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Theorem clwwlkn2 29297
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 12285 . . 3 2 ∈ β„•
2 eqid 2733 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 eqid 2733 . . . 4 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
42, 3isclwwlknx 29289 . . 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 1096 . . . 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 7416 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = 2 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (2 βˆ’ 1))
8 2m1e1 12338 . . . . . . . . . . . . 13 (2 βˆ’ 1) = 1
97, 8eqtrdi 2789 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) = 2 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = 1)
109oveq2d 7425 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = 2 β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) = (0..^1))
11 fzo01 13714 . . . . . . . . . . 11 (0..^1) = {0}
1210, 11eqtrdi 2789 . . . . . . . . . 10 ((β™―β€˜π‘Š) = 2 β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) = {0})
1312adantr 482 . . . . . . . . 9 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) = {0})
1413raleqdv 3326 . . . . . . . 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 11208 . . . . . . . . 9 0 ∈ V
16 fveq2 6892 . . . . . . . . . . 11 (𝑖 = 0 β†’ (π‘Šβ€˜π‘–) = (π‘Šβ€˜0))
17 fv0p1e1 12335 . . . . . . . . . . 11 (𝑖 = 0 β†’ (π‘Šβ€˜(𝑖 + 1)) = (π‘Šβ€˜1))
1816, 17preq12d 4746 . . . . . . . . . 10 (𝑖 = 0 β†’ {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} = {(π‘Šβ€˜0), (π‘Šβ€˜1)})
1918eleq1d 2819 . . . . . . . . 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 4686 . . . . . . . 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 4737 . . . . . . . . 9 {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} = {(π‘Šβ€˜0), (lastSβ€˜π‘Š)}
23 lsw 14514 . . . . . . . . . . 11 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
249fveq2d 6896 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = 2 β†’ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)) = (π‘Šβ€˜1))
2523, 24sylan9eqr 2795 . . . . . . . . . 10 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜1))
2625preq2d 4745 . . . . . . . . 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 2785 . . . . . . . 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 2819 . . . . . . 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 632 . . . . . 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 566 . . . . . 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 580 . . . 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 577 . 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 1096 . . 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 462 . . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {csn 4629  {cpr 4631  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113   βˆ’ cmin 11444  β„•cn 12212  2c2 12267  ..^cfzo 13627  β™―chash 14290  Word cword 14464  lastSclsw 14512  Vtxcvtx 28256  Edgcedg 28307   ClWWalksN cclwwlkn 29277
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-clwwlk 29235  df-clwwlkn 29278
This theorem is referenced by:  clwwlknon2x  29356  2clwwlk2clwwlk  29603
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