Theorem rusgrnumwwlkb0 29802

Description: Induction base 0 for rusgrnumwwlk 29806. 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 483	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → 𝑃 ∈ 𝑉)
2		0nn0 12525	. . 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 29801	. . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 0 ∈ ℕ0) → (𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
6	1, 2, 5	sylancl 584	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
7		df-rab 3431	. . . . 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 29692	. . . . . . . . 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 2815	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ 𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1}))
12		rabid 3451	. . . . . . 7 ⊢ (𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1} ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1))
13	11, 12	bitrdi 286	. . . . . 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 2797	. . . 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 14604	. . . . . . . . 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 287	. . . . . . . 8 ⊢ (𝑃 ∈ (Vtx‘𝐺) → (((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃) ↔ 𝑣 = ⟨“𝑃”⟩))
19		vex 3477	. . . . . . . . 9 ⊢ 𝑣 ∈ V
20		eleq1w 2812	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑣 ∈ Word (Vtx‘𝐺)))
21		fveqeq2 6911	. . . . . . . . . . 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 6901	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → (𝑤‘0) = (𝑣‘0))
24	23	eqeq1d 2730	. . . . . . . . . 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 3669	. . . . . . . 8 ⊢ (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
27		velsn 4648	. . . . . . . 8 ⊢ (𝑣 ∈ {⟨“𝑃”⟩} ↔ 𝑣 = ⟨“𝑃”⟩)
28	18, 26, 27	3bitr4g 313	. . . . . . 7 ⊢ (𝑃 ∈ (Vtx‘𝐺) → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
29	28, 3	eleq2s 2847	. . . . . 6 ⊢ (𝑃 ∈ 𝑉 → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
30	29	eqrdv 2726	. . . . 5 ⊢ (𝑃 ∈ 𝑉 → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
31	30	adantl 480	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
32	8, 15, 31	3eqtrd 2772	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {⟨“𝑃”⟩})
33	32	fveq2d 6906	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (♯‘{⟨“𝑃”⟩}))
34		s1cl 14592	. . . 4 ⊢ (𝑃 ∈ 𝑉 → ⟨“𝑃”⟩ ∈ Word 𝑉)
35		hashsng 14368	. . . 4 ⊢ (⟨“𝑃”⟩ ∈ Word 𝑉 → (♯‘{⟨“𝑃”⟩}) = 1)
36	34, 35	syl 17	. . 3 ⊢ (𝑃 ∈ 𝑉 → (♯‘{⟨“𝑃”⟩}) = 1)
37	36	adantl 480	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (♯‘{⟨“𝑃”⟩}) = 1)
38	6, 33, 37	3eqtrd 2772	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2705  {crab 3430  {csn 4632  ‘cfv 6553  (class class class)co 7426   ∈ cmpo 7428  0cc0 11146  1c1 11147  ℕ₀cn0 12510  ♯chash 14329  Word cword 14504  ⟨“cs1 14585  Vtxcvtx 28829  USPGraphcuspgr 28981   WWalksN cwwlksn 29657

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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-n0 12511  df-xnn0 12583  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-hash 14330  df-word 14505  df-s1 14586  df-wwlks 29661  df-wwlksn 29662

This theorem is referenced by:  rusgrnumwwlk  29806