𝐶)) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶))))
281, 27syl 17 . 2 (𝐶 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘(1st𝐶)) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶))))
2928imp 407 1 ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝐶))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  0cc0 10525  1c1 10526   + caddc 10528  cle 10664  cmin 10858  2c2 11680  0cn0 11885  ...cfz 12880  chash 13678  Word cword 13849   prefix cpfx 14020  Vtxcvtx 26708  Walkscwlks 27305  ClWalkscclwlks 27478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ifp 1055  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-substr 13991  df-pfx 14021  df-wlks 27308  df-clwlks 27479
This theorem is referenced by:  clwlknf1oclwwlkn  27790
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