Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clwlksf1clwwlk Structured version   Visualization version   GIF version

Theorem clwlksf1clwwlk 41276
 Description: There is a one-to-one function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)
Hypotheses
Ref Expression
clwlksfclwwlk.1 𝐴 = (1st𝑐)
clwlksfclwwlk.2 𝐵 = (2nd𝑐)
clwlksfclwwlk.c 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
clwlksfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
Assertion
Ref Expression
clwlksf1clwwlk ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶1-1→(𝑁 ClWWalkSN 𝐺))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝐶,𝑐   𝐹,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlksf1clwwlk
Dummy variables 𝑖 𝑤 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwlksfclwwlk.1 . . 3 𝐴 = (1st𝑐)
2 clwlksfclwwlk.2 . . 3 𝐵 = (2nd𝑐)
3 clwlksfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
4 clwlksfclwwlk.f . . 3 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
51, 2, 3, 4clwlksfclwwlk 41269 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalkSN 𝐺))
6 simprl 790 . . . . . 6 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → 𝑢𝐶)
7 ovex 6577 . . . . . 6 ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) ∈ V
8 fveq2 6103 . . . . . . . . 9 (𝑐 = 𝑢 → (2nd𝑐) = (2nd𝑢))
92, 8syl5eq 2656 . . . . . . . 8 (𝑐 = 𝑢𝐵 = (2nd𝑢))
10 fveq2 6103 . . . . . . . . . . 11 (𝑐 = 𝑢 → (1st𝑐) = (1st𝑢))
111, 10syl5eq 2656 . . . . . . . . . 10 (𝑐 = 𝑢𝐴 = (1st𝑢))
1211fveq2d 6107 . . . . . . . . 9 (𝑐 = 𝑢 → (#‘𝐴) = (#‘(1st𝑢)))
1312opeq2d 4347 . . . . . . . 8 (𝑐 = 𝑢 → ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st𝑢))⟩)
149, 13oveq12d 6567 . . . . . . 7 (𝑐 = 𝑢 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩))
1514, 4fvmptg 6189 . . . . . 6 ((𝑢𝐶 ∧ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) ∈ V) → (𝐹𝑢) = ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩))
166, 7, 15sylancl 693 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → (𝐹𝑢) = ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩))
17 simpr 476 . . . . . . . 8 ((𝑢𝐶𝑤𝐶) → 𝑤𝐶)
18 ovex 6577 . . . . . . . 8 ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩) ∈ V
1917, 18jctir 559 . . . . . . 7 ((𝑢𝐶𝑤𝐶) → (𝑤𝐶 ∧ ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩) ∈ V))
2019adantl 481 . . . . . 6 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → (𝑤𝐶 ∧ ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩) ∈ V))
21 fveq2 6103 . . . . . . . . 9 (𝑐 = 𝑤 → (2nd𝑐) = (2nd𝑤))
222, 21syl5eq 2656 . . . . . . . 8 (𝑐 = 𝑤𝐵 = (2nd𝑤))
23 fveq2 6103 . . . . . . . . . . 11 (𝑐 = 𝑤 → (1st𝑐) = (1st𝑤))
241, 23syl5eq 2656 . . . . . . . . . 10 (𝑐 = 𝑤𝐴 = (1st𝑤))
2524fveq2d 6107 . . . . . . . . 9 (𝑐 = 𝑤 → (#‘𝐴) = (#‘(1st𝑤)))
2625opeq2d 4347 . . . . . . . 8 (𝑐 = 𝑤 → ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st𝑤))⟩)
2722, 26oveq12d 6567 . . . . . . 7 (𝑐 = 𝑤 → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩))
2827, 4fvmptg 6189 . . . . . 6 ((𝑤𝐶 ∧ ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩) ∈ V) → (𝐹𝑤) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩))
2920, 28syl 17 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → (𝐹𝑤) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩))
3016, 29eqeq12d 2625 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → ((𝐹𝑢) = (𝐹𝑤) ↔ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩)))
311, 2, 3, 4clwlksfclwwlk1hashn 41266 . . . . . . . . 9 (𝑤𝐶 → (#‘(1st𝑤)) = 𝑁)
3231eqcomd 2616 . . . . . . . 8 (𝑤𝐶𝑁 = (#‘(1st𝑤)))
3332adantl 481 . . . . . . 7 ((𝑢𝐶𝑤𝐶) → 𝑁 = (#‘(1st𝑤)))
3433ad2antlr 759 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) ∧ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩)) → 𝑁 = (#‘(1st𝑤)))
35 prmnn 15226 . . . . . . . . 9 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
3635ad2antlr 759 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → 𝑁 ∈ ℕ)
3717adantl 481 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → 𝑤𝐶)
381, 2, 3, 4clwlksf1clwwlklem 41275 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑢𝐶𝑤𝐶) → (((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩) → ∀𝑖 ∈ (0...𝑁)((2nd𝑢)‘𝑖) = ((2nd𝑤)‘𝑖)))
