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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  ℝcr 10271  0cc0 10272  1c1 10273   + caddc 10275   ≤ cle 10412   − cmin 10606  ℕcn 11374  2c2 11430  ℕ0cn0 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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