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Theorem wlkv0 28905
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 (((Vtxβ€˜πΊ) = βˆ… ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ ((1st β€˜π‘Š) = βˆ… ∧ (2nd β€˜π‘Š) = βˆ…))

Proof of Theorem wlkv0
StepHypRef Expression
1 wlkcpr 28883 . . 3 (π‘Š ∈ (Walksβ€˜πΊ) ↔ (1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š))
2 eqid 2732 . . . . . 6 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
32wlkf 28868 . . . . 5 ((1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š) β†’ (1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ))
4 eqid 2732 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
54wlkp 28870 . . . . 5 ((1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š) β†’ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ))
63, 5jca 512 . . . 4 ((1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š) β†’ ((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ)))
7 feq3 6700 . . . . . . 7 ((Vtxβ€˜πΊ) = βˆ… β†’ ((2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ) ↔ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))βŸΆβˆ…))
8 f00 6773 . . . . . . 7 ((2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))βŸΆβˆ… ↔ ((2nd β€˜π‘Š) = βˆ… ∧ (0...(β™―β€˜(1st β€˜π‘Š))) = βˆ…))
97, 8bitrdi 286 . . . . . 6 ((Vtxβ€˜πΊ) = βˆ… β†’ ((2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ) ↔ ((2nd β€˜π‘Š) = βˆ… ∧ (0...(β™―β€˜(1st β€˜π‘Š))) = βˆ…)))
10 0z 12568 . . . . . . . . . . . . 13 0 ∈ β„€
11 nn0z 12582 . . . . . . . . . . . . 13 ((β™―β€˜(1st β€˜π‘Š)) ∈ β„•0 β†’ (β™―β€˜(1st β€˜π‘Š)) ∈ β„€)
12 fzn 13516 . . . . . . . . . . . . 13 ((0 ∈ β„€ ∧ (β™―β€˜(1st β€˜π‘Š)) ∈ β„€) β†’ ((β™―β€˜(1st β€˜π‘Š)) < 0 ↔ (0...(β™―β€˜(1st β€˜π‘Š))) = βˆ…))
1310, 11, 12sylancr 587 . . . . . . . . . . . 12 ((β™―β€˜(1st β€˜π‘Š)) ∈ β„•0 β†’ ((β™―β€˜(1st β€˜π‘Š)) < 0 ↔ (0...(β™―β€˜(1st β€˜π‘Š))) = βˆ…))
14 nn0nlt0 12497 . . . . . . . . . . . . 13 ((β™―β€˜(1st β€˜π‘Š)) ∈ β„•0 β†’ Β¬ (β™―β€˜(1st β€˜π‘Š)) < 0)
1514pm2.21d 121 . . . . . . . . . . . 12 ((β™―β€˜(1st β€˜π‘Š)) ∈ β„•0 β†’ ((β™―β€˜(1st β€˜π‘Š)) < 0 β†’ (1st β€˜π‘Š) = βˆ…))
1613, 15sylbird 259 . . . . . . . . . . 11 ((β™―β€˜(1st β€˜π‘Š)) ∈ β„•0 β†’ ((0...(β™―β€˜(1st β€˜π‘Š))) = βˆ… β†’ (1st β€˜π‘Š) = βˆ…))
1716com12 32 . . . . . . . . . 10 ((0...(β™―β€˜(1st β€˜π‘Š))) = βˆ… β†’ ((β™―β€˜(1st β€˜π‘Š)) ∈ β„•0 β†’ (1st β€˜π‘Š) = βˆ…))
1817adantl 482 . . . . . . . . 9 (((2nd β€˜π‘Š) = βˆ… ∧ (0...(β™―β€˜(1st β€˜π‘Š))) = βˆ…) β†’ ((β™―β€˜(1st β€˜π‘Š)) ∈ β„•0 β†’ (1st β€˜π‘Š) = βˆ…))
19 lencl 14482 . . . . . . . . 9 ((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) β†’ (β™―β€˜(1st β€˜π‘Š)) ∈ β„•0)
2018, 19impel 506 . . . . . . . 8 ((((2nd β€˜π‘Š) = βˆ… ∧ (0...(β™―β€˜(1st β€˜π‘Š))) = βˆ…) ∧ (1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ)) β†’ (1st β€˜π‘Š) = βˆ…)
21 simpll 765 . . . . . . . 8 ((((2nd β€˜π‘Š) = βˆ… ∧ (0...(β™―β€˜(1st β€˜π‘Š))) = βˆ…) ∧ (1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ)) β†’ (2nd β€˜π‘Š) = βˆ…)
2220, 21jca 512 . . . . . . 7 ((((2nd β€˜π‘Š) = βˆ… ∧ (0...(β™―β€˜(1st β€˜π‘Š))) = βˆ…) ∧ (1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ)) β†’ ((1st β€˜π‘Š) = βˆ… ∧ (2nd β€˜π‘Š) = βˆ…))
2322ex 413 . . . . . 6 (((2nd β€˜π‘Š) = βˆ… ∧ (0...(β™―β€˜(1st β€˜π‘Š))) = βˆ…) β†’ ((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) β†’ ((1st β€˜π‘Š) = βˆ… ∧ (2nd β€˜π‘Š) = βˆ…)))
249, 23syl6bi 252 . . . . 5 ((Vtxβ€˜πΊ) = βˆ… β†’ ((2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ) β†’ ((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) β†’ ((1st β€˜π‘Š) = βˆ… ∧ (2nd β€˜π‘Š) = βˆ…))))
2524impcomd 412 . . . 4 ((Vtxβ€˜πΊ) = βˆ… β†’ (((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ)) β†’ ((1st β€˜π‘Š) = βˆ… ∧ (2nd β€˜π‘Š) = βˆ…)))
266, 25syl5 34 . . 3 ((Vtxβ€˜πΊ) = βˆ… β†’ ((1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š) β†’ ((1st β€˜π‘Š) = βˆ… ∧ (2nd β€˜π‘Š) = βˆ…)))
271, 26biimtrid 241 . 2 ((Vtxβ€˜πΊ) = βˆ… β†’ (π‘Š ∈ (Walksβ€˜πΊ) β†’ ((1st β€˜π‘Š) = βˆ… ∧ (2nd β€˜π‘Š) = βˆ…)))
2827imp 407 1 (((Vtxβ€˜πΊ) = βˆ… ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ ((1st β€˜π‘Š) = βˆ… ∧ (2nd β€˜π‘Š) = βˆ…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ…c0 4322   class class class wbr 5148  dom cdm 5676  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  0cc0 11109   < clt 11247  β„•0cn0 12471  β„€cz 12557  ...cfz 13483  β™―chash 14289  Word cword 14463  Vtxcvtx 28253  iEdgciedg 28254  Walkscwlks 28850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-hash 14290  df-word 14464  df-wlks 28853
This theorem is referenced by:  g0wlk0  28906
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