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Mirrors > Home > MPE Home > Th. List > crctcshtrl | Structured version Visualization version GIF version |
Description: Cyclically shifting the indices of a circuit 〈𝐹, 𝑃〉 results in a trail 〈𝐻, 𝑄〉. (Contributed by AV, 14-Mar-2021.) (Proof shortened by AV, 30-Oct-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 |
---|---|
crctcshtrl | ⊢ (𝜑 → 𝐻(Trails‘𝐺)𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crctcsh.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | crctcsh.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | crctcsh.d | . . 3 ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) | |
4 | crctcsh.n | . . 3 ⊢ 𝑁 = (♯‘𝐹) | |
5 | crctcsh.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁)) | |
6 | crctcsh.h | . . 3 ⊢ 𝐻 = (𝐹 cyclShift 𝑆) | |
7 | crctcsh.q | . . 3 ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | crctcshwlk 29852 | . 2 ⊢ (𝜑 → 𝐻(Walks‘𝐺)𝑄) |
9 | crctistrl 29828 | . . . . 5 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
10 | 2 | trlf1 29731 | . . . . 5 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) |
11 | df-f1 6568 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun ◡𝐹)) | |
12 | iswrdi 14553 | . . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 → 𝐹 ∈ Word dom 𝐼) | |
13 | 12 | anim1i 615 | . . . . . 6 ⊢ ((𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ∧ Fun ◡𝐹) → (𝐹 ∈ Word dom 𝐼 ∧ Fun ◡𝐹)) |
14 | 11, 13 | sylbi 217 | . . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → (𝐹 ∈ Word dom 𝐼 ∧ Fun ◡𝐹)) |
15 | 3, 9, 10, 14 | 4syl 19 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ Word dom 𝐼 ∧ Fun ◡𝐹)) |
16 | elfzoelz 13696 | . . . . 5 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℤ) | |
17 | 5, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ ℤ) |
18 | df-3an 1088 | . . . 4 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ Fun ◡𝐹 ∧ 𝑆 ∈ ℤ) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ Fun ◡𝐹) ∧ 𝑆 ∈ ℤ)) | |
19 | 15, 17, 18 | sylanbrc 583 | . . 3 ⊢ (𝜑 → (𝐹 ∈ Word dom 𝐼 ∧ Fun ◡𝐹 ∧ 𝑆 ∈ ℤ)) |
20 | cshinj 14846 | . . 3 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ Fun ◡𝐹 ∧ 𝑆 ∈ ℤ) → (𝐻 = (𝐹 cyclShift 𝑆) → Fun ◡𝐻)) | |
21 | 19, 6, 20 | mpisyl 21 | . 2 ⊢ (𝜑 → Fun ◡𝐻) |
22 | istrl 29729 | . 2 ⊢ (𝐻(Trails‘𝐺)𝑄 ↔ (𝐻(Walks‘𝐺)𝑄 ∧ Fun ◡𝐻)) | |
23 | 8, 21, 22 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐻(Trails‘𝐺)𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5688 dom cdm 5689 Fun wfun 6557 ⟶wf 6559 –1-1→wf1 6560 ‘cfv 6563 (class class class)co 7431 0cc0 11153 + caddc 11156 ≤ cle 11294 − cmin 11490 ℤcz 12611 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Word cword 14549 cyclShift ccsh 14823 Vtxcvtx 29028 iEdgciedg 29029 Walkscwlks 29629 Trailsctrls 29723 Circuitsccrcts 29817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-hash 14367 df-word 14550 df-concat 14606 df-substr 14676 df-pfx 14706 df-csh 14824 df-wlks 29632 df-trls 29725 df-crcts 29819 |
This theorem is referenced by: crctcsh 29854 eucrctshift 30272 |
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