Theorem wwlksnextfun 27679

Description: Lemma for wwlksnextbij 27683. (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 6682	. . . . . 6 ⊢ (𝑤 = 𝑡 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑡) = (𝑁 + 2)))
2		oveq1 7166	. . . . . . 7 ⊢ (𝑤 = 𝑡 → (𝑤 prefix (𝑁 + 1)) = (𝑡 prefix (𝑁 + 1)))
3	2	eqeq1d 2826	. . . . . 6 ⊢ (𝑤 = 𝑡 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑡 prefix (𝑁 + 1)) = 𝑊))
4		fveq2 6673	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → (lastS‘𝑤) = (lastS‘𝑡))
5	4	preq2d 4679	. . . . . . 7 ⊢ (𝑤 = 𝑡 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑡)})
6	5	eleq1d 2900	. . . . . 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 3686	. . . 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 11909	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
12		2re 11714	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
13	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
14		nn0ge0 11925	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
15		2pos 11743	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
16	15	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 0 < 2)
17	11, 13, 14, 16	addgegt0d 11216	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
18	17	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 0 < (𝑁 + 2))
19		breq2 5073	. . . . . . . . . . . . . . 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 13729	. . . . . . . . . . . . 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 13923	. . . . . 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 4673	. . . . 5 ⊢ (𝑛 = (lastS‘𝑡) → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), (lastS‘𝑡)})
36	35	eleq1d 2900	. . . 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 3686	. . 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 6881	1 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  {crab 3145  ∅c0 4294  {cpr 4572   class class class wbr 5069   ↦ cmpt 5149  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  ℝcr 10539  0cc0 10540  1c1 10541   + caddc 10543   < clt 10678  2c2 11695  ℕ₀cn0 11900  ♯chash 13693  Word cword 13864  lastSclsw 13917   prefix cpfx 14035  Vtxcvtx 26784  Edgcedg 26835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-fz 12896  df-fzo 13037  df-hash 13694  df-word 13865  df-lsw 13918

This theorem is referenced by:  wwlksnextinj  27680  wwlksnextsurj  27681