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Theorem wlklnwwlkln1 29386
Description: The sequence of vertices in a walk of length 𝑁 is a walk as word of length 𝑁 in a pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
Assertion
Ref Expression
wlklnwwlkln1 (𝐺 ∈ UPGraph β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁) β†’ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
Proof of Theorem wlklnwwlkln1
StepHypRef Expression
1 wlkcl 29136 . . . 4 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
21adantr 480 . . 3 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁) β†’ (β™―β€˜πΉ) ∈ β„•0)
3 wlkiswwlks1 29385 . . . . . . . 8 (𝐺 ∈ UPGraph β†’ (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃 ∈ (WWalksβ€˜πΊ)))
43com12 32 . . . . . . 7 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (𝐺 ∈ UPGraph β†’ 𝑃 ∈ (WWalksβ€˜πΊ)))
54ad2antrl 725 . . . . . 6 (((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) β†’ (𝐺 ∈ UPGraph β†’ 𝑃 ∈ (WWalksβ€˜πΊ)))
65imp 406 . . . . 5 ((((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) ∧ 𝐺 ∈ UPGraph) β†’ 𝑃 ∈ (WWalksβ€˜πΊ))
7 wlklenvp1 29139 . . . . . . . 8 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1))
87ad2antrl 725 . . . . . . 7 (((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) β†’ (β™―β€˜π‘ƒ) = ((β™―β€˜πΉ) + 1))
9 oveq1 7419 . . . . . . . . 9 ((β™―β€˜πΉ) = 𝑁 β†’ ((β™―β€˜πΉ) + 1) = (𝑁 + 1))
109adantl 481 . . . . . . . 8 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁) β†’ ((β™―β€˜πΉ) + 1) = (𝑁 + 1))
1110adantl 481 . . . . . . 7 (((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) β†’ ((β™―β€˜πΉ) + 1) = (𝑁 + 1))
128, 11eqtrd 2771 . . . . . 6 (((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) β†’ (β™―β€˜π‘ƒ) = (𝑁 + 1))
1312adantr 480 . . . . 5 ((((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) ∧ 𝐺 ∈ UPGraph) β†’ (β™―β€˜π‘ƒ) = (𝑁 + 1))
14 eleq1 2820 . . . . . . . . 9 ((β™―β€˜πΉ) = 𝑁 β†’ ((β™―β€˜πΉ) ∈ β„•0 ↔ 𝑁 ∈ β„•0))
15 iswwlksn 29356 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))))
1614, 15syl6bi 252 . . . . . . . 8 ((β™―β€˜πΉ) = 𝑁 β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1)))))
1716adantl 481 . . . . . . 7 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁) β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1)))))
1817impcom 407 . . . . . 6 (((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))))
1918adantr 480 . . . . 5 ((((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) ∧ 𝐺 ∈ UPGraph) β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))))
206, 13, 19mpbir2and 710 . . . 4 ((((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) ∧ 𝐺 ∈ UPGraph) β†’ 𝑃 ∈ (𝑁 WWalksN 𝐺))
2120ex 412 . . 3 (((β™―β€˜πΉ) ∈ β„•0 ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁)) β†’ (𝐺 ∈ UPGraph β†’ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
222, 21mpancom 685 . 2 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁) β†’ (𝐺 ∈ UPGraph β†’ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
2322com12 32 1 (𝐺 ∈ UPGraph β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 𝑁) β†’ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7412  1c1 11114   + caddc 11116  β„•0cn0 12477  β™―chash 14295  UPGraphcupgr 28604  Walkscwlks 29117  WWalkscwwlks 29343   WWalksN cwwlksn 29344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-oadd 8473  df-er 8706  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9899  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-edg 28572  df-uhgr 28582  df-upgr 28606  df-wlks 29120  df-wwlks 29348  df-wwlksn 29349
This theorem is referenced by:  wlklnwwlkn  29402  wlklnwwlknupgr  29404
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