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Theorem rusgrnumwwlkb0 26750
 Description: Induction base 0 for rusgrnumwwlk 26754. 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 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
Assertion
Ref Expression
rusgrnumwwlkb0 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (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
StepHypRef Expression
1 simpr 477 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → 𝑃𝑉)
2 0nn0 11259 . . 3 0 ∈ ℕ0
3 rusgrnumwwlk.v . . . 4 𝑉 = (Vtx‘𝐺)
4 rusgrnumwwlk.l . . . 4 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
53, 4rusgrnumwwlklem 26749 . . 3 ((𝑃𝑉 ∧ 0 ∈ ℕ0) → (𝑃𝐿0) = (#‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
61, 2, 5sylancl 693 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = (#‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
7 df-rab 2916 . . . . 5 {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)}
87a1i 11 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)})
9 wwlksn0s 26632 . . . . . . . . 9 (0 WWalksN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1}
109a1i 11 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (0 WWalksN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1})
1110eleq2d 2684 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ 𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1}))
12 rabid 3109 . . . . . . 7 (𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1} ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1))
1311, 12syl6bb 276 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1)))
1413anbi1d 740 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → ((𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)))
1514abbidv 2738 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)} = {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)})
16 wrdl1s1 13341 . . . . . . . . 9 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 = ⟨“𝑃”⟩ ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃)))
17 df-3an 1038 . . . . . . . . 9 ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
1816, 17syl6rbb 277 . . . . . . . 8 (𝑃 ∈ (Vtx‘𝐺) → (((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃) ↔ 𝑣 = ⟨“𝑃”⟩))
19 vex 3192 . . . . . . . . 9 𝑣 ∈ V
20 eleq1 2686 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑣 ∈ Word (Vtx‘𝐺)))
21 fveq2 6153 . . . . . . . . . . . 12 (𝑤 = 𝑣 → (#‘𝑤) = (#‘𝑣))
2221eqeq1d 2623 . . . . . . . . . . 11 (𝑤 = 𝑣 → ((#‘𝑤) = 1 ↔ (#‘𝑣) = 1))
2320, 22anbi12d 746 . . . . . . . . . 10 (𝑤 = 𝑣 → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1)))
24 fveq1 6152 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤‘0) = (𝑣‘0))
2524eqeq1d 2623 . . . . . . . . . 10 (𝑤 = 𝑣 → ((𝑤‘0) = 𝑃 ↔ (𝑣‘0) = 𝑃))
2623, 25anbi12d 746 . . . . . . . . 9 (𝑤 = 𝑣 → (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃)))
2719, 26elab 3337 . . . . . . . 8 (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
28 velsn 4169 . . . . . . . 8 (𝑣 ∈ {⟨“𝑃”⟩} ↔ 𝑣 = ⟨“𝑃”⟩)
2918, 27, 283bitr4g 303 . . . . . . 7 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
3029, 3eleq2s 2716 . . . . . 6 (𝑃𝑉 → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
3130eqrdv 2619 . . . . 5 (𝑃𝑉 → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
3231adantl 482 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
338, 15, 323eqtrd 2659 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {⟨“𝑃”⟩})
3433fveq2d 6157 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (#‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (#‘{⟨“𝑃”⟩}))
35 s1cl 13329 . . . 4 (𝑃𝑉 → ⟨“𝑃”⟩ ∈ Word 𝑉)
36 hashsng 13107 . . . 4 (⟨“𝑃”⟩ ∈ Word 𝑉 → (#‘{⟨“𝑃”⟩}) = 1)
3735, 36syl 17 . . 3 (𝑃𝑉 → (#‘{⟨“𝑃”⟩}) = 1)
3837adantl 482 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (#‘{⟨“𝑃”⟩}) = 1)
396, 34, 383eqtrd 2659 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  {cab 2607  {crab 2911  {csn 4153  ‘cfv 5852  (class class class)co 6610   ↦ cmpt2 6612  0cc0 9888  1c1 9889  ℕ0cn0 11244  #chash 13065  Word cword 13238  ⟨“cs1 13241  Vtxcvtx 25791   USPGraph cuspgr 25953   WWalksN cwwlksn 26604 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-xnn0 11316  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-hash 13066  df-word 13246  df-s1 13249  df-wwlks 26608  df-wwlksn 26609 This theorem is referenced by:  rusgrnumwwlk  26754
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