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Theorem clwlknf1oclwwlknlem1 27852
 Description: Lemma 1 for clwlknf1oclwwlkn 27855. (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 27548 . . 3 (𝐶 ∈ (ClWalks‘𝐺) → 𝐶 ∈ (Walks‘𝐺))
2 wlkcpr 27402 . . . 4 (𝐶 ∈ (Walks‘𝐺) ↔ (1st𝐶)(Walks‘𝐺)(2nd𝐶))
3 eqid 2819 . . . . . . . 8 (Vtx‘𝐺) = (Vtx‘𝐺)
43wlkpwrd 27391 . . . . . . 7 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (2nd𝐶) ∈ Word (Vtx‘𝐺))
5 lencl 13875 . . . . . . . . 9 ((2nd𝐶) ∈ Word (Vtx‘𝐺) → (♯‘(2nd𝐶)) ∈ ℕ0)
64, 5syl 17 . . . . . . . 8 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (♯‘(2nd𝐶)) ∈ ℕ0)
7 wlklenvm1 27395 . . . . . . . . . . 11 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (♯‘(1st𝐶)) = ((♯‘(2nd𝐶)) − 1))
87breq2d 5069 . . . . . . . . . 10 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (1 ≤ (♯‘(1st𝐶)) ↔ 1 ≤ ((♯‘(2nd𝐶)) − 1)))
9 1red 10634 . . . . . . . . . . . . 13 ((♯‘(2nd𝐶)) ∈ ℕ0 → 1 ∈ ℝ)
10 nn0re 11898 . . . . . . . . . . . . 13 ((♯‘(2nd𝐶)) ∈ ℕ0 → (♯‘(2nd𝐶)) ∈ ℝ)
119, 9, 10leaddsub2d 11234 . . . . . . . . . . . 12 ((♯‘(2nd𝐶)) ∈ ℕ0 → ((1 + 1) ≤ (♯‘(2nd𝐶)) ↔ 1 ≤ ((♯‘(2nd𝐶)) − 1)))
12 1p1e2 11754 . . . . . . . . . . . . . 14 (1 + 1) = 2
1312breq1i 5064 . . . . . . . . . . . . 13 ((1 + 1) ≤ (♯‘(2nd𝐶)) ↔ 2 ≤ (♯‘(2nd𝐶)))
1413biimpi 218 . . . . . . . . . . . 12 ((1 + 1) ≤ (♯‘(2nd𝐶)) → 2 ≤ (♯‘(2nd𝐶)))
1511, 14syl6bir 256 . . . . . . . . . . 11 ((♯‘(2nd𝐶)) ∈ ℕ0 → (1 ≤ ((♯‘(2nd𝐶)) − 1) → 2 ≤ (♯‘(2nd𝐶))))
164, 5, 153syl 18 . . . . . . . . . 10 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (1 ≤ ((♯‘(2nd𝐶)) − 1) → 2 ≤ (♯‘(2nd𝐶))))
178, 16sylbid 242 . . . . . . . . 9 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (1 ≤ (♯‘(1st𝐶)) → 2 ≤ (♯‘(2nd𝐶))))
1817imp 409 . . . . . . . 8 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → 2 ≤ (♯‘(2nd𝐶)))
19 ige2m1fz 12989 . . . . . . . 8 (((♯‘(2nd𝐶)) ∈ ℕ0 ∧ 2 ≤ (♯‘(2nd𝐶))) → ((♯‘(2nd𝐶)) − 1) ∈ (0...(♯‘(2nd𝐶))))
206, 18, 19syl2an2r 683 . . . . . . 7 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → ((♯‘(2nd𝐶)) − 1) ∈ (0...(♯‘(2nd𝐶))))
21 pfxlen 14037 . . . . . . 7 (((2nd𝐶) ∈ Word (Vtx‘𝐺) ∧ ((♯‘(2nd𝐶)) − 1) ∈ (0...(♯‘(2nd𝐶)))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = ((♯‘(2nd𝐶)) − 1))
224, 20, 21syl2an2r 683 . . . . . 6 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = ((♯‘(2nd𝐶)) − 1))
237eqcomd 2825 . . . . . . 7 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → ((♯‘(2nd𝐶)) − 1) = (♯‘(1st𝐶)))
2423adantr 483 . . . . . 6 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → ((♯‘(2nd𝐶)) − 1) = (♯‘(1st𝐶)))
2522, 24eqtrd 2854 . . . . 5 (((1st𝐶)(Walks‘𝐺)(2nd𝐶) ∧ 1 ≤ (♯‘(1st𝐶))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶)))
2625ex 415 . . . 4 ((1st𝐶)(Walks‘𝐺)(2nd𝐶) → (1 ≤ (♯‘(1st𝐶)) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶))))
272, 26sylbi 219 . . 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 409 1 ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝐶))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108   class class class wbr 5057  ‘cfv 6348  (class class class)co 7148  1st c1st 7679  2nd c2nd 7680  0cc0 10529  1c1 10530   + caddc 10532   ≤ cle 10668   − cmin 10862  2c2 11684  ℕ0cn0 11889  ...cfz 12884  ♯chash 13682  Word cword 13853   prefix cpfx 14024  Vtxcvtx 26773  Walkscwlks 27370  ClWalkscclwlks 27543 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  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 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-hash 13683  df-word 13854  df-substr 13995  df-pfx 14025  df-wlks 27373  df-clwlks 27544 This theorem is referenced by:  clwlknf1oclwwlkn  27855
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