Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clwwlksnwwlkncl | Structured version Visualization version GIF version |
Description: Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) |
Ref | Expression |
---|---|
clwwlksnwwlkncl | ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (𝑁 ClWWalkSN 𝐺)) → (𝑃 ++ 〈“(𝑃‘0)”〉) ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2610 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | clwwlknp 41195 | . . . 4 ⊢ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) → ((𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))) |
4 | 1 | clwwlknbp0 41192 | . . . 4 ⊢ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) → ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁))) |
5 | simpr 476 | . . . . . 6 ⊢ (((((𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁))) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
6 | simpll1 1093 | . . . . . 6 ⊢ (((((𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁))) ∧ 𝑁 ∈ ℕ) → (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁)) | |
7 | 3simpc 1053 | . . . . . . 7 ⊢ (((𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))) | |
8 | 7 | ad2antrr 758 | . . . . . 6 ⊢ (((((𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁))) ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))) |
9 | 5, 6, 8 | 3jca 1235 | . . . . 5 ⊢ (((((𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁))) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)))) |
10 | 9 | ex 449 | . . . 4 ⊢ ((((𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)) ∧ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁))) → (𝑁 ∈ ℕ → (𝑁 ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))))) |
11 | 3, 4, 10 | syl2anc 691 | . . 3 ⊢ (𝑃 ∈ (𝑁 ClWWalkSN 𝐺) → (𝑁 ∈ ℕ → (𝑁 ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))))) |
12 | 11 | impcom 445 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (𝑁 ClWWalkSN 𝐺)) → (𝑁 ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺)))) |
13 | eqid 2610 | . . 3 ⊢ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)} = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)} | |
14 | 13 | clwwlksel 41221 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))) → (𝑃 ++ 〈“(𝑃‘0)”〉) ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}) |
15 | 12, 14 | syl 17 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (𝑁 ClWWalkSN 𝐺)) → (𝑃 ++ 〈“(𝑃‘0)”〉) ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 {cpr 4127 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕcn 10897 ..^cfzo 12334 #chash 12979 Word cword 13146 lastS clsw 13147 ++ cconcat 13148 〈“cs1 13149 Vtxcvtx 25673 Edgcedga 25792 WWalkSN cwwlksn 41029 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 |
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-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-wwlks 41033 df-wwlksn 41034 df-clwwlks 41185 df-clwwlksn 41186 |
This theorem is referenced by: (None) |
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