Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clwlksfclwwlk1hash | Structured version Visualization version GIF version |
Description: The size of the first component of a closed walk is an integer in the range between 0 and the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.) (Revised by AV, 2-May-2021.) |
Ref | Expression |
---|---|
clwlksfclwwlk.1 | ⊢ 𝐴 = (1st ‘𝑐) |
clwlksfclwwlk.2 | ⊢ 𝐵 = (2nd ‘𝑐) |
clwlksfclwwlk.c | ⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} |
clwlksfclwwlk.f | ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr 〈0, (#‘𝐴)〉)) |
Ref | Expression |
---|---|
clwlksfclwwlk1hash | ⊢ (𝑐 ∈ 𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlksfclwwlk.c | . . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁} | |
2 | 1 | rabeq2i 3170 | . 2 ⊢ (𝑐 ∈ 𝐶 ↔ (𝑐 ∈ (ClWalkS‘𝐺) ∧ (#‘𝐴) = 𝑁)) |
3 | eqid 2610 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
4 | eqid 2610 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | clwlksfclwwlk.1 | . . . . 5 ⊢ 𝐴 = (1st ‘𝑐) | |
6 | clwlksfclwwlk.2 | . . . . 5 ⊢ 𝐵 = (2nd ‘𝑐) | |
7 | 3, 4, 5, 6 | clWlkcompim 40987 | . . . 4 ⊢ (𝑐 ∈ (ClWalkS‘𝐺) → ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))if-((𝐵‘𝑖) = (𝐵‘(𝑖 + 1)), ((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖)}, {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐴‘𝑖))) ∧ (𝐵‘0) = (𝐵‘(#‘𝐴))))) |
8 | lencl 13179 | . . . . . 6 ⊢ (𝐴 ∈ Word dom (iEdg‘𝐺) → (#‘𝐴) ∈ ℕ0) | |
9 | ffn 5958 | . . . . . 6 ⊢ (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → 𝐵 Fn (0...(#‘𝐴))) | |
10 | fnfz0hash 13087 | . . . . . . 7 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ 𝐵 Fn (0...(#‘𝐴))) → (#‘𝐵) = ((#‘𝐴) + 1)) | |
11 | nn0fz0 12306 | . . . . . . . . . 10 ⊢ ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (0...(#‘𝐴))) | |
12 | fzelp1 12263 | . . . . . . . . . 10 ⊢ ((#‘𝐴) ∈ (0...(#‘𝐴)) → (#‘𝐴) ∈ (0...((#‘𝐴) + 1))) | |
13 | 11, 12 | sylbi 206 | . . . . . . . . 9 ⊢ ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ (0...((#‘𝐴) + 1))) |
14 | oveq2 6557 | . . . . . . . . . 10 ⊢ ((#‘𝐵) = ((#‘𝐴) + 1) → (0...(#‘𝐵)) = (0...((#‘𝐴) + 1))) | |
15 | 14 | eleq2d 2673 | . . . . . . . . 9 ⊢ ((#‘𝐵) = ((#‘𝐴) + 1) → ((#‘𝐴) ∈ (0...(#‘𝐵)) ↔ (#‘𝐴) ∈ (0...((#‘𝐴) + 1)))) |
16 | 13, 15 | syl5ibrcom 236 | . . . . . . . 8 ⊢ ((#‘𝐴) ∈ ℕ0 → ((#‘𝐵) = ((#‘𝐴) + 1) → (#‘𝐴) ∈ (0...(#‘𝐵)))) |
17 | 16 | adantr 480 | . . . . . . 7 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ 𝐵 Fn (0...(#‘𝐴))) → ((#‘𝐵) = ((#‘𝐴) + 1) → (#‘𝐴) ∈ (0...(#‘𝐵)))) |
18 | 10, 17 | mpd 15 | . . . . . 6 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ 𝐵 Fn (0...(#‘𝐴))) → (#‘𝐴) ∈ (0...(#‘𝐵))) |
19 | 8, 9, 18 | syl2an 493 | . . . . 5 ⊢ ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → (#‘𝐴) ∈ (0...(#‘𝐵))) |
20 | 19 | adantr 480 | . . . 4 ⊢ (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))if-((𝐵‘𝑖) = (𝐵‘(𝑖 + 1)), ((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖)}, {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐴‘𝑖))) ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → (#‘𝐴) ∈ (0...(#‘𝐵))) |
21 | 7, 20 | syl 17 | . . 3 ⊢ (𝑐 ∈ (ClWalkS‘𝐺) → (#‘𝐴) ∈ (0...(#‘𝐵))) |
22 | 21 | adantr 480 | . 2 ⊢ ((𝑐 ∈ (ClWalkS‘𝐺) ∧ (#‘𝐴) = 𝑁) → (#‘𝐴) ∈ (0...(#‘𝐵))) |
23 | 2, 22 | sylbi 206 | 1 ⊢ (𝑐 ∈ 𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 if-wif 1006 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ wss 3540 {csn 4125 {cpr 4127 〈cop 4131 ↦ cmpt 4643 dom cdm 5038 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 0cc0 9815 1c1 9816 + caddc 9818 ℕ0cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 substr csubstr 13150 Vtxcvtx 25673 iEdgciedg 25674 ClWalkScclwlks 40976 |
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-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-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-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-1wlks 40800 df-clwlks 40977 |
This theorem is referenced by: clwlksfclwwlk2sswd 41268 clwlksfclwwlk 41269 |
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