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Theorem wlkv0 28908
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 28886 . . 3 (π‘Š ∈ (Walksβ€˜πΊ) ↔ (1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š))
2 eqid 2733 . . . . . 6 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
32wlkf 28871 . . . . 5 ((1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š) β†’ (1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ))
4 eqid 2733 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
54wlkp 28873 . . . . 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 513 . . . 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 6701 . . . . . . 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 6774 . . . . . . 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 287 . . . . . 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 12569 . . . . . . . . . . . . 13 0 ∈ β„€
11 nn0z 12583 . . . . . . . . . . . . 13 ((β™―β€˜(1st β€˜π‘Š)) ∈ β„•0 β†’ (β™―β€˜(1st β€˜π‘Š)) ∈ β„€)
12 fzn 13517 . . . . . . . . . . . . 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 588 . . . . . . . . . . . 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 12498 . . . . . . . . . . . . 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 260 . . . . . . . . . . 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 483 . . . . . . . . 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 14483 . . . . . . . . 9 ((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) β†’ (β™―β€˜(1st β€˜π‘Š)) ∈ β„•0)
2018, 19impel 507 . . . . . . . 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 766 . . . . . . . 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 513 . . . . . . 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 414 . . . . . 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 253 . . . . 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 413 . . . 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 408 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 397   = wceq 1542   ∈ wcel 2107  βˆ…c0 4323   class class class wbr 5149  dom cdm 5677  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  0cc0 11110   < clt 11248  β„•0cn0 12472  β„€cz 12558  ...cfz 13484  β™―chash 14290  Word cword 14464  Vtxcvtx 28256  iEdgciedg 28257  Walkscwlks 28853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-wlks 28856
This theorem is referenced by:  g0wlk0  28909
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