Theorem clwlknf1oclwwlknlem1 28346

Description: Lemma 1 for clwlknf1oclwwlkn 28349. (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 28044	. . 3 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → 𝐶 ∈ (Walks‘𝐺))
2		wlkcpr 27898	. . . 4 ⊢ (𝐶 ∈ (Walks‘𝐺) ↔ (1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶))
3		eqid 2738	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkpwrd 27887	. . . . . . 7 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (2nd ‘𝐶) ∈ Word (Vtx‘𝐺))
5		lencl 14164	. . . . . . . . 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 27891	. . . . . . . . . . 11 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (♯‘(1st ‘𝐶)) = ((♯‘(2nd ‘𝐶)) − 1))
8	7	breq2d 5082	. . . . . . . . . 10 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
9		1red 10907	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → 1 ∈ ℝ)
10		nn0re 12172	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → (♯‘(2nd ‘𝐶)) ∈ ℝ)
11	9, 9, 10	leaddsub2d 11507	. . . . . . . . . . . 12 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
12		1p1e2 12028	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
13	12	breq1i 5077	. . . . . . . . . . . . 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 406	. . . . . . . 8 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → 2 ≤ (♯‘(2nd ‘𝐶)))
19		ige2m1fz 13275	. . . . . . . 8 ⊢ (((♯‘(2nd ‘𝐶)) ∈ ℕ0 ∧ 2 ≤ (♯‘(2nd ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
20	6, 18, 19	syl2an2r 681	. . . . . . 7 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
21		pfxlen 14324	. . . . . . 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 681	. . . . . 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 480	. . . . . 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 412	. . . 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 406	1 ⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  0cc0 10802  1c1 10803   + caddc 10805   ≤ cle 10941   − cmin 11135  2c2 11958  ℕ₀cn0 12163  ...cfz 13168  ♯chash 13972  Word cword 14145   prefix cpfx 14311  Vtxcvtx 27269  Walkscwlks 27866  ClWalkscclwlks 28039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-substr 14282  df-pfx 14312  df-wlks 27869  df-clwlks 28040

This theorem is referenced by:  clwlknf1oclwwlkn  28349