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Theorem clwlkclwwlkf 27396
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 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
clwlkclwwlkf.f 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
Assertion
Ref Expression
clwlkclwwlkf (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
Distinct variable groups:   𝑤,𝐺,𝑐   𝐶,𝑐
Allowed substitution hints:   𝐶(𝑤)   𝐹(𝑤,𝑐)

Proof of Theorem clwlkclwwlkf
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 clwlkclwwlkf.c . . . . 5 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
2 eqid 2778 . . . . 5 (1st𝑐) = (1st𝑐)
3 eqid 2778 . . . . 5 (2nd𝑐) = (2nd𝑐)
41, 2, 3clwlkclwwlkflem 27386 . . . 4 (𝑐𝐶 → ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ))
5 isclwlk 27125 . . . . . . . 8 ((1st𝑐)(ClWalks‘𝐺)(2nd𝑐) ↔ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐)))))
6 fvex 6459 . . . . . . . . 9 (1st𝑐) ∈ V
7 breq1 4889 . . . . . . . . 9 (𝑓 = (1st𝑐) → (𝑓(ClWalks‘𝐺)(2nd𝑐) ↔ (1st𝑐)(ClWalks‘𝐺)(2nd𝑐)))
86, 7spcev 3502 . . . . . . . 8 ((1st𝑐)(ClWalks‘𝐺)(2nd𝑐) → ∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐))
95, 8sylbir 227 . . . . . . 7 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐)))) → ∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐))
1093adant3 1123 . . . . . 6 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ) → ∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐))
1110adantl 475 . . . . 5 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → ∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐))
12 simpl 476 . . . . . 6 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → 𝐺 ∈ USPGraph)
13 eqid 2778 . . . . . . . . 9 (Vtx‘𝐺) = (Vtx‘𝐺)
1413wlkpwrd 26965 . . . . . . . 8 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
15143ad2ant1 1124 . . . . . . 7 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
1615adantl 475 . . . . . 6 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
17 elnnnn0c 11689 . . . . . . . . . 10 ((♯‘(1st𝑐)) ∈ ℕ ↔ ((♯‘(1st𝑐)) ∈ ℕ0 ∧ 1 ≤ (♯‘(1st𝑐))))
18 nn0re 11652 . . . . . . . . . . . . . 14 ((♯‘(1st𝑐)) ∈ ℕ0 → (♯‘(1st𝑐)) ∈ ℝ)
19 1e2m1 11509 . . . . . . . . . . . . . . . . 17 1 = (2 − 1)
2019breq1i 4893 . . . . . . . . . . . . . . . 16 (1 ≤ (♯‘(1st𝑐)) ↔ (2 − 1) ≤ (♯‘(1st𝑐)))
2120biimpi 208 . . . . . . . . . . . . . . 15 (1 ≤ (♯‘(1st𝑐)) → (2 − 1) ≤ (♯‘(1st𝑐)))
22 2re 11449 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
23 1re 10376 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
24 lesubadd 10847 . . . . . . . . . . . . . . . 16 ((2 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘(1st𝑐)) ∈ ℝ) → ((2 − 1) ≤ (♯‘(1st𝑐)) ↔ 2 ≤ ((♯‘(1st𝑐)) + 1)))
2522, 23, 24mp3an12 1524 . . . . . . . . . . . . . . 15 ((♯‘(1st𝑐)) ∈ ℝ → ((2 − 1) ≤ (♯‘(1st𝑐)) ↔ 2 ≤ ((♯‘(1st𝑐)) + 1)))
2621, 25syl5ib 236 . . . . . . . . . . . . . 14 ((♯‘(1st𝑐)) ∈ ℝ → (1 ≤ (♯‘(1st𝑐)) → 2 ≤ ((♯‘(1st𝑐)) + 1)))
2718, 26syl 17 . . . . . . . . . . . . 13 ((♯‘(1st𝑐)) ∈ ℕ0 → (1 ≤ (♯‘(1st𝑐)) → 2 ≤ ((♯‘(1st𝑐)) + 1)))
2827adantl 475 . . . . . . . . . . . 12 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) ∈ ℕ0) → (1 ≤ (♯‘(1st𝑐)) → 2 ≤ ((♯‘(1st𝑐)) + 1)))
