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Theorem clwlkclwwlkf 27162
 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.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
clwlkclwwlkf.f 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(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 2769 . . . . 5 (1st𝑐) = (1st𝑐)
3 eqid 2769 . . . . 5 (2nd𝑐) = (2nd𝑐)
41, 2, 3clwlkclwwlkflem 27158 . . . 4 (𝑐𝐶 → ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ))
5 isclwlk 26910 . . . . . . . 8 ((1st𝑐)(ClWalks‘𝐺)(2nd𝑐) ↔ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐)))))
6 fvex 6341 . . . . . . . . . 10 (1st𝑐) ∈ V
76jctl 566 . . . . . . . . 9 ((1st𝑐)(ClWalks‘𝐺)(2nd𝑐) → ((1st𝑐) ∈ V ∧ (1st𝑐)(ClWalks‘𝐺)(2nd𝑐)))
8 breq1 4786 . . . . . . . . . 10 (𝑓 = (1st𝑐) → (𝑓(ClWalks‘𝐺)(2nd𝑐) ↔ (1st𝑐)(ClWalks‘𝐺)(2nd𝑐)))
98rspcev 3457 . . . . . . . . 9 (((1st𝑐) ∈ V ∧ (1st𝑐)(ClWalks‘𝐺)(2nd𝑐)) → ∃𝑓 ∈ V 𝑓(ClWalks‘𝐺)(2nd𝑐))
10 rexex 3148 . . . . . . . . 9 (∃𝑓 ∈ V 𝑓(ClWalks‘𝐺)(2nd𝑐) → ∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐))
117, 9, 103syl 18 . . . . . . . 8 ((1st𝑐)(ClWalks‘𝐺)(2nd𝑐) → ∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐))
125, 11sylbir 225 . . . . . . 7 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐)))) → ∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐))
13123adant3 1124 . . . . . 6 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ) → ∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐))
1413adantl 474 . . . . 5 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → ∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐))
15 simpl 475 . . . . . 6 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → 𝐺 ∈ USPGraph)
16 eqid 2769 . . . . . . . . 9 (Vtx‘𝐺) = (Vtx‘𝐺)
1716wlkpwrd 26754 . . . . . . . 8 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
18173ad2ant1 1125 . . . . . . 7 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
1918adantl 474 . . . . . 6 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
20 elnnnn0c 11538 . . . . . . . . . 10 ((♯‘(1st𝑐)) ∈ ℕ ↔ ((♯‘(1st𝑐)) ∈ ℕ0 ∧ 1 ≤ (♯‘(1st𝑐))))
21 nn0re 11501 . . . . . . . . . . . . . 14 ((♯‘(1st𝑐)) ∈ ℕ0 → (♯‘(1st𝑐)) ∈ ℝ)
22 1e2m1 11336 . . . . . . . . . . . . . . . . 17 1 = (2 − 1)
2322breq1i 4790 . . . . . . . . . . . . . . . 16 (1 ≤ (♯‘(1st𝑐)) ↔ (2 − 1) ≤ (♯‘(1st𝑐)))
2423biimpi 206 . . . . . . . . . . . . . . 15 (1 ≤ (♯‘(1st𝑐)) → (2 − 1) ≤ (♯‘(1st𝑐)))
25 2re 11290 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
26 1re 10239 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
27 lesubadd 10700 . . . . . . . . . . . . . . . 16 ((2 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘(1st𝑐)) ∈ ℝ) → ((2 − 1) ≤ (♯‘(1st𝑐)) ↔ 2 ≤ ((♯‘(1st𝑐)) + 1)))
2825, 26, 27mp3an12 1560 . . . . . . . . . . . . . . 15 ((♯‘(1st𝑐)) ∈ ℝ → ((2 − 1) ≤ (♯‘(1st𝑐)) ↔ 2 ≤ ((♯‘(1st𝑐)) + 1)))
2924, 28syl5ib 234 . . . . . . . . . . . . . 14 ((♯‘(1st𝑐)) ∈ ℝ → (1 ≤ (♯‘(1st𝑐)) → 2 ≤ ((♯‘(1st𝑐)) + 1)))
3021, 29syl 17 . . . . . . . . . . . . 13 ((♯‘(1st𝑐)) ∈ ℕ0 → (1 ≤ (♯‘(1st𝑐)) → 2 ≤ ((♯‘(1st𝑐)) + 1)))
3130adantl 474 . . . . . . . . . . . 12 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) ∈ ℕ0) → (1 ≤ (♯‘(1st𝑐)) → 2 ≤ ((♯‘(1st𝑐)) + 1)))
32 wlklenvp1 26755 . . . . . . . . . . . . . 14 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(2nd𝑐)) = ((♯‘(1st𝑐)) + 1))
3332adantr 473 . . . . . . . . . . . . 13 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) ∈ ℕ0) → (♯‘(2nd𝑐)) = ((♯‘(1st𝑐)) + 1))
3433breq2d 4795 . . . . . . . . . . . 12 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) ∈ ℕ0) → (2 ≤ (♯‘(2nd𝑐)) ↔ 2 ≤ ((♯‘(1st𝑐)) + 1)))