3936, 6, 37, 38syl3anc 1318 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → (((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩) → ∀𝑖 ∈ (0...𝑁)((2nd𝑢)‘𝑖) = ((2nd𝑤)‘𝑖)))
4039imp 444 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) ∧ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩)) → ∀𝑖 ∈ (0...𝑁)((2nd𝑢)‘𝑖) = ((2nd𝑤)‘𝑖))
41 fusgrusgr 40541 . . . . . . . . 9 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
42 usgruspgr 40408 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
4341, 42syl 17 . . . . . . . 8 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph )
4443ad3antrrr 762 . . . . . . 7 ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) ∧ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩)) → 𝐺 ∈ USPGraph )
45 elrabi 3328 . . . . . . . . . . 11 (𝑢 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑢 ∈ (ClWalkS‘𝐺))
46 clwlk1wlk 40982 . . . . . . . . . . 11 (𝑢 ∈ (ClWalkS‘𝐺) → 𝑢 ∈ (1Walks‘𝐺))
4745, 46syl 17 . . . . . . . . . 10 (𝑢 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑢 ∈ (1Walks‘𝐺))
4847, 3eleq2s 2706 . . . . . . . . 9 (𝑢𝐶𝑢 ∈ (1Walks‘𝐺))
49 elrabi 3328 . . . . . . . . . . 11 (𝑤 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑤 ∈ (ClWalkS‘𝐺))
50 clwlk1wlk 40982 . . . . . . . . . . 11 (𝑤 ∈ (ClWalkS‘𝐺) → 𝑤 ∈ (1Walks‘𝐺))
5149, 50syl 17 . . . . . . . . . 10 (𝑤 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} → 𝑤 ∈ (1Walks‘𝐺))
5251, 3eleq2s 2706 . . . . . . . . 9 (𝑤𝐶𝑤 ∈ (1Walks‘𝐺))
5348, 52anim12i 588 . . . . . . . 8 ((𝑢𝐶𝑤𝐶) → (𝑢 ∈ (1Walks‘𝐺) ∧ 𝑤 ∈ (1Walks‘𝐺)))
5453ad2antlr 759 . . . . . . 7 ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) ∧ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩)) → (𝑢 ∈ (1Walks‘𝐺) ∧ 𝑤 ∈ (1Walks‘𝐺)))
551, 2, 3, 4clwlksfclwwlk1hashn 41266 . . . . . . . . . 10 (𝑢𝐶 → (#‘(1st𝑢)) = 𝑁)
5655eqcomd 2616 . . . . . . . . 9 (𝑢𝐶𝑁 = (#‘(1st𝑢)))
5756adantr 480 . . . . . . . 8 ((𝑢𝐶𝑤𝐶) → 𝑁 = (#‘(1st𝑢)))
5857ad2antlr 759 . . . . . . 7 ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) ∧ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩)) → 𝑁 = (#‘(1st𝑢)))
59 uspgr2wlkeq 40854 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝑢 ∈ (1Walks‘𝐺) ∧ 𝑤 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝑢))) → (𝑢 = 𝑤 ↔ (𝑁 = (#‘(1st𝑤)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd𝑢)‘𝑖) = ((2nd𝑤)‘𝑖))))
6044, 54, 58, 59syl3anc 1318 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) ∧ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩)) → (𝑢 = 𝑤 ↔ (𝑁 = (#‘(1st𝑤)) ∧ ∀𝑖 ∈ (0...𝑁)((2nd𝑢)‘𝑖) = ((2nd𝑤)‘𝑖))))
6134, 40, 60mpbir2and 959 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) ∧ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩)) → 𝑢 = 𝑤)
6261ex 449 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → (((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩) → 𝑢 = 𝑤))
6330, 62sylbid 229 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑢𝐶𝑤𝐶)) → ((𝐹𝑢) = (𝐹𝑤) → 𝑢 = 𝑤))
6463ralrimivva 2954 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ∀𝑢𝐶𝑤𝐶 ((𝐹𝑢) = (𝐹𝑤) → 𝑢 = 𝑤))
65 dff13 6416 . 2 (𝐹:𝐶1-1→(𝑁 ClWWalkSN 𝐺) ↔ (𝐹:𝐶⟶(𝑁 ClWWalkSN 𝐺) ∧ ∀𝑢𝐶𝑤𝐶 ((𝐹𝑢) = (𝐹𝑤) → 𝑢 = 𝑤)))
665, 64, 65sylanbrc 695 1 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶1-1→(𝑁 ClWWalkSN 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  ℕcn 10897  ...cfz 12197  #chash 12979   substr csubstr 13150  ℙcprime 15223   USPGraph cuspgr 40378   USGraph cusgr 40379   FinUSGraph cfusgr 40535  1Walksc1wlks 40796  ClWalkScclwlks 40976   ClWWalkSN cclwwlksn 41184 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  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  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-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-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-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-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-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-substr 13158  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-prm 15224  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-1wlks 40800  df-wlks 40801  df-clwlks 40977  df-clwwlks 41185  df-clwwlksn 41186 This theorem is referenced by:  clwlksf1oclwwlk  41277
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