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Theorem wwlksnextprop 27690
Description: Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
Hypotheses
Ref Expression
wwlksnextprop.x 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
wwlksnextprop.e 𝐸 = (Edg‘𝐺)
wwlksnextprop.y 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
Assertion
Ref Expression
wwlksnextprop (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑃   𝑦,𝐸   𝑥,𝑁,𝑦   𝑦,𝑃   𝑦,𝑋   𝑦,𝑌   𝑥,𝑤
Allowed substitution hints:   𝑃(𝑥)   𝐸(𝑥,𝑤)   𝐺(𝑥,𝑦)   𝑋(𝑥,𝑤)   𝑌(𝑥,𝑤)

Proof of Theorem wwlksnextprop
StepHypRef Expression
1 eqidd 2822 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
2 wwlksnextprop.x . . . . . . . . 9 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
32wwlksnextproplem1 27687 . . . . . . . 8 ((𝑥𝑋𝑁 ∈ ℕ0) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
43ancoms 461 . . . . . . 7 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
54adantr 483 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
6 eqeq2 2833 . . . . . . 7 ((𝑥‘0) = 𝑃 → (((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0) ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
76adantl 484 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0) ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
85, 7mpbid 234 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃)
9 wwlksnextprop.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
102, 9wwlksnextproplem2 27688 . . . . . . 7 ((𝑥𝑋𝑁 ∈ ℕ0) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
1110ancoms 461 . . . . . 6 ((𝑁 ∈ ℕ0𝑥𝑋) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
1211adantr 483 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
13 simpr 487 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑥𝑋) → 𝑥𝑋)
1413adantr 483 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑥𝑋)
15 simpr 487 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥‘0) = 𝑃)
16 simpll 765 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑁 ∈ ℕ0)
17 wwlksnextprop.y . . . . . . . 8 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
182, 9, 17wwlksnextproplem3 27689 . . . . . . 7 ((𝑥𝑋 ∧ (𝑥‘0) = 𝑃𝑁 ∈ ℕ0) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
1914, 15, 16, 18syl3anc 1367 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
20 eqeq2 2833 . . . . . . . 8 (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑥 prefix (𝑁 + 1)) = 𝑦 ↔ (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1))))
21 fveq1 6668 . . . . . . . . 9 (𝑦 = (𝑥 prefix (𝑁 + 1)) → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
2221eqeq1d 2823 . . . . . . . 8 (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
23 fveq2 6669 . . . . . . . . . 10 (𝑦 = (𝑥 prefix (𝑁 + 1)) → (lastS‘𝑦) = (lastS‘(𝑥 prefix (𝑁 + 1))))
2423preq1d 4674 . . . . . . . . 9 (𝑦 = (𝑥 prefix (𝑁 + 1)) → {(lastS‘𝑦), (lastS‘𝑥)} = {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)})
2524eleq1d 2897 . . . . . . . 8 (𝑦 = (𝑥 prefix (𝑁 + 1)) → ({(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸 ↔ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸))
2620, 22, 253anbi123d 1432 . . . . . . 7 (𝑦 = (𝑥 prefix (𝑁 + 1)) → (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) ↔ ((𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ∧ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 ∧ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)))
2726adantl 484 . . . . . 6 ((((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) ∧ 𝑦 = (𝑥 prefix (𝑁 + 1))) → (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) ↔ ((𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ∧ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 ∧ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)))
2819, 27rspcedv 3615 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ∧ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 ∧ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸) → ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
291, 8, 12, 28mp3and 1460 . . . 4 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸))
3029ex 415 . . 3 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥‘0) = 𝑃 → ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
3121eqcoms 2829 . . . . . . . . 9 ((𝑥 prefix (𝑁 + 1)) = 𝑦 → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
3231eqeq1d 2823 . . . . . . . 8 ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
333eqcomd 2827 . . . . . . . . . . 11 ((𝑥𝑋𝑁 ∈ ℕ0) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
3433ancoms 461 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑥𝑋) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
3534adantr 483 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
36 eqeq2 2833 . . . . . . . . . 10 (𝑃 = ((𝑥 prefix (𝑁 + 1))‘0) → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
3736eqcoms 2829 . . . . . . . . 9 (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
3835, 37syl5ibr 248 . . . . . . . 8 (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
3932, 38syl6bi 255 . . . . . . 7 ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃)))
4039imp 409 . . . . . 6 (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
41403adant3 1128 . . . . 5 (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
4241com12 32 . . . 4 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
4342rexlimdva 3284 . . 3 ((𝑁 ∈ ℕ0𝑥𝑋) → (∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
4430, 43impbid 214 . 2 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥‘0) = 𝑃 ↔ ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
4544rabbidva 3478 1 (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wrex 3139  {crab 3142  {cpr 4568  cfv 6354  (class class class)co 7155  0cc0 10536  1c1 10537   + caddc 10539  0cn0 11896  lastSclsw 13913   prefix cpfx 14031  Edgcedg 26831   WWalksN cwwlksn 27603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-fzo 13033  df-hash 13690  df-word 13861  df-lsw 13914  df-substr 14002  df-pfx 14032  df-wwlks 27607  df-wwlksn 27608
This theorem is referenced by: (None)
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