Theorem clwlknf1oclwwlknlem1 28445

Description: Lemma 1 for clwlknf1oclwwlkn 28448. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)

Assertion

Ref	Expression
clwlknf1oclwwlknlem1	⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶)))

Proof of Theorem clwlknf1oclwwlknlem1

Step	Hyp	Ref	Expression
1		clwlkwlk 28143	. . 3 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → 𝐶 ∈ (Walks‘𝐺))
2		wlkcpr 27996	. . . 4 ⊢ (𝐶 ∈ (Walks‘𝐺) ↔ (1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶))
3		eqid 2738	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkpwrd 27984	. . . . . . 7 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (2nd ‘𝐶) ∈ Word (Vtx‘𝐺))
5		lencl 14236	. . . . . . . . 9 ⊢ ((2nd ‘𝐶) ∈ Word (Vtx‘𝐺) → (♯‘(2nd ‘𝐶)) ∈ ℕ0)
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (♯‘(2nd ‘𝐶)) ∈ ℕ0)
7		wlklenvm1 27989	. . . . . . . . . . 11 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (♯‘(1st ‘𝐶)) = ((♯‘(2nd ‘𝐶)) − 1))
8	7	breq2d 5086	. . . . . . . . . 10 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
9		1red 10976	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → 1 ∈ ℝ)
10		nn0re 12242	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → (♯‘(2nd ‘𝐶)) ∈ ℝ)
11	9, 9, 10	leaddsub2d 11577	. . . . . . . . . . . 12 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
12		1p1e2 12098	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
13	12	breq1i 5081	. . . . . . . . . . . . 13 ⊢ ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) ↔ 2 ≤ (♯‘(2nd ‘𝐶)))
14	13	biimpi 215	. . . . . . . . . . . 12 ⊢ ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) → 2 ≤ (♯‘(2nd ‘𝐶)))
15	11, 14	syl6bir 253	. . . . . . . . . . 11 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → (1 ≤ ((♯‘(2nd ‘𝐶)) − 1) → 2 ≤ (♯‘(2nd ‘𝐶))))
16	4, 5, 15	3syl 18	. . . . . . . . . 10 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ ((♯‘(2nd ‘𝐶)) − 1) → 2 ≤ (♯‘(2nd ‘𝐶))))
17	8, 16	sylbid 239	. . . . . . . . 9 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) → 2 ≤ (♯‘(2nd ‘𝐶))))
18	17	imp 407	. . . . . . . 8 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → 2 ≤ (♯‘(2nd ‘𝐶)))
19		ige2m1fz 13346	. . . . . . . 8 ⊢ (((♯‘(2nd ‘𝐶)) ∈ ℕ0 ∧ 2 ≤ (♯‘(2nd ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
20	6, 18, 19	syl2an2r 682	. . . . . . 7 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
21		pfxlen 14396	. . . . . . 7 ⊢ (((2nd ‘𝐶) ∈ Word (Vtx‘𝐺) ∧ ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶)))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = ((♯‘(2nd ‘𝐶)) − 1))
22	4, 20, 21	syl2an2r 682	. . . . . 6 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = ((♯‘(2nd ‘𝐶)) − 1))
23	7	eqcomd 2744	. . . . . . 7 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → ((♯‘(2nd ‘𝐶)) − 1) = (♯‘(1st ‘𝐶)))
24	23	adantr 481	. . . . . 6 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) = (♯‘(1st ‘𝐶)))
25	22, 24	eqtrd 2778	. . . . 5 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶)))
26	25	ex 413	. . . 4 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶))))
27	2, 26	sylbi 216	. . 3 ⊢ (𝐶 ∈ (Walks‘𝐺) → (1 ≤ (♯‘(1st ‘𝐶)) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶))))
28	1, 27	syl 17	. 2 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘(1st ‘𝐶)) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶))))
29	28	imp 407	1 ⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  0cc0 10871  1c1 10872   + caddc 10874   ≤ cle 11010   − cmin 11205  2c2 12028  ℕ₀cn0 12233  ...cfz 13239  ♯chash 14044  Word cword 14217   prefix cpfx 14383  Vtxcvtx 27366  Walkscwlks 27963  ClWalkscclwlks 28138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-substr 14354  df-pfx 14384  df-wlks 27966  df-clwlks 28139

This theorem is referenced by:  clwlknf1oclwwlkn  28448