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Theorem wlkv0 26416
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 26394 . . 3 (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
2 eqid 2621 . . . . . 6 (iEdg‘𝐺) = (iEdg‘𝐺)
32wlkf 26380 . . . . 5 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (1st𝑊) ∈ Word dom (iEdg‘𝐺))
4 eqid 2621 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
54wlkp 26382 . . . . 5 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺))
63, 5jca 554 . . . 4 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)))
7 feq3 5985 . . . . . . . 8 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0...(#‘(1st𝑊)))⟶∅))
8 f00 6044 . . . . . . . 8 ((2nd𝑊):(0...(#‘(1st𝑊)))⟶∅ ↔ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅))
97, 8syl6bb 276 . . . . . . 7 ((Vtx‘𝐺) = ∅ → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ ((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅)))
10 0z 11332 . . . . . . . . . . . . . 14 0 ∈ ℤ
11 nn0z 11344 . . . . . . . . . . . . . 14 ((#‘(1st𝑊)) ∈ ℕ0 → (#‘(1st𝑊)) ∈ ℤ)
12 fzn 12299 . . . . . . . . . . . . . 14 ((0 ∈ ℤ ∧ (#‘(1st𝑊)) ∈ ℤ) → ((#‘(1st𝑊)) < 0 ↔ (0...(#‘(1st𝑊))) = ∅))
1310, 11, 12sylancr 694 . . . . . . . . . . . . 13 ((#‘(1st𝑊)) ∈ ℕ0 → ((#‘(1st𝑊)) < 0 ↔ (0...(#‘(1st𝑊))) = ∅))
14 nn0nlt0 11263 . . . . . . . . . . . . . 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 482 . . . . . . . . . 10 (((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) → ((#‘(1st𝑊)) ∈ ℕ0 → (1st𝑊) = ∅))
19 lencl 13263 . . . . . . . . . 10 ((1st𝑊) ∈ Word dom (iEdg‘𝐺) → (#‘(1st𝑊)) ∈ ℕ0)
2018, 19impel 485 . . . . . . . . 9 ((((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (1st𝑊) = ∅)
21 simpll 789 . . . . . . . . 9 ((((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → (2nd𝑊) = ∅)
2220, 21jca 554 . . . . . . . 8 ((((2nd𝑊) = ∅ ∧ (0...(#‘(1st𝑊))) = ∅) ∧ (1st𝑊) ∈ Word dom (iEdg‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
2322ex 450 . . . . . . 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 447 . . . 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 445 1 (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  c0 3891   class class class wbr 4613  dom cdm 5074  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  0cc0 9880   < clt 10018  0cn0 11236  cz 11321  ...cfz 12268  #chash 13057  Word cword 13230  Vtxcvtx 25774  iEdgciedg 25775  Walkscwlks 26362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-wlks 26365
This theorem is referenced by:  g0wlk0  26417
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