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Theorem clwwlknccat 29825
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.)
Assertion
Ref Expression
clwwlknccat ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π΄β€˜0) = (π΅β€˜0)) β†’ (𝐴 ++ 𝐡) ∈ ((𝑀 + 𝑁) ClWWalksN 𝐺))

Proof of Theorem clwwlknccat
StepHypRef Expression
1 isclwwlkn 29789 . . 3 (𝐴 ∈ (𝑀 ClWWalksN 𝐺) ↔ (𝐴 ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π΄) = 𝑀))
2 isclwwlkn 29789 . . 3 (𝐡 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝐡 ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π΅) = 𝑁))
3 biid 261 . . 3 ((π΄β€˜0) = (π΅β€˜0) ↔ (π΄β€˜0) = (π΅β€˜0))
4 simpl 482 . . . 4 ((𝐴 ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π΄) = 𝑀) β†’ 𝐴 ∈ (ClWWalksβ€˜πΊ))
5 simpl 482 . . . 4 ((𝐡 ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π΅) = 𝑁) β†’ 𝐡 ∈ (ClWWalksβ€˜πΊ))
6 id 22 . . . 4 ((π΄β€˜0) = (π΅β€˜0) β†’ (π΄β€˜0) = (π΅β€˜0))
7 clwwlkccat 29752 . . . 4 ((𝐴 ∈ (ClWWalksβ€˜πΊ) ∧ 𝐡 ∈ (ClWWalksβ€˜πΊ) ∧ (π΄β€˜0) = (π΅β€˜0)) β†’ (𝐴 ++ 𝐡) ∈ (ClWWalksβ€˜πΊ))
84, 5, 6, 7syl3an 1157 . . 3 (((𝐴 ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π΄) = 𝑀) ∧ (𝐡 ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π΅) = 𝑁) ∧ (π΄β€˜0) = (π΅β€˜0)) β†’ (𝐴 ++ 𝐡) ∈ (ClWWalksβ€˜πΊ))
91, 2, 3, 8syl3anb 1158 . 2 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π΄β€˜0) = (π΅β€˜0)) β†’ (𝐴 ++ 𝐡) ∈ (ClWWalksβ€˜πΊ))
10 eqid 2726 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
1110clwwlknwrd 29796 . . . . 5 (𝐴 ∈ (𝑀 ClWWalksN 𝐺) β†’ 𝐴 ∈ Word (Vtxβ€˜πΊ))
1210clwwlknwrd 29796 . . . . 5 (𝐡 ∈ (𝑁 ClWWalksN 𝐺) β†’ 𝐡 ∈ Word (Vtxβ€˜πΊ))
13 ccatlen 14531 . . . . 5 ((𝐴 ∈ Word (Vtxβ€˜πΊ) ∧ 𝐡 ∈ Word (Vtxβ€˜πΊ)) β†’ (β™―β€˜(𝐴 ++ 𝐡)) = ((β™―β€˜π΄) + (β™―β€˜π΅)))
1411, 12, 13syl2an 595 . . . 4 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (β™―β€˜(𝐴 ++ 𝐡)) = ((β™―β€˜π΄) + (β™―β€˜π΅)))
15 clwwlknlen 29794 . . . . 5 (𝐴 ∈ (𝑀 ClWWalksN 𝐺) β†’ (β™―β€˜π΄) = 𝑀)
16 clwwlknlen 29794 . . . . 5 (𝐡 ∈ (𝑁 ClWWalksN 𝐺) β†’ (β™―β€˜π΅) = 𝑁)
1715, 16oveqan12d 7424 . . . 4 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺)) β†’ ((β™―β€˜π΄) + (β™―β€˜π΅)) = (𝑀 + 𝑁))
1814, 17eqtrd 2766 . . 3 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (β™―β€˜(𝐴 ++ 𝐡)) = (𝑀 + 𝑁))
19183adant3 1129 . 2 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π΄β€˜0) = (π΅β€˜0)) β†’ (β™―β€˜(𝐴 ++ 𝐡)) = (𝑀 + 𝑁))
20 isclwwlkn 29789 . 2 ((𝐴 ++ 𝐡) ∈ ((𝑀 + 𝑁) ClWWalksN 𝐺) ↔ ((𝐴 ++ 𝐡) ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜(𝐴 ++ 𝐡)) = (𝑀 + 𝑁)))
219, 19, 20sylanbrc 582 1 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π΄β€˜0) = (π΅β€˜0)) β†’ (𝐴 ++ 𝐡) ∈ ((𝑀 + 𝑁) ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β€˜cfv 6537  (class class class)co 7405  0cc0 11112   + caddc 11115  β™―chash 14295  Word cword 14470   ++ cconcat 14526  Vtxcvtx 28764  ClWWalkscclwwlk 29743   ClWWalksN cclwwlkn 29786
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-rp 12981  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-lsw 14519  df-concat 14527  df-clwwlk 29744  df-clwwlkn 29787
This theorem is referenced by:  clwwlknonccat  29858
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