Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clwlkssizeeq | Structured version Visualization version GIF version |
Description: The size of the set of closed walks (defined as words) of length n corresponds to the size of the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.) |
Ref | Expression |
---|---|
clwlkssizeeq | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘(𝑁 ClWWalkSN 𝐺)) = (#‘{𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6113 | . . . . 5 ⊢ (ClWalkS‘𝐺) ∈ V | |
2 | 1 | rabex 4740 | . . . 4 ⊢ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁} ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁} ∈ V) |
4 | eqid 2610 | . . . 4 ⊢ (1st ‘𝑤) = (1st ‘𝑤) | |
5 | eqid 2610 | . . . 4 ⊢ (2nd ‘𝑤) = (2nd ‘𝑤) | |
6 | fveq2 6103 | . . . . . . 7 ⊢ (𝑐 = 𝑤 → (1st ‘𝑐) = (1st ‘𝑤)) | |
7 | 6 | fveq2d 6107 | . . . . . 6 ⊢ (𝑐 = 𝑤 → (#‘(1st ‘𝑐)) = (#‘(1st ‘𝑤))) |
8 | 7 | eqeq1d 2612 | . . . . 5 ⊢ (𝑐 = 𝑤 → ((#‘(1st ‘𝑐)) = 𝑁 ↔ (#‘(1st ‘𝑤)) = 𝑁)) |
9 | 8 | cbvrabv 3172 | . . . 4 ⊢ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁} = {𝑤 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑤)) = 𝑁} |
10 | fveq2 6103 | . . . . . 6 ⊢ (𝑢 = 𝑤 → (2nd ‘𝑢) = (2nd ‘𝑤)) | |
11 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑢 = 𝑤 → (1st ‘𝑢) = (1st ‘𝑤)) | |
12 | 11 | fveq2d 6107 | . . . . . . 7 ⊢ (𝑢 = 𝑤 → (#‘(1st ‘𝑢)) = (#‘(1st ‘𝑤))) |
13 | 12 | opeq2d 4347 | . . . . . 6 ⊢ (𝑢 = 𝑤 → 〈0, (#‘(1st ‘𝑢))〉 = 〈0, (#‘(1st ‘𝑤))〉) |
14 | 10, 13 | oveq12d 6567 | . . . . 5 ⊢ (𝑢 = 𝑤 → ((2nd ‘𝑢) substr 〈0, (#‘(1st ‘𝑢))〉) = ((2nd ‘𝑤) substr 〈0, (#‘(1st ‘𝑤))〉)) |
15 | 14 | cbvmptv 4678 | . . . 4 ⊢ (𝑢 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁} ↦ ((2nd ‘𝑢) substr 〈0, (#‘(1st ‘𝑢))〉)) = (𝑤 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁} ↦ ((2nd ‘𝑤) substr 〈0, (#‘(1st ‘𝑤))〉)) |
16 | 4, 5, 9, 15 | clwlksf1oclwwlk 41277 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝑢 ∈ {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁} ↦ ((2nd ‘𝑢) substr 〈0, (#‘(1st ‘𝑢))〉)):{𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁}–1-1-onto→(𝑁 ClWWalkSN 𝐺)) |
17 | 3, 16 | hasheqf1od 13006 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘{𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁}) = (#‘(𝑁 ClWWalkSN 𝐺))) |
18 | 17 | eqcomd 2616 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘(𝑁 ClWWalkSN 𝐺)) = (#‘{𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st ‘𝑐)) = 𝑁})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 0cc0 9815 #chash 12979 substr csubstr 13150 ℙcprime 15223 FinUSGraph cfusgr 40535 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-xnn0 11241 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-concat 13156 df-s1 13157 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: clwlksndivn 41279 |
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