29 wlklenvp1 26966 . . . . . . . . . . . . . 14 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(2nd𝑐)) = ((♯‘(1st𝑐)) + 1))
3029adantr 474 . . . . . . . . . . . . 13 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) ∈ ℕ0) → (♯‘(2nd𝑐)) = ((♯‘(1st𝑐)) + 1))
3130breq2d 4898 . . . . . . . . . . . 12 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) ∈ ℕ0) → (2 ≤ (♯‘(2nd𝑐)) ↔ 2 ≤ ((♯‘(1st𝑐)) + 1)))
3228, 31sylibrd 251 . . . . . . . . . . 11 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) ∈ ℕ0) → (1 ≤ (♯‘(1st𝑐)) → 2 ≤ (♯‘(2nd𝑐))))
3332expimpd 447 . . . . . . . . . 10 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (((♯‘(1st𝑐)) ∈ ℕ0 ∧ 1 ≤ (♯‘(1st𝑐))) → 2 ≤ (♯‘(2nd𝑐))))
3417, 33syl5bi 234 . . . . . . . . 9 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) ∈ ℕ → 2 ≤ (♯‘(2nd𝑐))))
3534a1d 25 . . . . . . . 8 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) → ((♯‘(1st𝑐)) ∈ ℕ → 2 ≤ (♯‘(2nd𝑐)))))
36353imp 1098 . . . . . . 7 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ) → 2 ≤ (♯‘(2nd𝑐)))
3736adantl 475 . . . . . 6 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → 2 ≤ (♯‘(2nd𝑐)))
38 eqid 2778 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
3913, 38clwlkclwwlk 27382 . . . . . 6 ((𝐺 ∈ USPGraph ∧ (2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ 2 ≤ (♯‘(2nd𝑐))) → (∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐) ↔ ((lastS‘(2nd𝑐)) = ((2nd𝑐)‘0) ∧ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) ∈ (ClWWalks‘𝐺))))
4012, 16, 37, 39syl3anc 1439 . . . . 5 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → (∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐) ↔ ((lastS‘(2nd𝑐)) = ((2nd𝑐)‘0) ∧ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) ∈ (ClWWalks‘𝐺))))
4111, 40mpbid 224 . . . 4 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → ((lastS‘(2nd𝑐)) = ((2nd𝑐)‘0) ∧ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) ∈ (ClWWalks‘𝐺)))
424, 41sylan2 586 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑐𝐶) → ((lastS‘(2nd𝑐)) = ((2nd𝑐)‘0) ∧ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) ∈ (ClWWalks‘𝐺)))
4342simprd 491 . 2 ((𝐺 ∈ USPGraph ∧ 𝑐𝐶) → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) ∈ (ClWWalks‘𝐺))
44 clwlkclwwlkf.f . 2 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
4543, 44fmptd 6648 1 (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wex 1823  wcel 2107  {crab 3094   class class class wbr 4886  cmpt 4965  wf 6131  cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  cr 10271  0cc0 10272  1c1 10273   + caddc 10275  cle 10412  cmin 10606  cn 11374  2c2 11430  0cn0 11642  chash 13435  Word cword 13599  lastSclsw 13652   prefix cpfx 13779  Vtxcvtx 26344  iEdgciedg 26345  USPGraphcuspgr 26497  Walkscwlks 26944  ClWalkscclwlks 27122  ClWWalkscclwwlk 27361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ifp 1047  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-lsw 13653  df-substr 13731  df-pfx 13780  df-edg 26396  df-uhgr 26406  df-upgr 26430  df-uspgr 26499  df-wlks 26947  df-clwlks 27123  df-clwwlk 27362
This theorem is referenced by:  clwlkclwwlkfo  27397  clwlkclwwlkf1  27398
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