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Theorem rusgrnumwwlkb0 30001
Description: Induction base 0 for rusgrnumwwlk 30005. 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 484 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → 𝑃𝑉)
2 0nn0 12539 . . 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 30000 . . 3 ((𝑃𝑉 ∧ 0 ∈ ℕ0) → (𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
61, 2, 5sylancl 586 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
7 df-rab 3434 . . . . 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 29891 . . . . . . . . 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 2825 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ 𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1}))
12 rabid 3455 . . . . . . 7 (𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1} ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1))
1311, 12bitrdi 287 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1)))
1413anbi1d 631 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → ((𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)))
1514abbidv 2806 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)} = {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)})
16 wrdl1s1 14649 . . . . . . . . 9 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 = ⟨“𝑃”⟩ ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃)))
17 df-3an 1088 . . . . . . . . 9 ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
1816, 17bitr2di 288 . . . . . . . 8 (𝑃 ∈ (Vtx‘𝐺) → (((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃) ↔ 𝑣 = ⟨“𝑃”⟩))
19 vex 3482 . . . . . . . . 9 𝑣 ∈ V
20 eleq1w 2822 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑣 ∈ Word (Vtx‘𝐺)))
21 fveqeq2 6916 . . . . . . . . . . 11 (𝑤 = 𝑣 → ((♯‘𝑤) = 1 ↔ (♯‘𝑣) = 1))
2220, 21anbi12d 632 . . . . . . . . . 10 (𝑤 = 𝑣 → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1)))
23 fveq1 6906 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤‘0) = (𝑣‘0))
2423eqeq1d 2737 . . . . . . . . . 10 (𝑤 = 𝑣 → ((𝑤‘0) = 𝑃 ↔ (𝑣‘0) = 𝑃))
2522, 24anbi12d 632 . . . . . . . . 9 (𝑤 = 𝑣 → (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃)))
2619, 25elab 3681 . . . . . . . 8 (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
27 velsn 4647 . . . . . . . 8 (𝑣 ∈ {⟨“𝑃”⟩} ↔ 𝑣 = ⟨“𝑃”⟩)
2818, 26, 273bitr4g 314 . . . . . . 7 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
2928, 3eleq2s 2857 . . . . . 6 (𝑃𝑉 → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
3029eqrdv 2733 . . . . 5 (𝑃𝑉 → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
3130adantl 481 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
328, 15, 313eqtrd 2779 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {⟨“𝑃”⟩})
3332fveq2d 6911 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (♯‘{⟨“𝑃”⟩}))
34 s1cl 14637 . . . 4 (𝑃𝑉 → ⟨“𝑃”⟩ ∈ Word 𝑉)
35 hashsng 14405 . . . 4 (⟨“𝑃”⟩ ∈ Word 𝑉 → (♯‘{⟨“𝑃”⟩}) = 1)
3634, 35syl 17 . . 3 (𝑃𝑉 → (♯‘{⟨“𝑃”⟩}) = 1)
3736adantl 481 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (♯‘{⟨“𝑃”⟩}) = 1)
386, 33, 373eqtrd 2779 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  {csn 4631  cfv 6563  (class class class)co 7431  cmpo 7433  0cc0 11153  1c1 11154  0cn0 12524  chash 14366  Word cword 14549  ⟨“cs1 14630  Vtxcvtx 29028  USPGraphcuspgr 29180   WWalksN cwwlksn 29856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-s1 14631  df-wwlks 29860  df-wwlksn 29861
This theorem is referenced by:  rusgrnumwwlk  30005
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