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Theorem dlwwlknondlwlknonf1olem1 30455

Description: Lemma 1 for dlwwlknondlwlknonf1o 30456. (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 29864	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
2		wlkcpr 29718	. . . . 5 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
3	1, 2	sylib 218	. . . 4 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
4		eqid 2737	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	wlkpwrd 29707	. . . 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 1135	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
8		eluzge2nn0 12817	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
9	8	3ad2ant3 1136	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝑁 ∈ ℕ₀)
10		eleq1 2825	. . . . . . 7 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
11	10	3ad2ant1 1134	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
12	9, 11	mpbird 257	. . . . 5 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ ℕ₀)
13		nn0fz0 13553	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ↔ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))))
14	12, 13	sylib 218	. . . 4 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(1^st ‘𝑐))))
15		fzelp1 13504	. . . 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 29708	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
18	17	eqcomd 2743	. . . . . . 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 7384	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (0...((♯‘(1^st ‘𝑐)) + 1)) = (0...(♯‘(2^nd ‘𝑐))))
21	20	eleq2d 2823	. . . 4 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1^st ‘𝑐)) ∈ (0...((♯‘(1^st ‘𝑐)) + 1)) ↔ (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐)))))
22	21	3ad2ant2 1135	. . 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 232	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(1^st ‘𝑐)) ∈ (0...(♯‘(2^nd ‘𝑐))))
24		2nn 12230	. . . . . . 7 ⊢ 2 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℕ)
26		eluz2nn 12813	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
27		eluzle 12776	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
28		elfz1b 13521	. . . . . 6 ⊢ (2 ∈ (1...𝑁) ↔ (2 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 2 ≤ 𝑁))
29	25, 26, 27, 28	syl3anbrc 1345	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ (1...𝑁))
30		ubmelfzo 13658	. . . . 5 ⊢ (2 ∈ (1...𝑁) → (𝑁 − 2) ∈ (0..^𝑁))
31	29, 30	syl 17	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 2) ∈ (0..^𝑁))
32	31	3ad2ant3 1136	. . 3 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 − 2) ∈ (0..^𝑁))
33		oveq2 7376	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (0..^(♯‘(1^st ‘𝑐))) = (0..^𝑁))
34	33	eleq2d 2823	. . . 4 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → ((𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐))) ↔ (𝑁 − 2) ∈ (0..^𝑁)))
35	34	3ad2ant1 1134	. . 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 257	. 2 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑁 − 2) ∈ (0..^(♯‘(1^st ‘𝑐))))
37		pfxfv 14618	. 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 1374	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 206 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 0cc0 11038 1c1 11039 + caddc 11041 ≤ cle 11179 − cmin 11376 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ℤ_≥cuz 12763 ...cfz 13435 ..^cfzo 13582 ♯chash 14265 Word cword 14448 prefix cpfx 14606 Vtxcvtx 29085 Walkscwlks 29686 ClWalkscclwlks 29859

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-substr 14577 df-pfx 14607 df-wlks 29689 df-clwlks 29860

This theorem is referenced by: dlwwlknondlwlknonf1o 30456

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