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Theorem rusgrnumwwlkb0 28965
Description: Induction base 0 for rusgrnumwwlk 28969. Here, we do not need the regularity of the graph yet. (Contributed by Alexander van der Vekens, 24-Jul-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
rusgrnumwwlkb0 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑃,𝑛,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀
Allowed substitution hints:   𝐿(𝑀,𝑣,𝑛)

Proof of Theorem rusgrnumwwlkb0
StepHypRef Expression
1 simpr 486 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → 𝑃 ∈ 𝑉)
2 0nn0 12436 . . 3 0 ∈ ℕ0
3 rusgrnumwwlk.v . . . 4 𝑉 = (Vtx‘𝐺)
4 rusgrnumwwlk.l . . . 4 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↩ (♯‘{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑣}))
53, 4rusgrnumwwlklem 28964 . . 3 ((𝑃 ∈ 𝑉 ∧ 0 ∈ ℕ0) → (𝑃𝐿0) = (♯‘{𝑀 ∈ (0 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃}))
61, 2, 5sylancl 587 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = (♯‘{𝑀 ∈ (0 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃}))
7 df-rab 3407 . . . . 5 {𝑀 ∈ (0 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃} = {𝑀 ∣ (𝑀 ∈ (0 WWalksN 𝐺) ∧ (𝑀‘0) = 𝑃)}
87a1i 11 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑀 ∈ (0 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃} = {𝑀 ∣ (𝑀 ∈ (0 WWalksN 𝐺) ∧ (𝑀‘0) = 𝑃)})
9 wwlksn0s 28855 . . . . . . . . 9 (0 WWalksN 𝐺) = {𝑀 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑀) = 1}
109a1i 11 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (0 WWalksN 𝐺) = {𝑀 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑀) = 1})
1110eleq2d 2820 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑀 ∈ (0 WWalksN 𝐺) ↔ 𝑀 ∈ {𝑀 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑀) = 1}))
12 rabid 3426 . . . . . . 7 (𝑀 ∈ {𝑀 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑀) = 1} ↔ (𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1))
1311, 12bitrdi 287 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑀 ∈ (0 WWalksN 𝐺) ↔ (𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1)))
1413anbi1d 631 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → ((𝑀 ∈ (0 WWalksN 𝐺) ∧ (𝑀‘0) = 𝑃) ↔ ((𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1) ∧ (𝑀‘0) = 𝑃)))
1514abbidv 2802 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑀 ∣ (𝑀 ∈ (0 WWalksN 𝐺) ∧ (𝑀‘0) = 𝑃)} = {𝑀 ∣ ((𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1) ∧ (𝑀‘0) = 𝑃)})
16 wrdl1s1 14511 . . . . . . . . 9 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 = ⟚“𝑃”⟩ ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃)))
17 df-3an 1090 . . . . . . . . 9 ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1 ∧ (𝑣‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
1816, 17bitr2di 288 . . . . . . . 8 (𝑃 ∈ (Vtx‘𝐺) → (((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃) ↔ 𝑣 = ⟚“𝑃”⟩))
19 vex 3451 . . . . . . . . 9 𝑣 ∈ V
20 eleq1w 2817 . . . . . . . . . . 11 (𝑀 = 𝑣 → (𝑀 ∈ Word (Vtx‘𝐺) ↔ 𝑣 ∈ Word (Vtx‘𝐺)))
21 fveqeq2 6855 . . . . . . . . . . 11 (𝑀 = 𝑣 → ((♯‘𝑀) = 1 ↔ (♯‘𝑣) = 1))
2220, 21anbi12d 632 . . . . . . . . . 10 (𝑀 = 𝑣 → ((𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1) ↔ (𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1)))
23 fveq1 6845 . . . . . . . . . . 11 (𝑀 = 𝑣 → (𝑀‘0) = (𝑣‘0))
2423eqeq1d 2735 . . . . . . . . . 10 (𝑀 = 𝑣 → ((𝑀‘0) = 𝑃 ↔ (𝑣‘0) = 𝑃))
2522, 24anbi12d 632 . . . . . . . . 9 (𝑀 = 𝑣 → (((𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1) ∧ (𝑀‘0) = 𝑃) ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃)))
2619, 25elab 3634 . . . . . . . 8 (𝑣 ∈ {𝑀 ∣ ((𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1) ∧ (𝑀‘0) = 𝑃)} ↔ ((𝑣 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑣) = 1) ∧ (𝑣‘0) = 𝑃))
27 velsn 4606 . . . . . . . 8 (𝑣 ∈ {⟚“𝑃”⟩} ↔ 𝑣 = ⟚“𝑃”⟩)
2818, 26, 273bitr4g 314 . . . . . . 7 (𝑃 ∈ (Vtx‘𝐺) → (𝑣 ∈ {𝑀 ∣ ((𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1) ∧ (𝑀‘0) = 𝑃)} ↔ 𝑣 ∈ {⟚“𝑃”⟩}))
2928, 3eleq2s 2852 . . . . . 6 (𝑃 ∈ 𝑉 → (𝑣 ∈ {𝑀 ∣ ((𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1) ∧ (𝑀‘0) = 𝑃)} ↔ 𝑣 ∈ {⟚“𝑃”⟩}))
3029eqrdv 2731 . . . . 5 (𝑃 ∈ 𝑉 → {𝑀 ∣ ((𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1) ∧ (𝑀‘0) = 𝑃)} = {⟚“𝑃”⟩})
3130adantl 483 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑀 ∣ ((𝑀 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑀) = 1) ∧ (𝑀‘0) = 𝑃)} = {⟚“𝑃”⟩})
328, 15, 313eqtrd 2777 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → {𝑀 ∈ (0 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃} = {⟚“𝑃”⟩})
3332fveq2d 6850 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑀 ∈ (0 WWalksN 𝐺) ∣ (𝑀‘0) = 𝑃}) = (♯‘{⟚“𝑃”⟩}))
34 s1cl 14499 . . . 4 (𝑃 ∈ 𝑉 → ⟚“𝑃”⟩ ∈ Word 𝑉)
35 hashsng 14278 . . . 4 (⟚“𝑃”⟩ ∈ Word 𝑉 → (♯‘{⟚“𝑃”⟩}) = 1)
3634, 35syl 17 . . 3 (𝑃 ∈ 𝑉 → (♯‘{⟚“𝑃”⟩}) = 1)
3736adantl 483 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (♯‘{⟚“𝑃”⟩}) = 1)
386, 33, 373eqtrd 2777 1 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  {crab 3406  {csn 4590  â€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  0cc0 11059  1c1 11060  â„•0cn0 12421  â™¯chash 14239  Word cword 14411  âŸšâ€œcs1 14492  Vtxcvtx 27996  USPGraphcuspgr 28148   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-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
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-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-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-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-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-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  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-nn 12162  df-n0 12422  df-xnn0 12494  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-s1 14493  df-wwlks 28824  df-wwlksn 28825
This theorem is referenced by:  rusgrnumwwlk  28969
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