MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwlkclwwlkf Structured version   Visualization version   GIF version

Theorem clwlkclwwlkf 29261
Description: 𝐹 is a function from the nonempty closed walks into the closed walks as word in a simple pseudograph. (Contributed by AV, 23-May-2022.) (Revised by AV, 29-Oct-2022.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}
clwlkclwwlkf.f 𝐹 = (𝑐 ∈ 𝐢 ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)))
Assertion
Ref Expression
clwlkclwwlkf (𝐺 ∈ USPGraph β†’ 𝐹:𝐢⟢(ClWWalksβ€˜πΊ))
Distinct variable groups:   𝑀,𝐺,𝑐   𝐢,𝑐
Allowed substitution hints:   𝐢(𝑀)   𝐹(𝑀,𝑐)

Proof of Theorem clwlkclwwlkf
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 clwlkclwwlkf.c . . . . 5 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}
2 eqid 2733 . . . . 5 (1st β€˜π‘) = (1st β€˜π‘)
3 eqid 2733 . . . . 5 (2nd β€˜π‘) = (2nd β€˜π‘)
41, 2, 3clwlkclwwlkflem 29257 . . . 4 (𝑐 ∈ 𝐢 β†’ ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•))
5 isclwlk 29030 . . . . . . . 8 ((1st β€˜π‘)(ClWalksβ€˜πΊ)(2nd β€˜π‘) ↔ ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘)))))
6 fvex 6905 . . . . . . . . 9 (1st β€˜π‘) ∈ V
7 breq1 5152 . . . . . . . . 9 (𝑓 = (1st β€˜π‘) β†’ (𝑓(ClWalksβ€˜πΊ)(2nd β€˜π‘) ↔ (1st β€˜π‘)(ClWalksβ€˜πΊ)(2nd β€˜π‘)))
86, 7spcev 3597 . . . . . . . 8 ((1st β€˜π‘)(ClWalksβ€˜πΊ)(2nd β€˜π‘) β†’ βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)(2nd β€˜π‘))
95, 8sylbir 234 . . . . . . 7 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘)))) β†’ βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)(2nd β€˜π‘))
1093adant3 1133 . . . . . 6 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•) β†’ βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)(2nd β€˜π‘))
1110adantl 483 . . . . 5 ((𝐺 ∈ USPGraph ∧ ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•)) β†’ βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)(2nd β€˜π‘))
12 simpl 484 . . . . . 6 ((𝐺 ∈ USPGraph ∧ ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•)) β†’ 𝐺 ∈ USPGraph)
13 eqid 2733 . . . . . . . . 9 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
1413wlkpwrd 28874 . . . . . . . 8 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
15143ad2ant1 1134 . . . . . . 7 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
1615adantl 483 . . . . . 6 ((𝐺 ∈ USPGraph ∧ ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•)) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
17 elnnnn0c 12517 . . . . . . . . . 10 ((β™―β€˜(1st β€˜π‘)) ∈ β„• ↔ ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 ∧ 1 ≀ (β™―β€˜(1st β€˜π‘))))
18 nn0re 12481 . . . . . . . . . . . . . 14 ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 β†’ (β™―β€˜(1st β€˜π‘)) ∈ ℝ)
19 1e2m1 12339 . . . . . . . . . . . . . . . . 17 1 = (2 βˆ’ 1)
2019breq1i 5156 . . . . . . . . . . . . . . . 16 (1 ≀ (β™―β€˜(1st β€˜π‘)) ↔ (2 βˆ’ 1) ≀ (β™―β€˜(1st β€˜π‘)))
2120biimpi 215 . . . . . . . . . . . . . . 15 (1 ≀ (β™―β€˜(1st β€˜π‘)) β†’ (2 βˆ’ 1) ≀ (β™―β€˜(1st β€˜π‘)))
22 2re 12286 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
23 1re 11214 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
24 lesubadd 11686 . . . . . . . . . . . . . . . 16 ((2 ∈ ℝ ∧ 1 ∈ ℝ ∧ (β™―β€˜(1st β€˜π‘)) ∈ ℝ) β†’ ((2 βˆ’ 1) ≀ (β™―β€˜(1st β€˜π‘)) ↔ 2 ≀ ((β™―β€˜(1st β€˜π‘)) + 1)))
2522, 23, 24mp3an12 1452 . . . . . . . . . . . . . . 15 ((β™―β€˜(1st β€˜π‘)) ∈ ℝ β†’ ((2 βˆ’ 1) ≀ (β™―β€˜(1st β€˜π‘)) ↔ 2 ≀ ((β™―β€˜(1st β€˜π‘)) + 1)))
2621, 25imbitrid 243 . . . . . . . . . . . . . 14 ((β™―β€˜(1st β€˜π‘)) ∈ ℝ β†’ (1 ≀ (β™―β€˜(1st β€˜π‘)) β†’ 2 ≀ ((β™―β€˜(1st β€˜π‘)) + 1)))
2718, 26syl 17 . . . . . . . . . . . . 13 ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 β†’ (1 ≀ (β™―β€˜(1st β€˜π‘)) β†’ 2 ≀ ((β™―β€˜(1st β€˜π‘)) + 1)))
