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Theorem clwwlknccat 29056
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 29020 . . 3 (𝐴 ∈ (𝑀 ClWWalksN 𝐺) ↔ (𝐴 ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π΄) = 𝑀))
2 isclwwlkn 29020 . . 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 484 . . . 4 ((𝐴 ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π΄) = 𝑀) β†’ 𝐴 ∈ (ClWWalksβ€˜πΊ))
5 simpl 484 . . . 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 28983 . . . 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 1161 . . 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 1162 . 2 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π΄β€˜0) = (π΅β€˜0)) β†’ (𝐴 ++ 𝐡) ∈ (ClWWalksβ€˜πΊ))
10 eqid 2733 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
1110clwwlknwrd 29027 . . . . 5 (𝐴 ∈ (𝑀 ClWWalksN 𝐺) β†’ 𝐴 ∈ Word (Vtxβ€˜πΊ))
1210clwwlknwrd 29027 . . . . 5 (𝐡 ∈ (𝑁 ClWWalksN 𝐺) β†’ 𝐡 ∈ Word (Vtxβ€˜πΊ))
13 ccatlen 14472 . . . . 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 597 . . . 4 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (β™―β€˜(𝐴 ++ 𝐡)) = ((β™―β€˜π΄) + (β™―β€˜π΅)))
15 clwwlknlen 29025 . . . . 5 (𝐴 ∈ (𝑀 ClWWalksN 𝐺) β†’ (β™―β€˜π΄) = 𝑀)
16 clwwlknlen 29025 . . . . 5 (𝐡 ∈ (𝑁 ClWWalksN 𝐺) β†’ (β™―β€˜π΅) = 𝑁)
1715, 16oveqan12d 7380 . . . 4 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺)) β†’ ((β™―β€˜π΄) + (β™―β€˜π΅)) = (𝑀 + 𝑁))
1814, 17eqtrd 2773 . . 3 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (β™―β€˜(𝐴 ++ 𝐡)) = (𝑀 + 𝑁))
19183adant3 1133 . 2 ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐡 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π΄β€˜0) = (π΅β€˜0)) β†’ (β™―β€˜(𝐴 ++ 𝐡)) = (𝑀 + 𝑁))
20 isclwwlkn 29020 . 2 ((𝐴 ++ 𝐡) ∈ ((𝑀 + 𝑁) ClWWalksN 𝐺) ↔ ((𝐴 ++ 𝐡) ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜(𝐴 ++ 𝐡)) = (𝑀 + 𝑁)))
219, 19, 20sylanbrc 584 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  β€˜cfv 6500  (class class class)co 7361  0cc0 11059   + caddc 11062  β™―chash 14239  Word cword 14411   ++ cconcat 14467  Vtxcvtx 27996  ClWWalkscclwwlk 28974   ClWWalksN cclwwlkn 29017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-oadd 8420  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-xnn0 12494  df-z 12508  df-uz 12772  df-rp 12924  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-lsw 14460  df-concat 14468  df-clwwlk 28975  df-clwwlkn 29018
This theorem is referenced by:  clwwlknonccat  29089
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