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Theorem dlwwlknondlwlknonf1olem1 30724

Description: Lemma 1 for dlwwlknondlwlknonf1o 30725. (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 30133	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
2		wlkcpr 29987	. . . . 5 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
3	1, 2	sylib 221	. . . 4 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
4		eqid 2763	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkpwrd 29976	. . . 4 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
6	3, 5	syl 18	. . 3 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
7	6	3ad2ant2 1152	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
8		eluzge2nn0 12920	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
9	8	3ad2ant3 1153	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝑁 ∈ ℕ₀)
10		eleq1 2851	. . . . . . 7 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
11	10	3ad2ant1 1151	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
12	9, 11	mpbird 260	. . . . 5 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ ℕ₀)
13		nn0fz0 13658	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))))
14	12, 13	sylib 221	. . . 4 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))))
15		fzelp1 13609	. . . 4 ⊢ ((♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))) → (♯‘(1^st ‘𝑐)) ∈ (0...((♯‘(1^st ‘𝑐)) + 1)))
16	14, 15	syl 18	. . 3 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...((♯‘(1^st ‘𝑐)) + 1)))
17		wlklenvp1 29977	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
18	17	eqcomd 2769	. . . . . . 7 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → ((♯‘(1^st ‘𝑐)) + 1) = (♯‘(2^nd ‘𝑐)))
19	3, 18	syl 18	. . . . . 6 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1^st ‘𝑐)) + 1) = (♯‘(2^nd ‘𝑐)))
20	19	oveq2d 7426	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (0...((♯‘(1^st ‘𝑐)) + 1)) = (0...(♯‘(2^nd ‘𝑐))))
21	20	eleq2d 2849	. . . 4 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1^st ‘𝑐)) ∈ (0...((♯‘(1^st ‘𝑐)) + 1)) ↔ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐)))))
22	21	3ad2ant2 1152	. . 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 235	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐))))
24		2nn 12318	. . . . . . 7 ⊢ 2 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℕ)
26		eluz2nn 12916	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
27		eluzle 12879	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
28		elfz1b 13626	. . . . . 6 ⊢ (2 ∈ (1...𝑁) ↔ (2 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 2 ≤ 𝑁))
29	25, 26, 27, 28	syl3anbrc 1362	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ (1...𝑁))
30		ubmelfzo 13764	. . . . 5 ⊢ (2 ∈ (1...𝑁) → (𝑁 − 2) ∈ (0..^𝑁))
31	29, 30	syl 18	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 2) ∈ (0..^𝑁))
32	31	3ad2ant3 1153	. . 3 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 − 2) ∈ (0..^𝑁))
33		oveq2 7418	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (0..^(♯‘(1^st ‘𝑐))) = (0..^𝑁))
34	33	eleq2d 2849	. . . 4 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐))) ↔ (𝑁 − 2) ∈ (0..^𝑁)))
35	34	3ad2ant1 1151	. . 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 260	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐))))
37		pfxfv 14725	. 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 1398	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 1^st c1st 7980 2^nd c2nd 7981 0cc0 11104 1c1 11105 + caddc 11107 ≤ cle 11248 − cmin 11445 ℕcn 12237 2c2 12299 ℕ₀cn0 12508 ℤ_≥cuz 12866 ...cfz 13539 ..^cfzo 13687 ♯chash 14371 Word cword 14555 prefix cpfx 14713 Vtxcvtx 29355 Walkscwlks 29955 ClWalkscclwlks 30128

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-n0 12509 df-z 12596 df-uz 12867 df-fz 13540 df-fzo 13688 df-hash 14372 df-word 14556 df-substr 14684 df-pfx 14714 df-wlks 29958 df-clwlks 30129

This theorem is used by: dlwwlknondlwlknonf1o 30725

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