Theorem wwlksnextfun 29984

Description: Lemma for wwlksnextbij 29988. (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 6844	. . . . . 6 ⊢ (𝑤 = 𝑡 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑡) = (𝑁 + 2)))
2		oveq1 7368	. . . . . . 7 ⊢ (𝑤 = 𝑡 → (𝑤 prefix (𝑁 + 1)) = (𝑡 prefix (𝑁 + 1)))
3	2	eqeq1d 2739	. . . . . 6 ⊢ (𝑤 = 𝑡 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑡 prefix (𝑁 + 1)) = 𝑊))
4		fveq2 6835	. . . . . . . 8 ⊢ (𝑤 = 𝑡 → (lastS‘𝑤) = (lastS‘𝑡))
5	4	preq2d 4685	. . . . . . 7 ⊢ (𝑤 = 𝑡 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑡)})
6	5	eleq1d 2822	. . . . . 6 ⊢ (𝑤 = 𝑡 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
7	1, 3, 6	3anbi123d 1439	. . . . 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 3638	. . . 4 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)))
10		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 𝑡 ∈ Word 𝑉)
11		nn0re 12440	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
12		2re 12249	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
13	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
14		nn0ge0 12456	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
15		2pos 12278	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
16	15	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 0 < 2)
17	11, 13, 14, 16	addgegt0d 11717	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
18	17	ad2antlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 0 < (𝑁 + 2))
19		breq2 5090	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑡) = (𝑁 + 2) → (0 < (♯‘𝑡) ↔ 0 < (𝑁 + 2)))
20	19	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → (0 < (♯‘𝑡) ↔ 0 < (𝑁 + 2)))
21	18, 20	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 0 < (♯‘𝑡))
22		hashgt0n0 14321	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 0 < (♯‘𝑡)) → 𝑡 ≠ ∅)
23	10, 21, 22	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 𝑡 ≠ ∅)
24	10, 23	jca 511	. . . . . . . . . . 11 ⊢ (((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅))
25	24	expcom 413	. . . . . . . . . 10 ⊢ ((♯‘𝑡) = (𝑁 + 2) → ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉 ∧ 𝑡 ≠ ∅)))
26	25	3ad2ant1 1134	. . . . . . . . 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 415	. . . . . . . 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 407	. . . . . . 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 407	. . . . . 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 14524	. . . . . 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 1225	. . . . 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 511	. . . 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 4679	. . . . 5 ⊢ (𝑛 = (lastS‘𝑡) → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), (lastS‘𝑡)})
36	35	eleq1d 2822	. . . 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 3638	. . 3 ⊢ ((lastS‘𝑡) ∈ 𝑅 ↔ ((lastS‘𝑡) ∈ 𝑉 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
39	34, 38	sylibr 234	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑡 ∈ 𝐷) → (lastS‘𝑡) ∈ 𝑅)
40		wwlksnextbij0.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))
41	39, 40	fmptd 7061	1 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3390  ∅c0 4274  {cpr 4570   class class class wbr 5086   ↦ cmpt 5167  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  ℝcr 11031  0cc0 11032  1c1 11033   + caddc 11035   < clt 11173  2c2 12230  ℕ₀cn0 12431  ♯chash 14286  Word cword 14469  lastSclsw 14518   prefix cpfx 14627  Vtxcvtx 29082  Edgcedg 29133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-n0 12432  df-xnn0 12505  df-z 12519  df-uz 12783  df-fz 13456  df-fzo 13603  df-hash 14287  df-word 14470  df-lsw 14519

This theorem is referenced by:  wwlksnextinj  29985  wwlksnextsurj  29986