Theorem wwlksnextfun 26793

Description: Lemma for wwlksnextbij 26797. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)

Hypotheses

Ref	Expression
wwlksnextbij0.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e	⊢ 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
wwlksnextbij.r	⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ ( lastS ‘𝑡))

Assertion

Ref	Expression
wwlksnextfun	⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑊   𝑡,𝐷   𝑛,𝐸   𝑤,𝐸   𝑡,𝑁,𝑤   𝑡,𝑅   𝑛,𝑉   𝑤,𝑉   𝑛,𝑊   𝑡,𝑛

Allowed substitution hints:   𝐷(𝑤,𝑛)   𝑅(𝑤,𝑛)   𝐸(𝑡)   𝐹(𝑤,𝑡,𝑛)   𝐺(𝑡,𝑛)   𝑁(𝑛)   𝑉(𝑡)   𝑊(𝑡)

Proof of Theorem wwlksnextfun

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 ⊢ (𝑤 = 𝑡 → (#‘𝑤) = (#‘𝑡))
2	1	eqeq1d 2624	. . . . . 6 ⊢ (𝑤 = 𝑡 → ((#‘𝑤) = (𝑁 + 2) ↔ (#‘𝑡) = (𝑁 + 2)))
3		oveq1 6657	. . . . . . 7 ⊢ (𝑤 = 𝑡 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑡 substr ⟨0, (𝑁 + 1)⟩))
4	3	eqeq1d 2624	. . . . . 6 ⊢ (𝑤 = 𝑡 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
5		fveq2 6191	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → ( lastS ‘𝑤) = ( lastS ‘𝑡))
6	5	preq2d 4275	. . . . . . 7 ⊢ (𝑤 = 𝑡 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑡)})
7	6	eleq1d 2686	. . . . . 6 ⊢ (𝑤 = 𝑡 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
8	2, 4, 7	3anbi123d 1399	. . . . 5 ⊢ (𝑤 = 𝑡 → (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) ↔ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)))
9		wwlksnextbij0.d	. . . . 5 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
10	8, 9	elrab2 3366	. . . 4 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)))
11		simpll 790	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 𝑡 ∈ Word 𝑉)
12		nn0re 11301	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
13		2re 11090	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
14	13	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
15		nn0ge0 11318	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
16		2pos 11112	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
17	16	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 0 < 2)
18	12, 14, 15, 17	addgegt0d 10601	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
19	18	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 0 < (𝑁 + 2))
20		breq2 4657	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑡) = (𝑁 + 2) → (0 < (#‘𝑡) ↔ 0 < (𝑁 + 2)))
21	20	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → (0 < (#‘𝑡) ↔ 0 < (𝑁 + 2)))
22	19, 21	mpbird 247	. . . . . . . . . . . . 13 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 0 < (#‘𝑡))
23		hashgt0n0 13156	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 0 < (#‘𝑡)) → 𝑡 ≠ ∅)
24	11, 22, 23	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 𝑡 ≠ ∅)
25	11, 24	jca 554	. . . . . . . . . . 11 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅))
26	25	expcom 451	. . . . . . . . . 10 ⊢ ((#‘𝑡) = (𝑁 + 2) → ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅)))
27	26	3ad2ant1 1082	. . . . . . . . 9 ⊢ (((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸) → ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅)))
28	27	expd 452	. . . . . . . 8 ⊢ (((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸) → (𝑡 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅))))
29	28	impcom 446	. . . . . . 7 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅)))
30	29	impcom 446	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅))
31		lswcl 13355	. . . . . 6 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅) → ( lastS ‘𝑡) ∈ 𝑉)
32	30, 31	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))) → ( lastS ‘𝑡) ∈ 𝑉)
33		simprr3 1111	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))) → {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)
34	32, 33	jca 554	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))) → (( lastS ‘𝑡) ∈ 𝑉 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
35	10, 34	sylan2b 492	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑡 ∈ 𝐷) → (( lastS ‘𝑡) ∈ 𝑉 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
36		preq2 4269	. . . . 5 ⊢ (𝑛 = ( lastS ‘𝑡) → {( lastS ‘𝑊), 𝑛} = {( lastS ‘𝑊), ( lastS ‘𝑡)})
37	36	eleq1d 2686	. . . 4 ⊢ (𝑛 = ( lastS ‘𝑡) → ({( lastS ‘𝑊), 𝑛} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
38		wwlksnextbij.r	. . . 4 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}
39	37, 38	elrab2 3366	. . 3 ⊢ (( lastS ‘𝑡) ∈ 𝑅 ↔ (( lastS ‘𝑡) ∈ 𝑉 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
40	35, 39	sylibr 224	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑡 ∈ 𝐷) → ( lastS ‘𝑡) ∈ 𝑅)
41		wwlksnextbij.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ ( lastS ‘𝑡))
42	40, 41	fmptd 6385	1 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {crab 2916  ∅c0 3915  {cpr 4179  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074  2c2 11070  ℕ₀cn0 11292  #chash 13117  Word cword 13291   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300

This theorem is referenced by:  wwlksnextinj  26794  wwlksnextsur  26795