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Theorem wlkv0 26603
Description: If there is a walk in the null graph (a class without vertices), it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
Assertion
Ref Expression
wlkv0 (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))

Proof of Theorem wlkv0
StepHypRef Expression
1 wlkcpr 26580 . . 3 (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
2 eqid 2651 . . . . . 6 (iEdg‘𝐺) = (iEdg‘𝐺)
32wlkf 26566 . . . . 5 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (1st𝑊) ∈ Word dom (iEdg‘𝐺))
4 eqid 2651 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
54wlkp 26568 . . . . 5 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺))
63, 5jca 553 . . . 4 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)))
7 feq3 6066 . . . . . . . 8 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0...(#‘(1st𝑊)))⟶∅))
8 f00 6125 . . . . . . . 8 ((2nd𝑊):(0...(#‘(1st𝑊)))⟶∅ ↔ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅))
97, 8syl6bb 276 . . . . . . 7 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅)))
10 0z 11426 . . . . . . . . . . . . . 14 0 ∈ ℤ
11 nn0z 11438 . . . . . . . . . . . . . 14 ((#‘(1st𝑊)) ∈ ℕ0 → (#‘(1st𝑊)) ∈ ℤ)
12 fzn 12395 . . . . . . . . . . . . . 14 ((0 ∈ ℤ ∧ (#‘(1st𝑊)) ∈ ℤ) → ((#‘(1st𝑊)) < 0 ↔ (0...(#‘(1st𝑊))) = ∅))
1310, 11, 12sylancr 696 . . . . . . . . . . . . 13 ((#‘(1st𝑊)) ∈ ℕ0 → ((#‘(1st𝑊)) < 0 ↔ (0...(#‘(1st𝑊))) = ∅))
14 nn0nlt0 11357 . . . . . . . . . . . . . 14 ((#‘(1st𝑊)) ∈ ℕ0 → ¬ (#‘(1st𝑊)) < 0)
1514pm2.21d 118 . . . . . . . . . . . . 13 ((#‘(1st𝑊)) ∈ ℕ0 → ((#‘(1st𝑊)) < 0 → (1st𝑊) = ∅))
1613, 15sylbird 250 . . . . . . . . . . . 12 ((#‘(1st𝑊)) ∈ ℕ0 → ((0...(#‘(1st𝑊))) = ∅ → (1st𝑊) = ∅))
1716com12 32 . . . . . . . . . . 11 ((0...(#‘(1st𝑊))) = ∅ → ((#‘(1st𝑊)) ∈ ℕ0 → (1st𝑊) = ∅))
1817adantl 481 . . . . . . . . . 10 (((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) → ((#‘(1st𝑊)) ∈ ℕ0 → (1st𝑊) = ∅))
19 lencl 13356 . . . . . . . . . 10 ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → (#‘(1st𝑊)) ∈ ℕ0)
2018, 19impel 484 . . . . . . . . 9 ((((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (1st𝑊) = ∅)
21 simpll 805 . . . . . . . . 9 ((((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (2nd𝑊) = ∅)
2220, 21jca 553 . . . . . . . 8 ((((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
2322ex 449 . . . . . . 7 (((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
249, 23syl6bi 243 . . . . . 6 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))))
2524com23 86 . . . . 5 ((Vtx‘𝐺) = ∅ → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))))
2625impd 446 . . . 4 ((Vtx‘𝐺) = ∅ → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
276, 26syl5 34 . . 3 ((Vtx‘𝐺) = ∅ → ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
281, 27syl5bi 232 . 2 ((Vtx‘𝐺) = ∅ → (𝑊 ∈ (Walks‘𝐺) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅)))
2928imp 444 1 (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  c0 3948   class class class wbr 4685  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  0cc0 9974   < clt 10112  0cn0 11330  cz 11415  ...cfz 12364  #chash 13157  Word cword 13323  Vtxcvtx 25919  iEdgciedg 25920  Walkscwlks 26548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-wlks 26551
This theorem is referenced by:  g0wlk0  26604
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