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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 29879 | . . 3 β’ (π΄ β (π ClWWalksN πΊ) β (π΄ β (ClWWalksβπΊ) β§ (β―βπ΄) = π)) | |
| 2 | isclwwlkn 29879 | . . 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 481 | . . . 4 β’ ((π΄ β (ClWWalksβπΊ) β§ (β―βπ΄) = π) β π΄ β (ClWWalksβπΊ)) | |
| 5 | simpl 481 | . . . 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 29842 | . . . 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 1157 | . . 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 1158 | . 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 2725 | . . . . . 6 β’ (VtxβπΊ) = (VtxβπΊ) | |
| 11 | 10 | clwwlknwrd 29886 | . . . . 5 β’ (π΄ β (π ClWWalksN πΊ) β π΄ β Word (VtxβπΊ)) |
| 12 | 10 | clwwlknwrd 29886 | . . . . 5 β’ (π΅ β (π ClWWalksN πΊ) β π΅ β Word (VtxβπΊ)) |
| 13 | ccatlen 14555 | . . . . 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 594 | . . . 4 β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ)) β (β―β(π΄ ++ π΅)) = ((β―βπ΄) + (β―βπ΅))) |
| 15 | clwwlknlen 29884 | . . . . 5 β’ (π΄ β (π ClWWalksN πΊ) β (β―βπ΄) = π) | |
| 16 | clwwlknlen 29884 | . . . . 5 β’ (π΅ β (π ClWWalksN πΊ) β (β―βπ΅) = π) | |
| 17 | 15, 16 | oveqan12d 7434 | . . . 4 β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ)) β ((β―βπ΄) + (β―βπ΅)) = (π + π)) |
| 18 | 14, 17 | eqtrd 2765 | . . 3 β’ ((π΄ β (π ClWWalksN πΊ) β§ π΅ β (π ClWWalksN πΊ)) β (β―β(π΄ ++ π΅)) = (π + π)) |
| 19 | 18 | 3adant3 1129 | . 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 29879 | . 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 581 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6542 (class class class)co 7415 0cc0 11136 + caddc 11139 β―chash 14319 Word cword 14494 ++ cconcat 14550 Vtxcvtx 28851 ClWWalkscclwwlk 29833 ClWWalksN cclwwlkn 29876 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-oadd 8487 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-hash 14320 df-word 14495 df-lsw 14543 df-concat 14551 df-clwwlk 29834 df-clwwlkn 29877 |
| This theorem is referenced by: clwwlknonccat 29948 |
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