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Mirrors > Home > MPE Home > Th. List > crctcshwlk | Structured version Visualization version GIF version |
Description: Cyclically shifting the indices of a circuit 〈𝐹, 𝑃〉 results in a walk 〈𝐻, 𝑄〉. (Contributed by AV, 10-Mar-2021.) |
Ref | Expression |
---|---|
crctcsh.v | ⊢ 𝑉 = (Vtx‘𝐺) |
crctcsh.i | ⊢ 𝐼 = (iEdg‘𝐺) |
crctcsh.d | ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) |
crctcsh.n | ⊢ 𝑁 = (♯‘𝐹) |
crctcsh.s | ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁)) |
crctcsh.h | ⊢ 𝐻 = (𝐹 cyclShift 𝑆) |
crctcsh.q | ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) |
Ref | Expression |
---|---|
crctcshwlk | ⊢ (𝜑 → 𝐻(Walks‘𝐺)𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crctcsh.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | crctcsh.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | crctcsh.d | . . . 4 ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) | |
4 | crctcsh.n | . . . 4 ⊢ 𝑁 = (♯‘𝐹) | |
5 | crctcsh.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁)) | |
6 | crctcsh.h | . . . 4 ⊢ 𝐻 = (𝐹 cyclShift 𝑆) | |
7 | crctcsh.q | . . . 4 ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | crctcshlem4 28086 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝐻 = 𝐹 ∧ 𝑄 = 𝑃)) |
9 | crctistrl 28064 | . . . . . 6 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
10 | trliswlk 27967 | . . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
11 | 3, 9, 10 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
12 | breq12 5075 | . . . . 5 ⊢ ((𝐻 = 𝐹 ∧ 𝑄 = 𝑃) → (𝐻(Walks‘𝐺)𝑄 ↔ 𝐹(Walks‘𝐺)𝑃)) | |
13 | 11, 12 | syl5ibrcom 246 | . . . 4 ⊢ (𝜑 → ((𝐻 = 𝐹 ∧ 𝑄 = 𝑃) → 𝐻(Walks‘𝐺)𝑄)) |
14 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = 0) → ((𝐻 = 𝐹 ∧ 𝑄 = 𝑃) → 𝐻(Walks‘𝐺)𝑄)) |
15 | 8, 14 | mpd 15 | . 2 ⊢ ((𝜑 ∧ 𝑆 = 0) → 𝐻(Walks‘𝐺)𝑄) |
16 | 1, 2, 3, 4, 5, 6, 7 | crctcshwlkn0 28087 | . 2 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝐻(Walks‘𝐺)𝑄) |
17 | 15, 16 | pm2.61dane 3031 | 1 ⊢ (𝜑 → 𝐻(Walks‘𝐺)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ifcif 4456 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 0cc0 10802 + caddc 10805 ≤ cle 10941 − cmin 11135 ...cfz 13168 ..^cfzo 13311 ♯chash 13972 cyclShift ccsh 14429 Vtxcvtx 27269 iEdgciedg 27270 Walkscwlks 27866 Trailsctrls 27960 Circuitsccrcts 28053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-hash 13973 df-word 14146 df-concat 14202 df-substr 14282 df-pfx 14312 df-csh 14430 df-wlks 27869 df-trls 27962 df-crcts 28055 |
This theorem is referenced by: crctcshtrl 28089 |
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