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Mirrors > Home > MPE Home > Th. List > clwwlknclwwlkdif | Structured version Visualization version GIF version |
Description: The set π΄ of walks of length π starting with a fixed vertex π and ending not at this vertex is the difference between the set πΆ of walks of length π starting with this vertex π and the set π΅ of closed walks of length π anchored at this vertex π. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 16-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknclwwlkdif.a | β’ π΄ = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)} |
clwwlknclwwlkdif.b | β’ π΅ = (π(π WWalksNOn πΊ)π) |
clwwlknclwwlkdif.c | β’ πΆ = {π€ β (π WWalksN πΊ) β£ (π€β0) = π} |
Ref | Expression |
---|---|
clwwlknclwwlkdif | β’ π΄ = (πΆ β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlknclwwlkdif.a | . 2 β’ π΄ = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)} | |
2 | clwwlknclwwlkdif.c | . . . 4 β’ πΆ = {π€ β (π WWalksN πΊ) β£ (π€β0) = π} | |
3 | clwwlknclwwlkdif.b | . . . . 5 β’ π΅ = (π(π WWalksNOn πΊ)π) | |
4 | eqid 2736 | . . . . . 6 β’ (VtxβπΊ) = (VtxβπΊ) | |
5 | 4 | iswwlksnon 28263 | . . . . 5 β’ (π(π WWalksNOn πΊ)π) = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (π€βπ) = π)} |
6 | 3, 5 | eqtri 2764 | . . . 4 β’ π΅ = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (π€βπ) = π)} |
7 | 2, 6 | difeq12i 4061 | . . 3 β’ (πΆ β π΅) = ({π€ β (π WWalksN πΊ) β£ (π€β0) = π} β {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (π€βπ) = π)}) |
8 | difrab 4248 | . . 3 β’ ({π€ β (π WWalksN πΊ) β£ (π€β0) = π} β {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (π€βπ) = π)}) = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ Β¬ ((π€β0) = π β§ (π€βπ) = π))} | |
9 | annotanannot 833 | . . . . 5 β’ (((π€β0) = π β§ Β¬ ((π€β0) = π β§ (π€βπ) = π)) β ((π€β0) = π β§ Β¬ (π€βπ) = π)) | |
10 | df-ne 2942 | . . . . . . 7 β’ ((π€βπ) β π β Β¬ (π€βπ) = π) | |
11 | wwlknlsw 28257 | . . . . . . . 8 β’ (π€ β (π WWalksN πΊ) β (π€βπ) = (lastSβπ€)) | |
12 | 11 | neeq1d 3001 | . . . . . . 7 β’ (π€ β (π WWalksN πΊ) β ((π€βπ) β π β (lastSβπ€) β π)) |
13 | 10, 12 | bitr3id 285 | . . . . . 6 β’ (π€ β (π WWalksN πΊ) β (Β¬ (π€βπ) = π β (lastSβπ€) β π)) |
14 | 13 | anbi2d 630 | . . . . 5 β’ (π€ β (π WWalksN πΊ) β (((π€β0) = π β§ Β¬ (π€βπ) = π) β ((π€β0) = π β§ (lastSβπ€) β π))) |
15 | 9, 14 | bitrid 283 | . . . 4 β’ (π€ β (π WWalksN πΊ) β (((π€β0) = π β§ Β¬ ((π€β0) = π β§ (π€βπ) = π)) β ((π€β0) = π β§ (lastSβπ€) β π))) |
16 | 15 | rabbiia 3414 | . . 3 β’ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ Β¬ ((π€β0) = π β§ (π€βπ) = π))} = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)} |
17 | 7, 8, 16 | 3eqtrri 2769 | . 2 β’ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)} = (πΆ β π΅) |
18 | 1, 17 | eqtri 2764 | 1 β’ π΄ = (πΆ β π΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β§ wa 397 = wceq 1539 β wcel 2104 β wne 2941 {crab 3284 β cdif 3889 βcfv 6458 (class class class)co 7307 0cc0 10917 lastSclsw 14310 Vtxcvtx 27411 WWalksN cwwlksn 28236 WWalksNOn cwwlksnon 28237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-fzo 13429 df-hash 14091 df-word 14263 df-lsw 14311 df-wwlks 28240 df-wwlksn 28241 df-wwlksnon 28242 |
This theorem is referenced by: clwwlknclwwlkdifnum 28389 |
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