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Mirrors > Home > MPE Home > Th. List > wwlknp | Structured version Visualization version GIF version |
Description: Properties of a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 9-Apr-2021.) |
Ref | Expression |
---|---|
wwlkbp.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wwlknp.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
wwlknp | ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlkbp.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | wwlknbp 27628 | . 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) |
3 | iswwlksn 27624 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))) | |
4 | wwlknp.e | . . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺) | |
5 | 1, 4 | iswwlks 27622 | . . . . . . 7 ⊢ (𝑊 ∈ (WWalks‘𝐺) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
6 | simpl2 1189 | . . . . . . . . 9 ⊢ (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0)) → 𝑊 ∈ Word 𝑉) | |
7 | simprl 770 | . . . . . . . . 9 ⊢ (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0)) → (♯‘𝑊) = (𝑁 + 1)) | |
8 | oveq1 7142 | . . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((♯‘𝑊) − 1) = ((𝑁 + 1) − 1)) | |
9 | nn0cn 11895 | . . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
10 | pncan1 11053 | . . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁) | |
11 | 9, 10 | syl 17 | . . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) − 1) = 𝑁) |
12 | 8, 11 | sylan9eq 2853 | . . . . . . . . . . . . . 14 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑊) − 1) = 𝑁) |
13 | 12 | oveq2d 7151 | . . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0) → (0..^((♯‘𝑊) − 1)) = (0..^𝑁)) |
14 | 13 | raleqdv 3364 | . . . . . . . . . . . 12 ⊢ (((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0) → (∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
15 | 14 | biimpcd 252 | . . . . . . . . . . 11 ⊢ (∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → (((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0) → ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
16 | 15 | 3ad2ant3 1132 | . . . . . . . . . 10 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0) → ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
17 | 16 | imp 410 | . . . . . . . . 9 ⊢ (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0)) → ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) |
18 | 6, 7, 17 | 3jca 1125 | . . . . . . . 8 ⊢ (((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) ∧ ((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0)) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
19 | 18 | ex 416 | . . . . . . 7 ⊢ ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))) |
20 | 5, 19 | sylbi 220 | . . . . . 6 ⊢ (𝑊 ∈ (WWalks‘𝐺) → (((♯‘𝑊) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))) |
21 | 20 | expdimp 456 | . . . . 5 ⊢ ((𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))) |
22 | 21 | com12 32 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))) |
23 | 3, 22 | sylbid 243 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))) |
24 | 23 | 3ad2ant2 1131 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))) |
25 | 2, 24 | mpcom 38 | 1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 Vcvv 3441 ∅c0 4243 {cpr 4527 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 − cmin 10859 ℕ0cn0 11885 ..^cfzo 13028 ♯chash 13686 Word cword 13857 Vtxcvtx 26789 Edgcedg 26840 WWalkscwwlks 27611 WWalksN cwwlksn 27612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-wwlks 27616 df-wwlksn 27617 |
This theorem is referenced by: wwlknbp1 27630 wwlksnext 27679 wwlksnextbi 27680 wwlksnredwwlkn 27681 wwlksnredwwlkn0 27682 wwlksnextwrd 27683 wwlksnextsurj 27686 wwlksnextproplem2 27696 wwlksnextproplem3 27697 rusgrnumwwlks 27760 clwwlkinwwlk 27825 clwwlkf1 27834 wwlksext2clwwlk 27842 clwwlknonwwlknonb 27891 clwwlkvbij 27898 numclwwlk2lem1 28161 |
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