2827adantl 483 . . . . . . . . . . . 12 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•0) β†’ (1 ≀ (β™―β€˜(1st β€˜π‘)) β†’ 2 ≀ ((β™―β€˜(1st β€˜π‘)) + 1)))
29 wlklenvp1 28875 . . . . . . . . . . . . . 14 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1))
3029adantr 482 . . . . . . . . . . . . 13 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•0) β†’ (β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1))
3130breq2d 5161 . . . . . . . . . . . 12 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•0) β†’ (2 ≀ (β™―β€˜(2nd β€˜π‘)) ↔ 2 ≀ ((β™―β€˜(1st β€˜π‘)) + 1)))
3228, 31sylibrd 259 . . . . . . . . . . 11 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•0) β†’ (1 ≀ (β™―β€˜(1st β€˜π‘)) β†’ 2 ≀ (β™―β€˜(2nd β€˜π‘))))
3332expimpd 455 . . . . . . . . . 10 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (((β™―β€˜(1st β€˜π‘)) ∈ β„•0 ∧ 1 ≀ (β™―β€˜(1st β€˜π‘))) β†’ 2 ≀ (β™―β€˜(2nd β€˜π‘))))
3417, 33biimtrid 241 . . . . . . . . 9 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ ((β™―β€˜(1st β€˜π‘)) ∈ β„• β†’ 2 ≀ (β™―β€˜(2nd β€˜π‘))))
3534a1d 25 . . . . . . . 8 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) β†’ ((β™―β€˜(1st β€˜π‘)) ∈ β„• β†’ 2 ≀ (β™―β€˜(2nd β€˜π‘)))))
36353imp 1112 . . . . . . 7 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•) β†’ 2 ≀ (β™―β€˜(2nd β€˜π‘)))
3736adantl 483 . . . . . 6 ((𝐺 ∈ USPGraph ∧ ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•)) β†’ 2 ≀ (β™―β€˜(2nd β€˜π‘)))
38 eqid 2733 . . . . . . 7 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
3913, 38clwlkclwwlk 29255 . . . . . 6 ((𝐺 ∈ USPGraph ∧ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ 2 ≀ (β™―β€˜(2nd β€˜π‘))) β†’ (βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)(2nd β€˜π‘) ↔ ((lastSβ€˜(2nd β€˜π‘)) = ((2nd β€˜π‘)β€˜0) ∧ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ))))
4012, 16, 37, 39syl3anc 1372 . . . . 5 ((𝐺 ∈ USPGraph ∧ ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•)) β†’ (βˆƒπ‘“ 𝑓(ClWalksβ€˜πΊ)(2nd β€˜π‘) ↔ ((lastSβ€˜(2nd β€˜π‘)) = ((2nd β€˜π‘)β€˜0) ∧ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ))))
4111, 40mpbid 231 . . . 4 ((𝐺 ∈ USPGraph ∧ ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ ((2nd β€˜π‘)β€˜0) = ((2nd β€˜π‘)β€˜(β™―β€˜(1st β€˜π‘))) ∧ (β™―β€˜(1st β€˜π‘)) ∈ β„•)) β†’ ((lastSβ€˜(2nd β€˜π‘)) = ((2nd β€˜π‘)β€˜0) ∧ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ)))
424, 41sylan2 594 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑐 ∈ 𝐢) β†’ ((lastSβ€˜(2nd β€˜π‘)) = ((2nd β€˜π‘)β€˜0) ∧ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ)))
4342simprd 497 . 2 ((𝐺 ∈ USPGraph ∧ 𝑐 ∈ 𝐢) β†’ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)) ∈ (ClWWalksβ€˜πΊ))
44 clwlkclwwlkf.f . 2 𝐹 = (𝑐 ∈ 𝐢 ↦ ((2nd β€˜π‘) prefix ((β™―β€˜(2nd β€˜π‘)) βˆ’ 1)))
4543, 44fmptd 7114 1 (𝐺 ∈ USPGraph β†’ 𝐹:𝐢⟢(ClWWalksβ€˜πΊ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {crab 3433   class class class wbr 5149   ↦ cmpt 5232  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   ≀ cle 11249   βˆ’ cmin 11444  β„•cn 12212  2c2 12267  β„•0cn0 12472  β™―chash 14290  Word cword 14464  lastSclsw 14512   prefix cpfx 14620  Vtxcvtx 28256  iEdgciedg 28257  USPGraphcuspgr 28408  Walkscwlks 28853  ClWalkscclwlks 29027  ClWWalkscclwwlk 29234
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-ifp 1063  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-2o 8467  df-oadd 8470  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  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-substr 14591  df-pfx 14621  df-edg 28308  df-uhgr 28318  df-upgr 28342  df-uspgr 28410  df-wlks 28856  df-clwlks 29028  df-clwwlk 29235
This theorem is referenced by:  clwlkclwwlkfo  29262  clwlkclwwlkf1  29263
  Copyright terms: Public domain W3C validator