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Theorem clwwlkn2 29882
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 12325 . . 3 2 ∈ β„•
2 eqid 2728 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
3 eqid 2728 . . . 4 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
42, 3isclwwlknx 29874 . . 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 7433 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = 2 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (2 βˆ’ 1))
8 2m1e1 12378 . . . . . . . . . . . . 13 (2 βˆ’ 1) = 1
97, 8eqtrdi 2784 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) = 2 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = 1)
109oveq2d 7442 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = 2 β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) = (0..^1))
11 fzo01 13756 . . . . . . . . . . 11 (0..^1) = {0}
1210, 11eqtrdi 2784 . . . . . . . . . 10 ((β™―β€˜π‘Š) = 2 β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) = {0})
1312adantr 479 . . . . . . . . 9 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ (0..^((β™―β€˜π‘Š) βˆ’ 1)) = {0})
1413raleqdv 3323 . . . . . . . 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 11248 . . . . . . . . 9 0 ∈ V
16 fveq2 6902 . . . . . . . . . . 11 (𝑖 = 0 β†’ (π‘Šβ€˜π‘–) = (π‘Šβ€˜0))
17 fv0p1e1 12375 . . . . . . . . . . 11 (𝑖 = 0 β†’ (π‘Šβ€˜(𝑖 + 1)) = (π‘Šβ€˜1))
1816, 17preq12d 4750 . . . . . . . . . 10 (𝑖 = 0 β†’ {(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} = {(π‘Šβ€˜0), (π‘Šβ€˜1)})
1918eleq1d 2814 . . . . . . . . 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 4690 . . . . . . . 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 286 . . . . . . 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 4741 . . . . . . . . 9 {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} = {(π‘Šβ€˜0), (lastSβ€˜π‘Š)}
23 lsw 14556 . . . . . . . . . . 11 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
249fveq2d 6906 . . . . . . . . . . 11 ((β™―β€˜π‘Š) = 2 β†’ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)) = (π‘Šβ€˜1))
2523, 24sylan9eqr 2790 . . . . . . . . . 10 (((β™―β€˜π‘Š) = 2 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ)) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜1))
2625preq2d 4749 . . . . . . . . 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 2780 . . . . . . . 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 2814 . . . . . . 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 563 . . . . . 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 286 . . . . 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 577 . . . 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 282 . . 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 574 . 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 459 . . 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 275 . 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 296 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {csn 4632  {cpr 4634  β€˜cfv 6553  (class class class)co 7426  0cc0 11148  1c1 11149   + caddc 11151   βˆ’ cmin 11484  β„•cn 12252  2c2 12307  ..^cfzo 13669  β™―chash 14331  Word cword 14506  lastSclsw 14554  Vtxcvtx 28837  Edgcedg 28888   ClWWalksN cclwwlkn 29862
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-map 8855  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-card 9972  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-n0 12513  df-xnn0 12585  df-z 12599  df-uz 12863  df-fz 13527  df-fzo 13670  df-hash 14332  df-word 14507  df-lsw 14555  df-clwwlk 29820  df-clwwlkn 29863
This theorem is referenced by:  clwwlknon2x  29941  2clwwlk2clwwlk  30188
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