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| Mirrors > Home > MPE Home > Th. List > clwwlknccat | Structured version Visualization version GIF version | ||
| Description: The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk with a length which is the sum of the lengths of the two walks. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 24-Apr-2022.) |
| Ref | Expression |
|---|---|
| clwwlknccat | β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ) β§ (π΄β0) = (π΅β0)) β (π΄ ++ π΅) β ((π + π) ClWWalksN πΊ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isclwwlkn 29277 | . . 3 β’ (π΄ β (π ClWWalksN πΊ) β (π΄ β (ClWWalksβπΊ) β§ (β―βπ΄) = π)) | |
| 2 | isclwwlkn 29277 | . . 3 β’ (π΅ β (π ClWWalksN πΊ) β (π΅ β (ClWWalksβπΊ) β§ (β―βπ΅) = π)) | |
| 3 | biid 260 | . . 3 β’ ((π΄β0) = (π΅β0) β (π΄β0) = (π΅β0)) | |
| 4 | simpl 483 | . . . 4 β’ ((π΄ β (ClWWalksβπΊ) β§ (β―βπ΄) = π) β π΄ β (ClWWalksβπΊ)) | |
| 5 | simpl 483 | . . . 4 β’ ((π΅ β (ClWWalksβπΊ) β§ (β―βπ΅) = π) β π΅ β (ClWWalksβπΊ)) | |
| 6 | id 22 | . . . 4 β’ ((π΄β0) = (π΅β0) β (π΄β0) = (π΅β0)) | |
| 7 | clwwlkccat 29240 | . . . 4 β’ ((π΄ β (ClWWalksβπΊ) β§ π΅ β (ClWWalksβπΊ) β§ (π΄β0) = (π΅β0)) β (π΄ ++ π΅) β (ClWWalksβπΊ)) | |
| 8 | 4, 5, 6, 7 | syl3an 1160 | . . 3 β’ (((π΄ β (ClWWalksβπΊ) β§ (β―βπ΄) = π) β§ (π΅ β (ClWWalksβπΊ) β§ (β―βπ΅) = π) β§ (π΄β0) = (π΅β0)) β (π΄ ++ π΅) β (ClWWalksβπΊ)) |
| 9 | 1, 2, 3, 8 | syl3anb 1161 | . 2 β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ) β§ (π΄β0) = (π΅β0)) β (π΄ ++ π΅) β (ClWWalksβπΊ)) |
| 10 | eqid 2732 | . . . . . 6 β’ (VtxβπΊ) = (VtxβπΊ) | |
| 11 | 10 | clwwlknwrd 29284 | . . . . 5 β’ (π΄ β (π ClWWalksN πΊ) β π΄ β Word (VtxβπΊ)) |
| 12 | 10 | clwwlknwrd 29284 | . . . . 5 β’ (π΅ β (π ClWWalksN πΊ) β π΅ β Word (VtxβπΊ)) |
| 13 | ccatlen 14524 | . . . . 5 β’ ((π΄ β Word (VtxβπΊ) β§ π΅ β Word (VtxβπΊ)) β (β―β(π΄ ++ π΅)) = ((β―βπ΄) + (β―βπ΅))) | |
| 14 | 11, 12, 13 | syl2an 596 | . . . 4 β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ)) β (β―β(π΄ ++ π΅)) = ((β―βπ΄) + (β―βπ΅))) |
| 15 | clwwlknlen 29282 | . . . . 5 β’ (π΄ β (π ClWWalksN πΊ) β (β―βπ΄) = π) | |
| 16 | clwwlknlen 29282 | . . . . 5 β’ (π΅ β (π ClWWalksN πΊ) β (β―βπ΅) = π) | |
| 17 | 15, 16 | oveqan12d 7427 | . . . 4 β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ)) β ((β―βπ΄) + (β―βπ΅)) = (π + π)) |
| 18 | 14, 17 | eqtrd 2772 | . . 3 β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ)) β (β―β(π΄ ++ π΅)) = (π + π)) |
| 19 | 18 | 3adant3 1132 | . 2 β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ) β§ (π΄β0) = (π΅β0)) β (β―β(π΄ ++ π΅)) = (π + π)) |
| 20 | isclwwlkn 29277 | . 2 β’ ((π΄ ++ π΅) β ((π + π) ClWWalksN πΊ) β ((π΄ ++ π΅) β (ClWWalksβπΊ) β§ (β―β(π΄ ++ π΅)) = (π + π))) | |
| 21 | 9, 19, 20 | sylanbrc 583 | 1 β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ) β§ (π΄β0) = (π΅β0)) β (π΄ ++ π΅) β ((π + π) ClWWalksN πΊ)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7408 0cc0 11109 + caddc 11112 β―chash 14289 Word cword 14463 ++ cconcat 14519 Vtxcvtx 28253 ClWWalkscclwwlk 29231 ClWWalksN cclwwlkn 29274 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-xnn0 12544 df-z 12558 df-uz 12822 df-rp 12974 df-fz 13484 df-fzo 13627 df-hash 14290 df-word 14464 df-lsw 14512 df-concat 14520 df-clwwlk 29232 df-clwwlkn 29275 |
| This theorem is referenced by: clwwlknonccat 29346 |
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