dlwwlknondlwlknonf1olem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dlwwlknondlwlknonf1olem1		Structured version Visualization version GIF version

Theorem dlwwlknondlwlknonf1olem1 30980

Description: Lemma 1 for dlwwlknondlwlknonf1o 30981. (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 30377	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
2		wlkcpr 30224	. . . . 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 30213	. . . 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 13032	. . . . . . 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 2852	. . . . . . 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 13772	. . . . 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 13723	. . . 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 30214	. . . . . . . 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 7446	. . . . 5 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (0...((♯‘(1^st ‘𝑐)) + 1)) = (0...(♯‘(2^nd ‘𝑐))))
21	20	eleq2d 2850	. . . 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 12429	. . . . . . 7 ⊢ 2 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℕ)
26		eluz2nn 13028	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
27		eluzle 12991	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
28		elfz1b 13740	. . . . . 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 13878	. . . . 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 7438	. . . . 5 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (0..^(♯‘(1^st ‘𝑐))) = (0..^𝑁))
34	33	eleq2d 2850	. . . 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 14846	. 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 1569 ∈ wcel 2146 class class class wbr 5106 ‘cfv 6547 (class class class)co 7430 1^st c1st 8009 2^nd c2nd 8010 0cc0 11213 1c1 11214 + caddc 11216 ≤ cle 11357 − cmin 11554 ℕcn 12348 2c2 12410 ℕ₀cn0 12619 ℤ_≥cuz 12978 ...cfz 13652 ..^cfzo 13801 ♯chash 14487 Word cword 14671 prefix cpfx 14834 Vtxcvtx 29589 Walkscwlks 30192 ClWalkscclwlks 30372

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-er 8731 df-map 8863 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-card 10033 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-nn 12349 df-2 12418 df-n0 12620 df-z 12707 df-uz 12979 df-fz 13653 df-fzo 13802 df-hash 14488 df-word 14672 df-substr 14803 df-pfx 14835 df-wlks 30195 df-clwlks 30373

This theorem is used by: dlwwlknondlwlknonf1o 30981

W3C validator