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Theorem wlkonl1iedg 29189
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 2730 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
21wlkonprop 29182 . . 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 6890 . . . . . . . . . . 11 (π‘˜ = 0 β†’ (π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜0))
4 fv0p1e1 12339 . . . . . . . . . . 11 (π‘˜ = 0 β†’ (π‘ƒβ€˜(π‘˜ + 1)) = (π‘ƒβ€˜1))
53, 4preq12d 4744 . . . . . . . . . 10 (π‘˜ = 0 β†’ {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
65sseq1d 4012 . . . . . . . . 9 (π‘˜ = 0 β†’ ({(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† 𝑒 ↔ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† 𝑒))
76rexbidv 3176 . . . . . . . 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 29149 . . . . . . . . . 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 479 . . . . . . . . 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 479 . . . . . . . 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 29139 . . . . . . . . . . 11 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
13 elnnne0 12490 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) ∈ β„• ↔ ((β™―β€˜πΉ) ∈ β„•0 ∧ (β™―β€˜πΉ) β‰  0))
1413simplbi2 499 . . . . . . . . . . . 12 ((β™―β€˜πΉ) ∈ β„•0 β†’ ((β™―β€˜πΉ) β‰  0 β†’ (β™―β€˜πΉ) ∈ β„•))
15 lbfzo0 13676 . . . . . . . . . . . 12 (0 ∈ (0..^(β™―β€˜πΉ)) ↔ (β™―β€˜πΉ) ∈ β„•)
1614, 15imbitrrdi 251 . . . . . . . . . . 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 479 . . . . . . . . 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 405 . . . . . . . 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 3612 . . . . . . 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 6903 . . . . . . . . . . 11 (π‘ƒβ€˜0) ∈ V
22 fvex 6903 . . . . . . . . . . 11 (π‘ƒβ€˜1) ∈ V
2321, 22prss 4822 . . . . . . . . . 10 (((π‘ƒβ€˜0) ∈ 𝑒 ∧ (π‘ƒβ€˜1) ∈ 𝑒) ↔ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} βŠ† 𝑒)
24 eleq1 2819 . . . . . . . . . . . . 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 252 . . . . . . . . . . . 12 ((π‘ƒβ€˜0) = 𝐴 β†’ ((π‘ƒβ€˜0) ∈ 𝑒 β†’ ((π‘ƒβ€˜1) ∈ 𝑒 β†’ 𝐴 ∈ 𝑒)))
2726adantl 480 . . . . . . . . . . 11 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) β†’ ((π‘ƒβ€˜0) ∈ 𝑒 β†’ ((π‘ƒβ€˜1) ∈ 𝑒 β†’ 𝐴 ∈ 𝑒)))
2827impd 409 . . . . . . . . . 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 3168 . . . . . . . 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 479 . . . . . . 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 411 . . . . 5 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴) β†’ ((β™―β€˜πΉ) β‰  0 β†’ βˆƒπ‘’ ∈ ran 𝐼 𝐴 ∈ 𝑒))
34333adant3 1130 . . . 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 1133 . . 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 405 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βŠ† wss 3947  {cpr 4629   class class class wbr 5147  ran crn 5676  β€˜cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115  β„•cn 12216  β„•0cn0 12476  ..^cfzo 13631  β™―chash 14294  Vtxcvtx 28523  iEdgciedg 28524  Walkscwlks 29120  WalksOncwlkson 29121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-wlks 29123  df-wlkson 29124
This theorem is referenced by:  conngrv2edg  29715
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