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Theorem clwlkclwwlkf 29528
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 2730 . . . . 5 (1st β€˜π‘) = (1st β€˜π‘)
3 eqid 2730 . . . . 5 (2nd β€˜π‘) = (2nd β€˜π‘)
41, 2, 3clwlkclwwlkflem 29524 . . . 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 29297 . . . . . . . 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 6903 . . . . . . . . 9 (1st β€˜π‘) ∈ V
7 breq1 5150 . . . . . . . . 9 (𝑓 = (1st β€˜π‘) β†’ (𝑓(ClWalksβ€˜πΊ)(2nd β€˜π‘) ↔ (1st β€˜π‘)(ClWalksβ€˜πΊ)(2nd β€˜π‘)))
86, 7spcev 3595 . . . . . . . 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 1130 . . . . . 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 480 . . . . 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 481 . . . . . 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 2730 . . . . . . . . 9 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
1413wlkpwrd 29141 . . . . . . . 8 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
15143ad2ant1 1131 . . . . . . 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 480 . . . . . 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 12521 . . . . . . . . . 10 ((β™―β€˜(1st β€˜π‘)) ∈ β„• ↔ ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 ∧ 1 ≀ (β™―β€˜(1st β€˜π‘))))
18 nn0re 12485 . . . . . . . . . . . . . 14 ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 β†’ (β™―β€˜(1st β€˜π‘)) ∈ ℝ)
19 1e2m1 12343 . . . . . . . . . . . . . . . . 17 1 = (2 βˆ’ 1)
2019breq1i 5154 . . . . . . . . . . . . . . . 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 12290 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
23 1re 11218 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
24 lesubadd 11690 . . . . . . . . . . . . . . . 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 1449 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 29142 . . . . . . . . . . . . . 14 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1))
3029adantr 479 . . . . . . . . . . . . 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 5159 . . . . . . . . . . . 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 258 . . . . . . . . . . 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 452 . . . . . . . . . 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 1109 . . . . . . 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 480 . . . . . 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 2730 . . . . . . 7 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
3913, 38clwlkclwwlk 29522 . . . . . 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 1369 . . . . 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 591 . . 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 494 . 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 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  {crab 3430   class class class wbr 5147   ↦ cmpt 5230  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   ≀ cle 11253   βˆ’ cmin 11448  β„•cn 12216  2c2 12271  β„•0cn0 12476  β™―chash 14294  Word cword 14468  lastSclsw 14516   prefix cpfx 14624  Vtxcvtx 28523  iEdgciedg 28524  USPGraphcuspgr 28675  Walkscwlks 29120  ClWalkscclwlks 29294  ClWWalkscclwwlk 29501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  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 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  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 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-lsw 14517  df-substr 14595  df-pfx 14625  df-edg 28575  df-uhgr 28585  df-upgr 28609  df-uspgr 28677  df-wlks 29123  df-clwlks 29295  df-clwwlk 29502
This theorem is referenced by:  clwlkclwwlkfo  29529  clwlkclwwlkf1  29530
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