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Theorem clwlkclwwlkf 29001
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 28997 . . . 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 28770 . . . . . . . 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 6859 . . . . . . . . 9 (1st β€˜π‘) ∈ V
7 breq1 5112 . . . . . . . . 9 (𝑓 = (1st β€˜π‘) β†’ (𝑓(ClWalksβ€˜πΊ)(2nd β€˜π‘) ↔ (1st β€˜π‘)(ClWalksβ€˜πΊ)(2nd β€˜π‘)))
86, 7spcev 3567 . . . . . . . 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 28614 . . . . . . . 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 12466 . . . . . . . . . 10 ((β™―β€˜(1st β€˜π‘)) ∈ β„• ↔ ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 ∧ 1 ≀ (β™―β€˜(1st β€˜π‘))))
18 nn0re 12430 . . . . . . . . . . . . . 14 ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 β†’ (β™―β€˜(1st β€˜π‘)) ∈ ℝ)
19 1e2m1 12288 . . . . . . . . . . . . . . . . 17 1 = (2 βˆ’ 1)
2019breq1i 5116 . . . . . . . . . . . . . . . 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 12235 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
23 1re 11163 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
24 lesubadd 11635 . . . . . . . . . . . . . . . 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 28615 . . . . . . . . . . . . . 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 5121 . . . . . . . . . . . 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 28995 . . . . . 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 7066 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 3406   class class class wbr 5109   ↦ cmpt 5192  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  β„cr 11058  0cc0 11059  1c1 11060   + caddc 11062   ≀ cle 11198   βˆ’ cmin 11393  β„•cn 12161  2c2 12216  β„•0cn0 12421  β™―chash 14239  Word cword 14411  lastSclsw 14459   prefix cpfx 14567  Vtxcvtx 27996  iEdgciedg 27997  USPGraphcuspgr 28148  Walkscwlks 28593  ClWalkscclwlks 28767  ClWWalkscclwwlk 28974
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-oadd 8420  df-er 8654  df-map 8773  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-dju 9845  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-n0 12422  df-xnn0 12494  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-lsw 14460  df-substr 14538  df-pfx 14568  df-edg 28048  df-uhgr 28058  df-upgr 28082  df-uspgr 28150  df-wlks 28596  df-clwlks 28768  df-clwwlk 28975
This theorem is referenced by:  clwlkclwwlkfo  29002  clwlkclwwlkf1  29003
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