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Mirrors > Home > MPE Home > Th. List > hashwwlksnext | Structured version Visualization version GIF version |
Description: Number of walks (as words) extended by an edge as a sum over the prefixes. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 26-Oct-2022.) |
Ref | Expression |
---|---|
wwlksnextprop.x | β’ π = ((π + 1) WWalksN πΊ) |
wwlksnextprop.e | β’ πΈ = (EdgβπΊ) |
wwlksnextprop.y | β’ π = {π€ β (π WWalksN πΊ) β£ (π€β0) = π} |
Ref | Expression |
---|---|
hashwwlksnext | β’ ((VtxβπΊ) β Fin β (β―β{π₯ β π β£ βπ¦ β π ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)}) = Ξ£π¦ β π (β―β{π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlksnextprop.y | . . 3 β’ π = {π€ β (π WWalksN πΊ) β£ (π€β0) = π} | |
2 | wwlksnfi 29704 | . . . 4 β’ ((VtxβπΊ) β Fin β (π WWalksN πΊ) β Fin) | |
3 | ssrab2 4073 | . . . 4 β’ {π€ β (π WWalksN πΊ) β£ (π€β0) = π} β (π WWalksN πΊ) | |
4 | ssfi 9189 | . . . 4 β’ (((π WWalksN πΊ) β Fin β§ {π€ β (π WWalksN πΊ) β£ (π€β0) = π} β (π WWalksN πΊ)) β {π€ β (π WWalksN πΊ) β£ (π€β0) = π} β Fin) | |
5 | 2, 3, 4 | sylancl 585 | . . 3 β’ ((VtxβπΊ) β Fin β {π€ β (π WWalksN πΊ) β£ (π€β0) = π} β Fin) |
6 | 1, 5 | eqeltrid 2832 | . 2 β’ ((VtxβπΊ) β Fin β π β Fin) |
7 | wwlksnextprop.x | . . . . 5 β’ π = ((π + 1) WWalksN πΊ) | |
8 | wwlksnfi 29704 | . . . . 5 β’ ((VtxβπΊ) β Fin β ((π + 1) WWalksN πΊ) β Fin) | |
9 | 7, 8 | eqeltrid 2832 | . . . 4 β’ ((VtxβπΊ) β Fin β π β Fin) |
10 | rabfi 9285 | . . . 4 β’ (π β Fin β {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} β Fin) | |
11 | 9, 10 | syl 17 | . . 3 β’ ((VtxβπΊ) β Fin β {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} β Fin) |
12 | 11 | adantr 480 | . 2 β’ (((VtxβπΊ) β Fin β§ π¦ β π) β {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} β Fin) |
13 | wwlksnextprop.e | . . . 4 β’ πΈ = (EdgβπΊ) | |
14 | 7, 13, 1 | disjxwwlkn 29711 | . . 3 β’ Disj π¦ β π {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)} |
15 | 14 | a1i 11 | . 2 β’ ((VtxβπΊ) β Fin β Disj π¦ β π {π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)}) |
16 | 6, 12, 15 | hashrabrex 15795 | 1 β’ ((VtxβπΊ) β Fin β (β―β{π₯ β π β£ βπ¦ β π ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)}) = Ξ£π¦ β π (β―β{π₯ β π β£ ((π₯ prefix π) = π¦ β§ (π¦β0) = π β§ {(lastSβπ¦), (lastSβπ₯)} β πΈ)})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 βwrex 3065 {crab 3427 β wss 3944 {cpr 4626 Disj wdisj 5107 βcfv 6542 (class class class)co 7414 Fincfn 8955 0cc0 11130 1c1 11131 + caddc 11133 β―chash 14313 lastSclsw 14536 prefix cpfx 14644 Ξ£csu 15656 Vtxcvtx 28796 Edgcedg 28847 WWalksN cwwlksn 29624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-oi 9525 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-word 14489 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-clim 15456 df-sum 15657 df-wwlks 29628 df-wwlksn 29629 |
This theorem is referenced by: rusgrnumwwlks 29772 |
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