Theorem rusgrnumwwlkb0 29734

Description: Induction base 0 for rusgrnumwwlk 29738. Here, we do not need the regularity of the graph yet. (Contributed by Alexander van der Vekens, 24-Jul-2018.) (Revised by AV, 7-May-2021.)

Hypotheses

Ref	Expression
rusgrnumwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
rusgrnumwwlk.l	⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))

Assertion

Ref	Expression
rusgrnumwwlkb0	⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)

Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑃,𝑛,𝑣,𝑤   𝑛,𝑉,𝑣,𝑤

Allowed substitution hints:   𝐿(𝑤,𝑣,𝑛)

Proof of Theorem rusgrnumwwlkb0

Step	Hyp	Ref	Expression
1		simpr 484	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → 𝑃 ∈ 𝑉)
2		0nn0 12491	. . 3 ⊢ 0 ∈ ℕ0
3		rusgrnumwwlk.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4		rusgrnumwwlk.l	. . . 4 ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
5	3, 4	rusgrnumwwlklem 29733	. . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 0 ∈ ℕ0) → (𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
6	1, 2, 5	sylancl 585	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
7		df-rab 3427	. . . . 5 ⊢ {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)}
8	7	a1i 11	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)})
9		wwlksn0s 29624	. . . . . . . . 9 ⊢ (0 WWalksN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1}
10	9	a1i 11	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (0 WWalksN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1})
11	10	eleq2d 2813	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ 𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1}))
12		rabid 3446	. . . . . . 7 ⊢ (𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1} ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1))
13	11, 12	bitrdi 287	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1)))
14	13	anbi1d 629	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → ((𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)))
15	14	abbidv 2795	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)} = {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)})
16		wrdl1s1 14570	. . . . . . . . 9 ⊢ (𝑃 ∈ (Vtx‘𝐺) → (𝑣 = ⟨“𝑃”⟩ ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃)))
17		df-3an 1086	. . . . . . . . 9 ⊢ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
18	16, 17	bitr2di 288	. . . . . . . 8 ⊢ (𝑃 ∈ (Vtx‘𝐺) → (((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃) ↔ 𝑣 = ⟨“𝑃”⟩))
19		vex 3472	. . . . . . . . 9 ⊢ 𝑣 ∈ V
20		eleq1w 2810	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑣 ∈ Word (Vtx‘𝐺)))
21		fveqeq2 6894	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → ((♯‘𝑤) = 1 ↔ (♯‘𝑣) = 1))
22	20, 21	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑤 = 𝑣 → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1)))
23		fveq1 6884	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → (𝑤‘0) = (𝑣‘0))
24	23	eqeq1d 2728	. . . . . . . . . 10 ⊢ (𝑤 = 𝑣 → ((𝑤‘0) = 𝑃 ↔ (𝑣‘0) = 𝑃))
25	22, 24	anbi12d 630	. . . . . . . . 9 ⊢ (𝑤 = 𝑣 → (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃)))
26	19, 25	elab 3663	. . . . . . . 8 ⊢ (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
27		velsn 4639	. . . . . . . 8 ⊢ (𝑣 ∈ {⟨“𝑃”⟩} ↔ 𝑣 = ⟨“𝑃”⟩)
28	18, 26, 27	3bitr4g 314	. . . . . . 7 ⊢ (𝑃 ∈ (Vtx‘𝐺) → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
29	28, 3	eleq2s 2845	. . . . . 6 ⊢ (𝑃 ∈ 𝑉 → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
30	29	eqrdv 2724	. . . . 5 ⊢ (𝑃 ∈ 𝑉 → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
31	30	adantl 481	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
32	8, 15, 31	3eqtrd 2770	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {⟨“𝑃”⟩})
33	32	fveq2d 6889	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (♯‘{⟨“𝑃”⟩}))
34		s1cl 14558	. . . 4 ⊢ (𝑃 ∈ 𝑉 → ⟨“𝑃”⟩ ∈ Word 𝑉)
35		hashsng 14334	. . . 4 ⊢ (⟨“𝑃”⟩ ∈ Word 𝑉 → (♯‘{⟨“𝑃”⟩}) = 1)
36	34, 35	syl 17	. . 3 ⊢ (𝑃 ∈ 𝑉 → (♯‘{⟨“𝑃”⟩}) = 1)
37	36	adantl 481	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (♯‘{⟨“𝑃”⟩}) = 1)
38	6, 33, 37	3eqtrd 2770	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  {crab 3426  {csn 4623  ‘cfv 6537  (class class class)co 7405   ∈ cmpo 7407  0cc0 11112  1c1 11113  ℕ₀cn0 12476  ♯chash 14295  Word cword 14470  ⟨“cs1 14551  Vtxcvtx 28764  USPGraphcuspgr 28916   WWalksN cwwlksn 29589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-s1 14552  df-wwlks 29593  df-wwlksn 29594

This theorem is referenced by:  rusgrnumwwlk  29738