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Theorem rusgrnumwwlk 28969
Description: In a 𝐟-regular graph, the number of walks of a fixed length 𝑁 from a fixed vertex is 𝐟 to the power of 𝑁. By definition, (𝑁 WWalksN 𝐺) is the set of walks (as words) with length 𝑁, and (𝑃𝐿𝑁) is the number of walks with length 𝑁 starting at the vertex 𝑃. Because of the 𝐟-regularity, a walk can be continued in 𝐟 different ways at the end vertex of the walk, and this repeated 𝑁 times.

This theorem even holds for 𝑁 = 0: in this case, the walk consists of only one vertex 𝑃, so the number of walks of length 𝑁 = 0 starting with 𝑃 is (𝐟↑0) = 1. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.)

Hypotheses
Ref Expression
rusgrnumwwlk.v 𝑉 = (Vtx‘𝐺)
rusgrnumwwlk.l 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↩ (♯‘{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑣}))
Assertion
Ref Expression
rusgrnumwwlk ((𝐺 RegUSGraph 𝐟 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐟↑𝑁))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑃,𝑛,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀   𝑀,𝐟
Allowed substitution hints:   𝐟(𝑣,𝑛)   𝐿(𝑀,𝑣,𝑛)

Proof of Theorem rusgrnumwwlk
Dummy variables 𝑥 𝑊 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7369 . . . . . . . 8 (𝑥 = 0 → (𝑃𝐿𝑥) = (𝑃𝐿0))
2 oveq2 7369 . . . . . . . 8 (𝑥 = 0 → (𝐟↑𝑥) = (𝐟↑0))
31, 2eqeq12d 2749 . . . . . . 7 (𝑥 = 0 → ((𝑃𝐿𝑥) = (𝐟↑𝑥) ↔ (𝑃𝐿0) = (𝐟↑0)))
43imbi2d 341 . . . . . 6 (𝑥 = 0 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑥) = (𝐟↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿0) = (𝐟↑0))))
5 oveq2 7369 . . . . . . . 8 (𝑥 = 𝑊 → (𝑃𝐿𝑥) = (𝑃𝐿𝑊))
6 oveq2 7369 . . . . . . . 8 (𝑥 = 𝑊 → (𝐟↑𝑥) = (𝐟↑𝑊))
75, 6eqeq12d 2749 . . . . . . 7 (𝑥 = 𝑊 → ((𝑃𝐿𝑥) = (𝐟↑𝑥) ↔ (𝑃𝐿𝑊) = (𝐟↑𝑊)))
87imbi2d 341 . . . . . 6 (𝑥 = 𝑊 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑥) = (𝐟↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑊) = (𝐟↑𝑊))))
9 oveq2 7369 . . . . . . . 8 (𝑥 = (𝑊 + 1) → (𝑃𝐿𝑥) = (𝑃𝐿(𝑊 + 1)))
10 oveq2 7369 . . . . . . . 8 (𝑥 = (𝑊 + 1) → (𝐟↑𝑥) = (𝐟↑(𝑊 + 1)))
119, 10eqeq12d 2749 . . . . . . 7 (𝑥 = (𝑊 + 1) → ((𝑃𝐿𝑥) = (𝐟↑𝑥) ↔ (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1))))
1211imbi2d 341 . . . . . 6 (𝑥 = (𝑊 + 1) → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑥) = (𝐟↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1)))))
13 oveq2 7369 . . . . . . . 8 (𝑥 = 𝑁 → (𝑃𝐿𝑥) = (𝑃𝐿𝑁))
14 oveq2 7369 . . . . . . . 8 (𝑥 = 𝑁 → (𝐟↑𝑥) = (𝐟↑𝑁))
1513, 14eqeq12d 2749 . . . . . . 7 (𝑥 = 𝑁 → ((𝑃𝐿𝑥) = (𝐟↑𝑥) ↔ (𝑃𝐿𝑁) = (𝐟↑𝑁)))
1615imbi2d 341 . . . . . 6 (𝑥 = 𝑁 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑥) = (𝐟↑𝑥)) ↔ (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑁) = (𝐟↑𝑁))))
17 rusgrusgr 28561 . . . . . . . . 9 (𝐺 RegUSGraph 𝐟 → 𝐺 ∈ USGraph)
18 usgruspgr 28178 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
1917, 18syl 17 . . . . . . . 8 (𝐺 RegUSGraph 𝐟 → 𝐺 ∈ USPGraph)
20 simpr 486 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → 𝑃 ∈ 𝑉)
21 rusgrnumwwlk.v . . . . . . . . 9 𝑉 = (Vtx‘𝐺)
22 rusgrnumwwlk.l . . . . . . . . 9 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↩ (♯‘{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑣}))
2321, 22rusgrnumwwlkb0 28965 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
2419, 20, 23syl2anr 598 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿0) = 1)
25 simpl 484 . . . . . . . . . . 11 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → 𝑉 ∈ Fin)
2625, 17anim12ci 615 . . . . . . . . . 10 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2721isfusgr 28315 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2826, 27sylibr 233 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → 𝐺 ∈ FinUSGraph)
29 simpr 486 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → 𝐺 RegUSGraph 𝐟)
30 ne0i 4298 . . . . . . . . . 10 (𝑃 ∈ 𝑉 → 𝑉 ≠ ∅)
