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Theorem wwlksnextproplem1 29163

Description: Lemma 1 for wwlksnextprop 29166. (Contributed by Alexander van der Vekens, 31-Jul-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)

**Hypothesis**
Ref	Expression
wwlksnextprop.x	⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)

**Assertion**
Ref	Expression
wwlksnextproplem1	⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑊‘0))

**Proof of Theorem wwlksnextproplem1**
Step	Hyp	Ref	Expression
1		wwlknbp1 29098	. . . . 5 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)))
2		simpl2 1193	. . . . . . 7 ⊢ ((((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑊 ∈ Word (Vtx‘𝐺))
3		peano2nn0 12512	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ₀)
4	3	3ad2ant1 1134	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑁 + 1) + 1) ∈ ℕ₀)
5		eleq1 2822	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((♯‘𝑊) ∈ ℕ₀ ↔ ((𝑁 + 1) + 1) ∈ ℕ₀))
6	5	3ad2ant3 1136	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((♯‘𝑊) ∈ ℕ₀ ↔ ((𝑁 + 1) + 1) ∈ ℕ₀))
7	4, 6	mpbird 257	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (♯‘𝑊) ∈ ℕ₀)
8	7	adantr 482	. . . . . . . 8 ⊢ ((((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (♯‘𝑊) ∈ ℕ₀)
9		simpr 486	. . . . . . . . 9 ⊢ ((((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
10		nn0re 12481	. . . . . . . . . . . . 13 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
11	10	lep1d 12145	. . . . . . . . . . . 12 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
12	11	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ (((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
13		breq2 5153	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ≤ (♯‘𝑊) ↔ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))
14	13	3ad2ant3 1136	. . . . . . . . . . 11 ⊢ (((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑁 + 1) ≤ (♯‘𝑊) ↔ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))
15	12, 14	mpbird 257	. . . . . . . . . 10 ⊢ (((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 + 1) ≤ (♯‘𝑊))
16	15	adantr 482	. . . . . . . . 9 ⊢ ((((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ≤ (♯‘𝑊))
17		nn0p1elfzo 13675	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ₀ ∧ (𝑁 + 1) ≤ (♯‘𝑊)) → 𝑁 ∈ (0..^(♯‘𝑊)))
18	9, 8, 16, 17	syl3anc 1372	. . . . . . . 8 ⊢ ((((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (0..^(♯‘𝑊)))
19		fz0add1fz1 13702	. . . . . . . 8 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑁 + 1) ∈ (1...(♯‘𝑊)))
20	8, 18, 19	syl2anc 585	. . . . . . 7 ⊢ ((((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ (1...(♯‘𝑊)))
21	2, 20	jca 513	. . . . . 6 ⊢ ((((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
22	21	ex 414	. . . . 5 ⊢ (((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊)))))
23	1, 22	syl 17	. . . 4 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊)))))
24		wwlksnextprop.x	. . . 4 ⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
25	23, 24	eleq2s 2852	. . 3 ⊢ (𝑊 ∈ 𝑋 → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊)))))
26	25	imp 408	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
27		pfxfv0 14642	. 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑊‘0))
28	26, 27	syl 17	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑊‘0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 ≤ cle 11249 ℕ₀cn0 12472 ...cfz 13484 ..^cfzo 13627 ♯chash 14290 Word cword 14464 prefix cpfx 14620 Vtxcvtx 28256 WWalksN cwwlksn 29080

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-substr 14591 df-pfx 14621 df-wwlks 29084 df-wwlksn 29085

This theorem is referenced by: wwlksnextproplem3 29165 wwlksnextprop 29166

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