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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 27876 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝐻 = 𝐹 ∧ 𝑄 = 𝑃)) |
9 | crctistrl 27854 | . . . . . 6 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
10 | trliswlk 27757 | . . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
11 | 3, 9, 10 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) |
12 | breq12 5048 | . . . . 5 ⊢ ((𝐻 = 𝐹 ∧ 𝑄 = 𝑃) → (𝐻(Walks‘𝐺)𝑄 ↔ 𝐹(Walks‘𝐺)𝑃)) | |
13 | 11, 12 | syl5ibrcom 250 | . . . 4 ⊢ (𝜑 → ((𝐻 = 𝐹 ∧ 𝑄 = 𝑃) → 𝐻(Walks‘𝐺)𝑄)) |
14 | 13 | adantr 484 | . . 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 27877 | . 2 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝐻(Walks‘𝐺)𝑄) |
17 | 15, 16 | pm2.61dane 3022 | 1 ⊢ (𝜑 → 𝐻(Walks‘𝐺)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ifcif 4429 class class class wbr 5043 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 0cc0 10712 + caddc 10715 ≤ cle 10851 − cmin 11045 ...cfz 13078 ..^cfzo 13221 ♯chash 13879 cyclShift ccsh 14336 Vtxcvtx 27059 iEdgciedg 27060 Walkscwlks 27656 Trailsctrls 27750 Circuitsccrcts 27843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-hash 13880 df-word 14053 df-concat 14109 df-substr 14189 df-pfx 14219 df-csh 14337 df-wlks 27659 df-trls 27752 df-crcts 27845 |
This theorem is referenced by: crctcshtrl 27879 |
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