3130ad2antlr 726 . . . . . . . . 9 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → 𝑉 ≠ ∅)
3221frusgrnn0 28568 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐟 ∧ 𝑉 ≠ ∅) → 𝐟 ∈ ℕ0)
3332nn0cnd 12483 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐟 ∧ 𝑉 ≠ ∅) → 𝐟 ∈ ℂ)
3428, 29, 31, 33syl3anc 1372 . . . . . . . 8 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → 𝐟 ∈ ℂ)
3534exp0d 14054 . . . . . . 7 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝐟↑0) = 1)
3624, 35eqtr4d 2776 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿0) = (𝐟↑0))
37 simpl 484 . . . . . . . . . . 11 (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉))
3837anim1i 616 . . . . . . . . . 10 ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) ∧ 𝑊 ∈ ℕ0) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝑊 ∈ ℕ0))
39 df-3an 1090 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑊 ∈ ℕ0) ↔ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝑊 ∈ ℕ0))
4038, 39sylibr 233 . . . . . . . . 9 ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) ∧ 𝑊 ∈ ℕ0) → (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑊 ∈ ℕ0))
4121, 22rusgrnumwwlks 28968 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐟 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑊 ∈ ℕ0)) → ((𝑃𝐿𝑊) = (𝐟↑𝑊) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1))))
4229, 40, 41syl2an2r 684 . . . . . . . 8 ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) ∧ 𝑊 ∈ ℕ0) → ((𝑃𝐿𝑊) = (𝐟↑𝑊) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1))))
4342expcom 415 . . . . . . 7 (𝑊 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → ((𝑃𝐿𝑊) = (𝐟↑𝑊) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1)))))
4443a2d 29 . . . . . 6 (𝑊 ∈ ℕ0 → ((((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑊) = (𝐟↑𝑊)) → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿(𝑊 + 1)) = (𝐟↑(𝑊 + 1)))))
454, 8, 12, 16, 36, 44nn0ind 12606 . . . . 5 (𝑁 ∈ ℕ0 → (((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) ∧ 𝐺 RegUSGraph 𝐟) → (𝑃𝐿𝑁) = (𝐟↑𝑁)))
4645expd 417 . . . 4 (𝑁 ∈ ℕ0 → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → (𝐺 RegUSGraph 𝐟 → (𝑃𝐿𝑁) = (𝐟↑𝑁))))
4746com12 32 . . 3 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉) → (𝑁 ∈ ℕ0 → (𝐺 RegUSGraph 𝐟 → (𝑃𝐿𝑁) = (𝐟↑𝑁))))
48473impia 1118 . 2 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐺 RegUSGraph 𝐟 → (𝑃𝐿𝑁) = (𝐟↑𝑁)))
4948impcom 409 1 ((𝐺 RegUSGraph 𝐟 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐟↑𝑁))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2940  {crab 3406  âˆ…c0 4286   class class class wbr 5109  â€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Fincfn 8889  â„‚cc 11057  0cc0 11059  1c1 11060   + caddc 11062  â„•0cn0 12421  â†‘cexp 13976  â™¯chash 14239  Vtxcvtx 27996  USPGraphcuspgr 28148  USGraphcusgr 28149  FinUSGraphcfusgr 28313   RegUSGraph crusgr 28553   WWalksN cwwlksn 28820
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-disj 5075  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-oadd 8420  df-er 8654  df-map 8773  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-oi 9454  df-dju 9845  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-xnn0 12494  df-z 12508  df-uz 12772  df-rp 12924  df-xadd 13042  df-fz 13434  df-fzo 13577  df-seq 13916  df-exp 13977  df-hash 14240  df-word 14412  df-lsw 14460  df-concat 14468  df-s1 14493  df-substr 14538  df-pfx 14568  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-clim 15379  df-sum 15580  df-vtx 27998  df-iedg 27999  df-edg 28048  df-uhgr 28058  df-ushgr 28059  df-upgr 28082  df-umgr 28083  df-uspgr 28150  df-usgr 28151  df-fusgr 28314  df-nbgr 28330  df-vtxdg 28463  df-rgr 28554  df-rusgr 28555  df-wwlks 28824  df-wwlksn 28825
This theorem is referenced by:  rusgrnumwwlkg  28970
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