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Theorem dlwwlknondlwlknonf1olem1 30194

Description: Lemma 1 for dlwwlknondlwlknonf1o 30195. (Contributed by AV, 29-May-2022.) (Revised by AV, 1-Nov-2022.)

**Assertion**
Ref	Expression
dlwwlknondlwlknonf1olem1	⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))

**Proof of Theorem dlwwlknondlwlknonf1olem1**
Step	Hyp	Ref	Expression
1		clwlkwlk 29609	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
2		wlkcpr 29463	. . . . 5 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
3	1, 2	sylib 217	. . . 4 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
4		eqid 2728	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkpwrd 29451	. . . 4 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
6	3, 5	syl 17	. . 3 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
7	6	3ad2ant2 1131	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
8		eluzge2nn0 12909	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
9	8	3ad2ant3 1132	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝑁 ∈ ℕ₀)
10		eleq1 2817	. . . . . . 7 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
11	10	3ad2ant1 1130	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
12	9, 11	mpbird 256	. . . . 5 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ ℕ₀)
13		nn0fz0 13639	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))))
14	12, 13	sylib 217	. . . 4 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))))
15		fzelp1 13593	. . . 4 ⊢ ((♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))) → (♯‘(1^st ‘𝑐)) ∈ (0...((♯‘(1^st ‘𝑐)) + 1)))
16	14, 15	syl 17	. . 3 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...((♯‘(1^st ‘𝑐)) + 1)))
17		wlklenvp1 29452	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
18	17	eqcomd 2734	. . . . . . 7 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → ((♯‘(1^st ‘𝑐)) + 1) = (♯‘(2^nd ‘𝑐)))
19	3, 18	syl 17	. . . . . 6 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1^st ‘𝑐)) + 1) = (♯‘(2^nd ‘𝑐)))
20	19	oveq2d 7442	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (0...((♯‘(1^st ‘𝑐)) + 1)) = (0...(♯‘(2^nd ‘𝑐))))
21	20	eleq2d 2815	. . . 4 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1^st ‘𝑐)) ∈ (0...((♯‘(1^st ‘𝑐)) + 1)) ↔ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐)))))
22	21	3ad2ant2 1131	. . 3 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → ((♯‘(1^st ‘𝑐)) ∈ (0...((♯‘(1^st ‘𝑐)) + 1)) ↔ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐)))))
23	16, 22	mpbid 231	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐))))
24		2nn 12323	. . . . . . 7 ⊢ 2 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℕ)
26		eluz2nn 12906	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
27		eluzle 12873	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
28		elfz1b 13610	. . . . . 6 ⊢ (2 ∈ (1...𝑁) ↔ (2 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 2 ≤ 𝑁))
29	25, 26, 27, 28	syl3anbrc 1340	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ (1...𝑁))
30		ubmelfzo 13737	. . . . 5 ⊢ (2 ∈ (1...𝑁) → (𝑁 − 2) ∈ (0..^𝑁))
31	29, 30	syl 17	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 2) ∈ (0..^𝑁))
32	31	3ad2ant3 1132	. . 3 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 − 2) ∈ (0..^𝑁))
33		oveq2 7434	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (0..^(♯‘(1^st ‘𝑐))) = (0..^𝑁))
34	33	eleq2d 2815	. . . 4 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐))) ↔ (𝑁 − 2) ∈ (0..^𝑁)))
35	34	3ad2ant1 1130	. . 3 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → ((𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐))) ↔ (𝑁 − 2) ∈ (0..^𝑁)))
36	32, 35	mpbird 256	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐))))
37		pfxfv 14672	. 2 ⊢ (((2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐))) ∧ (𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐)))) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))
38	7, 23, 36, 37	syl3anc 1368	1 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 1^st c1st 7997 2^nd c2nd 7998 0cc0 11146 1c1 11147 + caddc 11149 ≤ cle 11287 − cmin 11482 ℕcn 12250 2c2 12305 ℕ₀cn0 12510 ℤ_≥cuz 12860 ...cfz 13524 ..^cfzo 13667 ♯chash 14329 Word cword 14504 prefix cpfx 14660 Vtxcvtx 28829 Walkscwlks 29430 ClWalkscclwlks 29604

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-substr 14631 df-pfx 14661 df-wlks 29433 df-clwlks 29605

This theorem is referenced by: dlwwlknondlwlknonf1o 30195

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