Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwlknclwlkdifnum Structured version   Visualization version   GIF version

Theorem clwlknclwlkdifnum 26488
 Description: In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
Hypotheses
Ref Expression
clwlknclwlkdif.a 𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
clwlknclwlkdif.b 𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
Assertion
Ref Expression
clwlknclwlkdifnum (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾𝑁) − (#‘𝐵)))
Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑉   𝑤,𝑋
Allowed substitution hints:   𝐴(𝑤)   𝐵(𝑤)   𝐾(𝑤)

Proof of Theorem clwlknclwlkdifnum
StepHypRef Expression
1 clwlknclwlkdif.a . . . 4 𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}
2 clwlknclwlkdif.b . . . 4 𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}
31, 2clwlknclwlkdifs 26487 . . 3 𝐴 = ({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)
43fveq2i 6106 . 2 (#‘𝐴) = (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵))
5 simpr 476 . . . . . 6 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) → 𝑉 ∈ Fin)
65adantr 480 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝑉 ∈ Fin)
7 wwlknfi 26266 . . . . 5 (𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)
8 rabfi 8070 . . . . 5 (((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
96, 7, 83syl 18 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∈ Fin)
10 simpr 476 . . . . . . 7 ((( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
1110a1i 11 . . . . . 6 (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋))
1211ss2rabi 3647 . . . . 5 {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)} ⊆ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}
132, 12eqsstri 3598 . . . 4 𝐵 ⊆ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}
14 hashssdif 13061 . . . 4 (({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∈ Fin ∧ 𝐵 ⊆ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) → (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
159, 13, 14sylancl 693 . . 3 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)))
16 simpl 472 . . . . . 6 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) → ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)
1716adantr 480 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾)
18 simpl 472 . . . . . 6 ((𝑋𝑉𝑁 ∈ ℕ) → 𝑋𝑉)
1918adantl 481 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝑋𝑉)
20 nnnn0 11176 . . . . . . 7 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2120adantl 481 . . . . . 6 ((𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
2221adantl 481 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → 𝑁 ∈ ℕ0)
23 rusgranumwwlkg 26486 . . . . 5 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) = (𝐾𝑁))
2417, 6, 19, 22, 23syl13anc 1320 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) = (𝐾𝑁))
2524oveq1d 6564 . . 3 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋}) − (#‘𝐵)) = ((𝐾𝑁) − (#‘𝐵)))
2615, 25eqtrd 2644 . 2 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)) = ((𝐾𝑁) − (#‘𝐵)))
274, 26syl5eq 2656 1 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾𝑁) − (#‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ↑cexp 12722  #chash 12979   lastS clsw 13147   WWalksN cwwlkn 26206   RegUSGrph crusgra 26450 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-inf2 8421  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  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-rmo 2904  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-disj 4554  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-se 4998  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-isom 5813  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-2o 7448  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-sup 8231  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-usgra 25862  df-nbgra 25949  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  numclwwlkqhash  26627
 Copyright terms: Public domain W3C validator