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Theorem clwwlkn2 29806
Description: A closed walk of length 2 represented as word is a word consisting of 2 symbols representing (not necessarily different) vertices connected by (at least) one edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Apr-2021.)
Assertion
Ref Expression
clwwlkn2 (π‘Š ∈ (2 ClWWalksN 𝐺) ↔ ((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)))
Proof of Theorem clwwlkn2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 2nn 12289 . . 3 2 ∈ β„•
2 eqid 2726 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 eqid 2726 . . . 4 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
42, 3isclwwlknx 29798 . . 3 (2 ∈ β„• β†’ (π‘Š ∈ (2 ClWWalksN 𝐺) ↔ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 2)))
51, 4ax-mp 5 . 2 (π‘Š ∈ (2 ClWWalksN 𝐺) ↔ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 2))
6 3anass 1092 . . . 4 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ↔ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ))))
7 oveq1 7412 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = 2 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (2 βˆ’ 1))
8 2m1e1 12342 . . . . . . . . . . . . 13 (2 βˆ’ 1) = 1
97, 8eqtrdi 2782 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) = 2 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = 1)
109oveq2d 7421 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = 2 β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) = (0..^1))
11 fzo01 13720 . . . . . . . . . . 11 (0..^1) = {0}
1210, 11eqtrdi 2782 . . . . . . . . . 10 ((β™―β€˜π‘Š) = 2 β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) = {0})
1312adantr 480 . . . . . . . . 9 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) = {0})
1413raleqdv 3319 . . . . . . . 8 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ {0} {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
15 c0ex 11212 . . . . . . . . 9 0 ∈ V
16 fveq2 6885 . . . . . . . . . . 11 (𝑖 = 0 β†’ (π‘Šβ€˜π‘–) = (π‘Šβ€˜0))
17 fv0p1e1 12339 . . . . . . . . . . 11 (𝑖 = 0 β†’ (π‘Šβ€˜(𝑖 + 1)) = (π‘Šβ€˜1))
1816, 17preq12d 4740 . . . . . . . . . 10 (𝑖 = 0 β†’ {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} = {(π‘Šβ€˜0), (π‘Šβ€˜1)})
1918eleq1d 2812 . . . . . . . . 9 (𝑖 = 0 β†’ ({(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)))
2015, 19ralsn 4680 . . . . . . . 8 (βˆ€π‘– ∈ {0} {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ))
2114, 20bitrdi 287 . . . . . . 7 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)))
22 prcom 4731 . . . . . . . . 9 {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} = {(π‘Šβ€˜0), (lastSβ€˜π‘Š)}
23 lsw 14520 . . . . . . . . . . 11 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
249fveq2d 6889 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = 2 β†’ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)) = (π‘Šβ€˜1))
2523, 24sylan9eqr 2788 . . . . . . . . . 10 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜1))
2625preq2d 4739 . . . . . . . . 9 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ {(π‘Šβ€˜0), (lastSβ€˜π‘Š)} = {(π‘Šβ€˜0), (π‘Šβ€˜1)})
2722, 26eqtrid 2778 . . . . . . . 8 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} = {(π‘Šβ€˜0), (π‘Šβ€˜1)})
2827eleq1d 2812 . . . . . . 7 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ) ↔ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)))
2921, 28anbi12d 630 . . . . . 6 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ↔ ({(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ))))
30 anidm 564 . . . . . 6 (({(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)) ↔ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ))
3129, 30bitrdi 287 . . . . 5 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ↔ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)))
3231pm5.32da 578 . . . 4 ((β™―β€˜π‘Š) = 2 β†’ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ))) ↔ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ))))
336, 32bitrid 283 . . 3 ((β™―β€˜π‘Š) = 2 β†’ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ↔ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ))))
3433pm5.32ri 575 . 2 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 2) ↔ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 2))
35 3anass 1092 . . 3 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)) ↔ ((β™―β€˜π‘Š) = 2 ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ))))
36 ancom 460 . . 3 (((β™―β€˜π‘Š) = 2 ∧ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ))) ↔ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 2))
3735, 36bitr2i 276 . 2 (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 2) ↔ ((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)))
385, 34, 373bitri 297 1 (π‘Š ∈ (2 ClWWalksN 𝐺) ↔ ((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ {(π‘Šβ€˜0), (π‘Šβ€˜1)} ∈ (Edgβ€˜πΊ)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {csn 4623  {cpr 4625  β€˜cfv 6537  (class class class)co 7405  0cc0 11112  1c1 11113   + caddc 11115   βˆ’ cmin 11448  β„•cn 12216  2c2 12271  ..^cfzo 13633  β™―chash 14295  Word cword 14470  lastSclsw 14518  Vtxcvtx 28764  Edgcedg 28815   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-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-2 12279  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-lsw 14519  df-clwwlk 29744  df-clwwlkn 29787
This theorem is referenced by:  clwwlknon2x  29865  2clwwlk2clwwlk  30112
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