Theorem wwlksnextfun 27675

Description: Lemma for wwlksnextbij 27679. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)

Hypotheses

Ref	Expression
wwlksnextbij0.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e	⊢ 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
wwlksnextbij0.r	⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij0.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		fveqeq2 6678	. . . . . 6 ⊢ (𝑤 = 𝑡 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑡) = (𝑁 + 2)))
2		oveq1 7162	. . . . . . 7 ⊢ (𝑤 = 𝑡 → (𝑤 prefix (𝑁 + 1)) = (𝑡 prefix (𝑁 + 1)))
3	2	eqeq1d 2823	. . . . . 6 ⊢ (𝑤 = 𝑡 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑡 prefix (𝑁 + 1)) = 𝑊))
4		fveq2 6669	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → (lastS‘𝑤) = (lastS‘𝑡))
5	4	preq2d 4675	. . . . . . 7 ⊢ (𝑤 = 𝑡 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑡)})
6	5	eleq1d 2897	. . . . . 6 ⊢ (𝑤 = 𝑡 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
7	1, 3, 6	3anbi123d 1432	. . . . 5 ⊢ (𝑤 = 𝑡 → (((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)))
8		wwlksnextbij0.d	. . . . 5 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
9	7, 8	elrab2 3682	. . . 4 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)))
10		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 𝑡 ∈ Word 𝑉)
11		nn0re 11905	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
12		2re 11710	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
13	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
14		nn0ge0 11921	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
15		2pos 11739	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
16	15	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 0 < 2)
17	11, 13, 14, 16	addgegt0d 11212	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
18	17	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 0 < (𝑁 + 2))
19		breq2 5069	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑡) = (𝑁 + 2) → (0 < (♯‘𝑡) ↔ 0 < (𝑁 + 2)))
20	19	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → (0 < (♯‘𝑡) ↔ 0 < (𝑁 + 2)))
21	18, 20	mpbird 259	. . . . . . . . . . . . 13 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 0 < (♯‘𝑡))
22		hashgt0n0 13725	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 0 < (♯‘𝑡)) → 𝑡 ≠ ∅)
23	10, 21, 22	syl2anc 586	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 𝑡 ≠ ∅)
24	10, 23	jca 514	. . . . . . . . . . 11 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅))
25	24	expcom 416	. . . . . . . . . 10 ⊢ ((♯‘𝑡) = (𝑁 + 2) → ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅)))
26	25	3ad2ant1 1129	. . . . . . . . 9 ⊢ (((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸) → ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅)))
27	26	expd 418	. . . . . . . 8 ⊢ (((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸) → (𝑡 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅))))
28	27	impcom 410	. . . . . . 7 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅)))
29	28	impcom 410	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅))
30		lswcl 13919	. . . . . 6 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅) → (lastS‘𝑡) ∈ 𝑉)
31	29, 30	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))) → (lastS‘𝑡) ∈ 𝑉)
32		simprr3 1219	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))) → {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)
33	31, 32	jca 514	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))) → ((lastS‘𝑡) ∈ 𝑉 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
34	9, 33	sylan2b 595	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑡 ∈ 𝐷) → ((lastS‘𝑡) ∈ 𝑉 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
35		preq2 4669	. . . . 5 ⊢ (𝑛 = (lastS‘𝑡) → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), (lastS‘𝑡)})
36	35	eleq1d 2897	. . . 4 ⊢ (𝑛 = (lastS‘𝑡) → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
37		wwlksnextbij0.r	. . . 4 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
38	36, 37	elrab2 3682	. . 3 ⊢ ((lastS‘𝑡) ∈ 𝑅 ↔ ((lastS‘𝑡) ∈ 𝑉 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
39	34, 38	sylibr 236	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑡 ∈ 𝐷) → (lastS‘𝑡) ∈ 𝑅)
40		wwlksnextbij0.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))
41	39, 40	fmptd 6877	1 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  {crab 3142  ∅c0 4290  {cpr 4568   class class class wbr 5065   ↦ cmpt 5145  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  ℝcr 10535  0cc0 10536  1c1 10537   + caddc 10539   < clt 10674  2c2 11691  ℕ₀cn0 11896  ♯chash 13689  Word cword 13860  lastSclsw 13913   prefix cpfx 14031  Vtxcvtx 26780  Edgcedg 26831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-n0 11897  df-xnn0 11967  df-z 11981  df-uz 12243  df-fz 12892  df-fzo 13033  df-hash 13690  df-word 13861  df-lsw 13914

This theorem is referenced by:  wwlksnextinj  27676  wwlksnextsurj  27677