Theorem clwlknf1oclwwlknlem1 30100

Description: Lemma 1 for clwlknf1oclwwlkn 30103. (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 29795	. . 3 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → 𝐶 ∈ (Walks‘𝐺))
2		wlkcpr 29647	. . . 4 ⊢ (𝐶 ∈ (Walks‘𝐺) ↔ (1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶))
3		eqid 2737	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkpwrd 29635	. . . . . . 7 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (2nd ‘𝐶) ∈ Word (Vtx‘𝐺))
5		lencl 14571	. . . . . . . . 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 29640	. . . . . . . . . . 11 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (♯‘(1st ‘𝐶)) = ((♯‘(2nd ‘𝐶)) − 1))
8	7	breq2d 5155	. . . . . . . . . 10 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
9		1red 11262	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → 1 ∈ ℝ)
10		nn0re 12535	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → (♯‘(2nd ‘𝐶)) ∈ ℝ)
11	9, 9, 10	leaddsub2d 11865	. . . . . . . . . . . 12 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
12		1p1e2 12391	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
13	12	breq1i 5150	. . . . . . . . . . . . 13 ⊢ ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) ↔ 2 ≤ (♯‘(2nd ‘𝐶)))
14	13	biimpi 216	. . . . . . . . . . . 12 ⊢ ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) → 2 ≤ (♯‘(2nd ‘𝐶)))
15	11, 14	biimtrrdi 254	. . . . . . . . . . 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 240	. . . . . . . . 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 13657	. . . . . . . 8 ⊢ (((♯‘(2nd ‘𝐶)) ∈ ℕ0 ∧ 2 ≤ (♯‘(2nd ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
20	6, 18, 19	syl2an2r 685	. . . . . . 7 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
21		pfxlen 14721	. . . . . . 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 685	. . . . . 6 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = ((♯‘(2nd ‘𝐶)) − 1))
23	7	eqcomd 2743	. . . . . . 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 2777	. . . . 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 217	. . 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 1540   ∈ wcel 2108   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  0cc0 11155  1c1 11156   + caddc 11158   ≤ cle 11296   − cmin 11492  2c2 12321  ℕ₀cn0 12526  ...cfz 13547  ♯chash 14369  Word cword 14552   prefix cpfx 14708  Vtxcvtx 29013  Walkscwlks 29614  ClWalkscclwlks 29790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-substr 14679  df-pfx 14709  df-wlks 29617  df-clwlks 29791

This theorem is referenced by:  clwlknf1oclwwlkn  30103