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Theorem wlkonl1iedg 28922
Description: If there is a walk between two vertices 𝐴 and 𝐡 at least of length 1, then the start vertex 𝐴 is incident with an edge. (Contributed by AV, 4-Apr-2021.)
Hypothesis
Ref Expression
wlkonl1iedg.i 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
wlkonl1iedg ((𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ∧ (β™―β€˜πΉ) β‰  0) β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒)
Distinct variable groups:   𝐴,𝑒   𝑒,𝐹   𝑒,𝐺   𝑒,𝐼   𝑃,𝑒
Allowed substitution hint:   𝐡(𝑒)
Proof of Theorem wlkonl1iedg
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
21wlkonprop 28915 . . 3 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
3 fveq2 6892 . . . . . . . . . . 11 (π‘˜ = 0 β†’ (π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜0))
4 fv0p1e1 12335 . . . . . . . . . . 11 (π‘˜ = 0 β†’ (π‘ƒβ€˜(π‘˜ + 1)) = (π‘ƒβ€˜1))
53, 4preq12d 4746 . . . . . . . . . 10 (π‘˜ = 0 β†’ {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
65sseq1d 4014 . . . . . . . . 9 (π‘˜ = 0 β†’ ({(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† 𝑒 ↔ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† 𝑒))
76rexbidv 3179 . . . . . . . 8 (π‘˜ = 0 β†’ (βˆƒπ‘’ ∈ ran 𝐼{(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† 𝑒 ↔ βˆƒπ‘’ ∈ ran 𝐼{(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† 𝑒))
8 wlkonl1iedg.i . . . . . . . . . . 11 𝐼 = (iEdgβ€˜πΊ)
98wlkvtxiedg 28882 . . . . . . . . . 10 (𝐹(Walksβ€˜πΊ)𝑃 β†’ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))βˆƒπ‘’ ∈ ran 𝐼{(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† 𝑒)
109adantr 482 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) β†’ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))βˆƒπ‘’ ∈ ran 𝐼{(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† 𝑒)
1110adantr 482 . . . . . . . 8 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) ∧ (β™―β€˜πΉ) β‰  0) β†’ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))βˆƒπ‘’ ∈ ran 𝐼{(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† 𝑒)
12 wlkcl 28872 . . . . . . . . . . 11 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
13 elnnne0 12486 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) ∈ β„• ↔ ((β™―β€˜πΉ) ∈ β„•0 ∧ (β™―β€˜πΉ) β‰  0))
1413simplbi2 502 . . . . . . . . . . . 12 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜πΉ) β‰  0 β†’ (β™―β€˜πΉ) ∈ β„•))
15 lbfzo0 13672 . . . . . . . . . . . 12 (0 ∈ (0..^(β™―β€˜πΉ)) ↔ (β™―β€˜πΉ) ∈ β„•)
1614, 15syl6ibr 252 . . . . . . . . . . 11 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜πΉ) β‰  0 β†’ 0 ∈ (0..^(β™―β€˜πΉ))))
1712, 16syl 17 . . . . . . . . . 10 (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((β™―β€˜πΉ) β‰  0 β†’ 0 ∈ (0..^(β™―β€˜πΉ))))
1817adantr 482 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) β†’ ((β™―β€˜πΉ) β‰  0 β†’ 0 ∈ (0..^(β™―β€˜πΉ))))
1918imp 408 . . . . . . . 8 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) ∧ (β™―β€˜πΉ) β‰  0) β†’ 0 ∈ (0..^(β™―β€˜πΉ)))
207, 11, 19rspcdva 3614 . . . . . . 7 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) ∧ (β™―β€˜πΉ) β‰  0) β†’ βˆƒπ‘’ ∈ ran 𝐼{(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† 𝑒)
21 fvex 6905 . . . . . . . . . . 11 (π‘ƒβ€˜0) ∈ V
22 fvex 6905 . . . . . . . . . . 11 (π‘ƒβ€˜1) ∈ V
2321, 22prss 4824 . . . . . . . . . 10 (((π‘ƒβ€˜0) ∈ 𝑒 ∧ (π‘ƒβ€˜1) ∈ 𝑒) ↔ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† 𝑒)
24 eleq1 2822 . . . . . . . . . . . . 13 ((π‘ƒβ€˜0) = 𝐴 β†’ ((π‘ƒβ€˜0) ∈ 𝑒 ↔ 𝐴 ∈ 𝑒))
25 ax-1 6 . . . . . . . . . . . . 13 (𝐴 ∈ 𝑒 β†’ ((π‘ƒβ€˜1) ∈ 𝑒 β†’ 𝐴 ∈ 𝑒))
2624, 25syl6bi 253 . . . . . . . . . . . 12 ((π‘ƒβ€˜0) = 𝐴 β†’ ((π‘ƒβ€˜0) ∈ 𝑒 β†’ ((π‘ƒβ€˜1) ∈ 𝑒 β†’ 𝐴 ∈ 𝑒)))
2726adantl 483 . . . . . . . . . . 11 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) β†’ ((π‘ƒβ€˜0) ∈ 𝑒 β†’ ((π‘ƒβ€˜1) ∈ 𝑒 β†’ 𝐴 ∈ 𝑒)))
2827impd 412 . . . . . . . . . 10 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) β†’ (((π‘ƒβ€˜0) ∈ 𝑒 ∧ (π‘ƒβ€˜1) ∈ 𝑒) β†’ 𝐴 ∈ 𝑒))
2923, 28biimtrrid 242 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) β†’ ({(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† 𝑒 β†’ 𝐴 ∈ 𝑒))
3029reximdv 3171 . . . . . . . 8 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) β†’ (βˆƒπ‘’ ∈ ran 𝐼{(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† 𝑒 β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒))
3130adantr 482 . . . . . . 7 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) ∧ (β™―β€˜πΉ) β‰  0) β†’ (βˆƒπ‘’ ∈ ran 𝐼{(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† 𝑒 β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒))
3220, 31mpd 15 . . . . . 6 (((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) ∧ (β™―β€˜πΉ) β‰  0) β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒)
3332ex 414 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) β†’ ((β™―β€˜πΉ) β‰  0 β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒))
34333adant3 1133 . . . 4 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((β™―β€˜πΉ) β‰  0 β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒))
35343ad2ant3 1136 . . 3 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ ((β™―β€˜πΉ) β‰  0 β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒))
362, 35syl 17 . 2 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ ((β™―β€˜πΉ) β‰  0 β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒))
3736imp 408 1 ((𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ∧ (β™―β€˜πΉ) β‰  0) β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  {cpr 4631   class class class wbr 5149  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  β„•cn 12212  β„•0cn0 12472  ..^cfzo 13627  β™―chash 14290  Vtxcvtx 28256  iEdgciedg 28257  Walkscwlks 28853  WalksOncwlkson 28854
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  df-wlkson 28857
This theorem is referenced by:  conngrv2edg  29448
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