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Theorem rusgrnumwwlkb0 27742
 Description: Induction base 0 for rusgrnumwwlk 27746. 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 487 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → 𝑃𝑉)
2 0nn0 11904 . . 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 27741 . . 3 ((𝑃𝑉 ∧ 0 ∈ ℕ0) → (𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
61, 2, 5sylancl 588 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
7 df-rab 3145 . . . . 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 27631 . . . . . . . . 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 2896 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑤 ∈ (0 WWalksN 𝐺) ↔ 𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1}))
12 rabid 3377 . . . . . . 7 (𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1} ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1))
1311, 12syl6bb 289 . . . . . 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 2883 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∣ (𝑤 ∈ (0 WWalksN 𝐺) ∧ (𝑤‘0) = 𝑃)} = {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)})
16 wrdl1s1 13960 . . . . . . . . 9 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 = ⟨“𝑃”⟩ ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃)))
17 df-3an 1083 . . . . . . . . 9 ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
1816, 17syl6rbb 290 . . . . . . . 8 (𝑃 ∈ (Vtx‘𝐺) → (((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃) ↔ 𝑣 = ⟨“𝑃”⟩))
19 vex 3496 . . . . . . . . 9 𝑣 ∈ V
20 eleq1w 2893 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑣 ∈ Word (Vtx‘𝐺)))
21 fveqeq2 6672 . . . . . . . . . . 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 6662 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤‘0) = (𝑣‘0))
2423eqeq1d 2821 . . . . . . . . . 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 3665 . . . . . . . 8 (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
27 velsn 4575 . . . . . . . 8 (𝑣 ∈ {⟨“𝑃”⟩} ↔ 𝑣 = ⟨“𝑃”⟩)
2818, 26, 273bitr4g 316 . . . . . . 7 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
2928, 3eleq2s 2929 . . . . . 6 (𝑃𝑉 → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
3029eqrdv 2817 . . . . 5 (𝑃𝑉 → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
3130adantl 484 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
328, 15, 313eqtrd 2858 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {⟨“𝑃”⟩})
3332fveq2d 6667 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (♯‘{𝑤 ∈ (0 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (♯‘{⟨“𝑃”⟩}))
34 s1cl 13948 . . . 4 (𝑃𝑉 → ⟨“𝑃”⟩ ∈ Word 𝑉)
35 hashsng 13722 . . . 4 (⟨“𝑃”⟩ ∈ Word 𝑉 → (♯‘{⟨“𝑃”⟩}) = 1)
3634, 35syl 17 . . 3 (𝑃𝑉 → (♯‘{⟨“𝑃”⟩}) = 1)
3736adantl 484 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (♯‘{⟨“𝑃”⟩}) = 1)
386, 33, 373eqtrd 2858 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  {cab 2797  {crab 3140  {csn 4559  ‘cfv 6348  (class class class)co 7148   ∈ cmpo 7150  0cc0 10529  1c1 10530  ℕ0cn0 11889  ♯chash 13682  Word cword 13853  ⟨“cs1 13941  Vtxcvtx 26773  USPGraphcuspgr 26925   WWalksN cwwlksn 27596 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-hash 13683  df-word 13854  df-s1 13942  df-wwlks 27600  df-wwlksn 27601 This theorem is referenced by:  rusgrnumwwlk  27746
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