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Theorem wwlksnextpropOLD 27188
Description: Obsolete version of wwlksnextprop 27187 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
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
wwlksnextpropOLD (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, (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 wwlksnextpropOLD
StepHypRef Expression
1 eqidd 2798 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
2 wwlksnextprop.x . . . . . . . . 9 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
32wwlksnextproplem1OLD 27182 . . . . . . . 8 ((𝑥𝑋𝑁 ∈ ℕ0) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0))
43ancoms 451 . . . . . . 7 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0))
54adantr 473 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0))
6 eqeq2 2808 . . . . . . 7 ((𝑥‘0) = 𝑃 → (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
76adantl 474 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
85, 7mpbid 224 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃)
9 wwlksnextprop.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
102, 9wwlksnextproplem2OLD 27184 . . . . . . 7 ((𝑥𝑋𝑁 ∈ ℕ0) → {(lastS‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑥)} ∈ 𝐸)
1110ancoms 451 . . . . . 6 ((𝑁 ∈ ℕ0𝑥𝑋) → {(lastS‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑥)} ∈ 𝐸)
1211adantr 473 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → {(lastS‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑥)} ∈ 𝐸)
13 simpr 478 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑥𝑋) → 𝑥𝑋)
1413adantr 473 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑥𝑋)
15 simpr 478 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥‘0) = 𝑃)
16 simpll 784 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑁 ∈ ℕ0)
17 wwlksnextprop.y . . . . . . . 8 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
182, 9, 17wwlksnextproplem3OLD 27186 . . . . . . 7 ((𝑥𝑋 ∧ (𝑥‘0) = 𝑃𝑁 ∈ ℕ0) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)
1914, 15, 16, 18syl3anc 1491 . . . . . 6 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)
20 eqeq2 2808 . . . . . . . 8 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ↔ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩)))
21 fveq1 6408 . . . . . . . . 9 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (𝑦‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
2221eqeq1d 2799 . . . . . . . 8 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ((𝑦‘0) = 𝑃 ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
23 fveq2 6409 . . . . . . . . . 10 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (lastS‘𝑦) = (lastS‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)))
2423preq1d 4461 . . . . . . . . 9 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → {(lastS‘𝑦), (lastS‘𝑥)} = {(lastS‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑥)})
2524eleq1d 2861 . . . . . . . 8 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ({(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸 ↔ {(lastS‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑥)} ∈ 𝐸))
2620, 22, 253anbi123d 1561 . . . . . . 7 (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∧ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 ∧ {(lastS‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑥)} ∈ 𝐸)))
2726adantl 474 . . . . . 6 ((((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) ∧ 𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩)) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∧ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 ∧ {(lastS‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑥)} ∈ 𝐸)))
2819, 27rspcedv 3499 . . . . 5 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∧ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 ∧ {(lastS‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), (lastS‘𝑥)} ∈ 𝐸) → ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
291, 8, 12, 28mp3and 1589 . . . 4 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ (𝑥‘0) = 𝑃) → ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸))
3029ex 402 . . 3 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥‘0) = 𝑃 → ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
3121eqcoms 2805 . . . . . . . . 9 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → (𝑦‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
3231eqeq1d 2799 . . . . . . . 8 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((𝑦‘0) = 𝑃 ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
333eqcomd 2803 . . . . . . . . . . 11 ((𝑥𝑋𝑁 ∈ ℕ0) → (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
3433ancoms 451 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑥𝑋) → (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
3534adantr 473 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
36 eqeq2 2808 . . . . . . . . . 10 (𝑃 = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0)))
3736eqcoms 2805 . . . . . . . . 9 (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0)))
3835, 37syl5ibr 238 . . . . . . . 8 (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
3932, 38syl6bi 245 . . . . . . 7 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((𝑦‘0) = 𝑃 → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃)))
4039imp 396 . . . . . 6 (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
41403adant3 1163 . . . . 5 (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (𝑥‘0) = 𝑃))
4241com12 32 . . . 4 (((𝑁 ∈ ℕ0𝑥𝑋) ∧ 𝑦𝑌) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
4342rexlimdva 3210 . . 3 ((𝑁 ∈ ℕ0𝑥𝑋) → (∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
4430, 43impbid 204 . 2 ((𝑁 ∈ ℕ0𝑥𝑋) → ((𝑥‘0) = 𝑃 ↔ ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
4544rabbidva 3370 1 (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wrex 3088  {crab 3091  {cpr 4368  cop 4372  cfv 6099  (class class class)co 6876  0cc0 10222  1c1 10223   + caddc 10225  0cn0 11576  lastSclsw 13578   substr csubstr 13660  Edgcedg 26273   WWalksN cwwlksn 27068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-oadd 7801  df-er 7980  df-map 8095  df-pm 8096  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-card 9049  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-nn 11311  df-2 11372  df-n0 11577  df-z 11663  df-uz 11927  df-fz 12577  df-fzo 12717  df-hash 13367  df-word 13531  df-lsw 13579  df-substr 13661  df-wwlks 27072  df-wwlksn 27073
This theorem is referenced by: (None)
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