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Theorem rusgrnumwwlkb0 41174
 Description: Induction base 0 for rusgrnumwwlk 41178. 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 476 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → 𝑃𝑉)
2 0nn0 11184 . . 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 41173 . . 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 2905 . . . . 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 41057 . . . . . . . . 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 2673 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑤 ∈ (0 WWalkSN 𝐺) ↔ 𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1}))
12 rabid 3095 . . . . . . 7 (𝑤 ∈ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1} ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1))
1311, 12syl6bb 275 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑤 ∈ (0 WWalkSN 𝐺) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1)))
1413anbi1d 737 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → ((𝑤 ∈ (0 WWalkSN 𝐺) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)))
1514abbidv 2728 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∣ (𝑤 ∈ (0 WWalkSN 𝐺) ∧ (𝑤‘0) = 𝑃)} = {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)})
16 wrdl1s1 13247 . . . . . . . . 9 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 = ⟨“𝑃”⟩ ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃)))
17 df-3an 1033 . . . . . . . . 9 ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
1816, 17syl6rbb 276 . . . . . . . 8 (𝑃 ∈ (Vtx‘𝐺) → (((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃) ↔ 𝑣 = ⟨“𝑃”⟩))
19 vex 3176 . . . . . . . . 9 𝑣 ∈ V
20 eleq1 2676 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑣 ∈ Word (Vtx‘𝐺)))
21 fveq2 6103 . . . . . . . . . . . 12 (𝑤 = 𝑣 → (#‘𝑤) = (#‘𝑣))
2221eqeq1d 2612 . . . . . . . . . . 11 (𝑤 = 𝑣 → ((#‘𝑤) = 1 ↔ (#‘𝑣) = 1))
2320, 22anbi12d 743 . . . . . . . . . 10 (𝑤 = 𝑣 → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1)))
24 fveq1 6102 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤‘0) = (𝑣‘0))
2524eqeq1d 2612 . . . . . . . . . 10 (𝑤 = 𝑣 → ((𝑤‘0) = 𝑃 ↔ (𝑣‘0) = 𝑃))
2623, 25anbi12d 743 . . . . . . . . 9 (𝑤 = 𝑣 → (((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃)))
2719, 26elab 3319 . . . . . . . 8 (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
28 velsn 4141 . . . . . . . 8 (𝑣 ∈ {⟨“𝑃”⟩} ↔ 𝑣 = ⟨“𝑃”⟩)
2918, 27, 283bitr4g 302 . . . . . . 7 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
3029, 3eleq2s 2706 . . . . . 6 (𝑃𝑉 → (𝑣 ∈ {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} ↔ 𝑣 ∈ {⟨“𝑃”⟩}))
3130eqrdv 2608 . . . . 5 (𝑃𝑉 → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
3231adantl 481 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∣ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 1) ∧ (𝑤‘0) = 𝑃)} = {⟨“𝑃”⟩})
338, 15, 323eqtrd 2648 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → {𝑤 ∈ (0 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} = {⟨“𝑃”⟩})
3433fveq2d 6107 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (#‘{𝑤 ∈ (0 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (#‘{⟨“𝑃”⟩}))
35 s1cl 13235 . . . 4 (𝑃𝑉 → ⟨“𝑃”⟩ ∈ Word 𝑉)
36 hashsng 13020 . . . 4 (⟨“𝑃”⟩ ∈ Word 𝑉 → (#‘{⟨“𝑃”⟩}) = 1)
3735, 36syl 17 . . 3 (𝑃𝑉 → (#‘{⟨“𝑃”⟩}) = 1)
3837adantl 481 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (#‘{⟨“𝑃”⟩}) = 1)
396, 34, 383eqtrd 2648 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  {csn 4125  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816  ℕ0cn0 11169  #chash 12979  Word cword 13146  ⟨“cs1 13149  Vtxcvtx 25673   USPGraph cuspgr 40378   WWalkSN cwwlksn 41029 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-s1 13157  df-wwlks 41033  df-wwlksn 41034 This theorem is referenced by:  rusgrnumwwlk  41178
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