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Theorem dlwwlknondlwlknonf1olem1 30566

Description: Lemma 1 for dlwwlknondlwlknonf1o 30567. (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 29975	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
2		wlkcpr 29829	. . . . 5 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
3	1, 2	sylib 220	. . . 4 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
4		eqid 2762	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkpwrd 29818	. . . 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 1147	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
8		eluzge2nn0 12893	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
9	8	3ad2ant3 1148	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝑁 ∈ ℕ₀)
10		eleq1 2850	. . . . . . 7 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
11	10	3ad2ant1 1146	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
12	9, 11	mpbird 259	. . . . 5 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ ℕ₀)
13		nn0fz0 13630	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))))
14	12, 13	sylib 220	. . . 4 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))))
15		fzelp1 13581	. . . 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 29819	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
18	17	eqcomd 2768	. . . . . . 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 7412	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (0...((♯‘(1^st ‘𝑐)) + 1)) = (0...(♯‘(2^nd ‘𝑐))))
21	20	eleq2d 2848	. . . 4 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1^st ‘𝑐)) ∈ (0...((♯‘(1^st ‘𝑐)) + 1)) ↔ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐)))))
22	21	3ad2ant2 1147	. . 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 234	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐))))
24		2nn 12291	. . . . . . 7 ⊢ 2 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℕ)
26		eluz2nn 12889	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
27		eluzle 12852	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
28		elfz1b 13598	. . . . . 6 ⊢ (2 ∈ (1...𝑁) ↔ (2 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 2 ≤ 𝑁))
29	25, 26, 27, 28	syl3anbrc 1357	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ (1...𝑁))
30		ubmelfzo 13736	. . . . 5 ⊢ (2 ∈ (1...𝑁) → (𝑁 − 2) ∈ (0..^𝑁))
31	29, 30	syl 17	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 2) ∈ (0..^𝑁))
32	31	3ad2ant3 1148	. . 3 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 − 2) ∈ (0..^𝑁))
33		oveq2 7404	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (0..^(♯‘(1^st ‘𝑐))) = (0..^𝑁))
34	33	eleq2d 2848	. . . 4 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐))) ↔ (𝑁 − 2) ∈ (0..^𝑁)))
35	34	3ad2ant1 1146	. . 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 259	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐))))
37		pfxfv 14696	. 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 1390	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 208 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 class class class wbr 5100 ‘cfv 6521 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 0cc0 11073 1c1 11074 + caddc 11076 ≤ cle 11217 − cmin 11414 ℕcn 12210 2c2 12272 ℕ₀cn0 12481 ℤ_≥cuz 12839 ...cfz 13512 ..^cfzo 13659 ♯chash 14343 Word cword 14526 prefix cpfx 14684 Vtxcvtx 29197 Walkscwlks 29797 ClWalkscclwlks 29970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-ifp 1075 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-fzo 13660 df-hash 14344 df-word 14527 df-substr 14655 df-pfx 14685 df-wlks 29800 df-clwlks 29971

This theorem is referenced by: dlwwlknondlwlknonf1o 30567

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