3531, 34sylibrd 249 . . . . . . . . . . 11 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) ∈ ℕ0) → (1 ≤ (♯‘(1st𝑐)) → 2 ≤ (♯‘(2nd𝑐))))
3635expimpd 629 . . . . . . . . . 10 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (((♯‘(1st𝑐)) ∈ ℕ0 ∧ 1 ≤ (♯‘(1st𝑐))) → 2 ≤ (♯‘(2nd𝑐))))
3720, 36syl5bi 232 . . . . . . . . 9 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) ∈ ℕ → 2 ≤ (♯‘(2nd𝑐))))
3837a1d 25 . . . . . . . 8 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) → ((♯‘(1st𝑐)) ∈ ℕ → 2 ≤ (♯‘(2nd𝑐)))))
39383imp 1099 . . . . . . 7 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ) → 2 ≤ (♯‘(2nd𝑐)))
4039adantl 474 . . . . . 6 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → 2 ≤ (♯‘(2nd𝑐)))
41 eqid 2769 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
4216, 41clwlkclwwlk 27156 . . . . . 6 ((𝐺 ∈ USPGraph ∧ (2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ 2 ≤ (♯‘(2nd𝑐))) → (∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐) ↔ ((lastS‘(2nd𝑐)) = ((2nd𝑐)‘0) ∧ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) ∈ (ClWWalks‘𝐺))))
4315, 19, 40, 42syl3anc 1474 . . . . 5 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → (∃𝑓 𝑓(ClWalks‘𝐺)(2nd𝑐) ↔ ((lastS‘(2nd𝑐)) = ((2nd𝑐)‘0) ∧ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) ∈ (ClWWalks‘𝐺))))
4414, 43mpbid 222 . . . 4 ((𝐺 ∈ USPGraph ∧ ((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ ((2nd𝑐)‘0) = ((2nd𝑐)‘(♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) ∈ ℕ)) → ((lastS‘(2nd𝑐)) = ((2nd𝑐)‘0) ∧ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) ∈ (ClWWalks‘𝐺)))
454, 44sylan2 493 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑐𝐶) → ((lastS‘(2nd𝑐)) = ((2nd𝑐)‘0) ∧ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) ∈ (ClWWalks‘𝐺)))
4645simprd 483 . 2 ((𝐺 ∈ USPGraph ∧ 𝑐𝐶) → ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) ∈ (ClWWalks‘𝐺))
47 clwlkclwwlkf.f . 2 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
4846, 47fmptd 6526 1 (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1069   = wceq 1629  ∃wex 1850   ∈ wcel 2143  ∃wrex 3060  {crab 3063  Vcvv 3348  ⟨cop 4319   class class class wbr 4783   ↦ cmpt 4860  ⟶wf 6026  ‘cfv 6030  (class class class)co 6791  1st c1st 7311  2nd c2nd 7312  ℝcr 10135  0cc0 10136  1c1 10137   + caddc 10139   ≤ cle 10275   − cmin 10466  ℕcn 11220  2c2 11270  ℕ0cn0 11492  ♯chash 13324  Word cword 13490  lastSclsw 13491   substr csubstr 13494  Vtxcvtx 26101  iEdgciedg 26102  USPGraphcuspgr 26271  Walkscwlks 26733  ClWalkscclwlks 26907  ClWWalkscclwwlk 27135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1048  df-3or 1070  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-nel 3045  df-ral 3064  df-rex 3065  df-reu 3066  df-rmo 3067  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-pss 3736  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4572  df-int 4609  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-tr 4884  df-id 5156  df-eprel 5161  df-po 5169  df-so 5170  df-fr 5207  df-we 5209  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-2o 7712  df-oadd 7715  df-er 7894  df-map 8009  df-pm 8010  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-card 8963  df-cda 9190  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-2 11279  df-n0 11493  df-xnn0 11564  df-z 11578  df-uz 11888  df-fz 12533  df-fzo 12673  df-hash 13325  df-word 13498  df-lsw 13499  df-substr 13502  df-edg 26167  df-uhgr 26180  df-upgr 26204  df-uspgr 26273  df-wlks 26736  df-clwlks 26908  df-clwwlk 27136 This theorem is referenced by:  clwlkclwwlkfo  27163  clwlkclwwlkf1  27164
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