MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numclwwlk2lem3 Structured version   Visualization version   GIF version

Theorem numclwwlk2lem3 26633
Description: In a friendship graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex equals the size of the set of all closed walks of length (n+2) starting with this vertex and not having this vertex as last but 2 vertex. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Proof shortened by AV, 5-May-2021.)
Hypotheses
Ref Expression
numclwwlk.c 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
numclwwlk.g 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
numclwwlk.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
numclwwlk.h 𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
Assertion
Ref Expression
numclwwlk2lem3 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (#‘(𝑋𝑄𝑁)) = (#‘(𝑋𝐻(𝑁 + 2))))
Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑤,𝐶   𝑤,𝑁   𝐶,𝑛,𝑣,𝑤   𝑣,𝑁   𝑛,𝑋,𝑣,𝑤   𝑣,𝑉   𝑤,𝐸   𝑤,𝑉   𝑤,𝐹   𝑤,𝑄   𝑤,𝐺   𝑣,𝐸   𝑣,𝐻,𝑤
Allowed substitution hints:   𝑄(𝑣,𝑛)   𝐹(𝑣,𝑛)   𝐺(𝑣,𝑛)   𝐻(𝑛)

Proof of Theorem numclwwlk2lem3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ovex 6577 . . . 4 (𝑋𝐻(𝑁 + 2)) ∈ V
21a1i 11 . . 3 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐻(𝑁 + 2)) ∈ V)
3 numclwwlk.c . . . 4 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
4 numclwwlk.f . . . 4 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
5 numclwwlk.g . . . 4 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
6 numclwwlk.q . . . 4 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
7 numclwwlk.h . . . 4 𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
8 eqid 2610 . . . 4 ( ∈ (𝑋𝐻(𝑁 + 2)) ↦ ( substr ⟨0, (𝑁 + 1)⟩)) = ( ∈ (𝑋𝐻(𝑁 + 2)) ↦ ( substr ⟨0, (𝑁 + 1)⟩))
93, 4, 5, 6, 7, 8numclwlk2lem2f1o 26632 . . 3 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → ( ∈ (𝑋𝐻(𝑁 + 2)) ↦ ( substr ⟨0, (𝑁 + 1)⟩)):(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁))
102, 9hasheqf1od 13006 . 2 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (#‘(𝑋𝐻(𝑁 + 2))) = (#‘(𝑋𝑄𝑁)))
1110eqcomd 2616 1 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (#‘(𝑋𝑄𝑁)) = (#‘(𝑋𝐻(𝑁 + 2))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  {crab 2900  Vcvv 3173  cop 4131   class class class wbr 4583  cmpt 4643  cfv 5804  (class class class)co 6549  cmpt2 6551  0cc0 9815  1c1 9816   + caddc 9818  cmin 10145  cn 10897  2c2 10947  0cn0 11169  cuz 11563  #chash 12979   lastS clsw 13147   substr csubstr 13150   WWalksN cwwlkn 26206   ClWWalksN cclwwlkn 26277   FriendGrph cfrgra 26515
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-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlk 26207  df-wwlkn 26208  df-clwwlk 26279  df-clwwlkn 26280  df-frgra 26516
This theorem is referenced by:  numclwwlk2  26634
  Copyright terms: Public domain W3C validator