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Theorem clwwlknwwlkncl 26328
Description: Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
Assertion
Ref Expression
clwwlknwwlkncl ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)})
Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑃   𝑤,𝑉

Proof of Theorem clwwlknwwlkncl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 clwwlknimp 26304 . . . 4 (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸))
2 clwwlknprop 26300 . . . 4 (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))
3 df-3an 1033 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ))
43simplbi2 653 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ ℕ → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ)))
543ad2ant1 1075 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)) → (𝑁 ∈ ℕ → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ)))
65adantl 481 . . . . . . 7 ((((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))) → (𝑁 ∈ ℕ → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ)))
76imp 444 . . . . . 6 (((((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))) ∧ 𝑁 ∈ ℕ) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ))
8 simpll1 1093 . . . . . 6 (((((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))) ∧ 𝑁 ∈ ℕ) → (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁))
9 3simpc 1053 . . . . . . 7 (((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸))
109ad2antrr 758 . . . . . 6 (((((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))) ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸))
117, 8, 103jca 1235 . . . . 5 (((((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))) ∧ 𝑁 ∈ ℕ) → ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)))
1211ex 449 . . . 4 ((((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁))) → (𝑁 ∈ ℕ → ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸))))
131, 2, 12syl2anc 691 . . 3 (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ → ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸))))
1413impcom 445 . 2 ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)))
15 eqid 2610 . . 3 {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
1615clwwlkel 26321 . 2 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)})
1714, 16syl 17 1 ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  {crab 2900  Vcvv 3173  {cpr 4127  ran crn 5039  cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  cmin 10145  cn 10897  0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   WWalksN cwwlkn 26206   ClWWalksN cclwwlkn 26277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-wwlk 26207  df-wwlkn 26208  df-clwwlk 26279  df-clwwlkn 26280
This theorem is referenced by: (None)
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