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Theorem wwlkn0 26217
 Description: A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
Assertion
Ref Expression
wwlkn0 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩)
Distinct variable groups:   𝑣,𝑉   𝑣,𝑊
Allowed substitution hint:   𝐸(𝑣)

Proof of Theorem wwlkn0
StepHypRef Expression
1 wwlknprop 26214 . 2 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0𝑊 ∈ Word 𝑉)))
2 simpl 472 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
3 simpr 476 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V)
4 0nn0 11184 . . . . . . 7 0 ∈ ℕ0
54a1i 11 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 0 ∈ ℕ0)
62, 3, 53jca 1235 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 0 ∈ ℕ0))
76adantr 480 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0𝑊 ∈ Word 𝑉)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 0 ∈ ℕ0))
8 iswwlkn 26212 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 0 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1))))
97, 8syl 17 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0𝑊 ∈ Word 𝑉)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1))))
10 0p1e1 11009 . . . . . . . . 9 (0 + 1) = 1
1110eqeq2i 2622 . . . . . . . 8 ((#‘𝑊) = (0 + 1) ↔ (#‘𝑊) = 1)
12 eqs1 13245 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → 𝑊 = ⟨“(𝑊‘0)”⟩)
13 wrdlen1 13198 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → ∃𝑣𝑉 (𝑊‘0) = 𝑣)
14 s1eq 13233 . . . . . . . . . . . . 13 ((𝑊‘0) = 𝑣 → ⟨“(𝑊‘0)”⟩ = ⟨“𝑣”⟩)
1514reximi 2994 . . . . . . . . . . . 12 (∃𝑣𝑉 (𝑊‘0) = 𝑣 → ∃𝑣𝑉 ⟨“(𝑊‘0)”⟩ = ⟨“𝑣”⟩)
1613, 15syl 17 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → ∃𝑣𝑉 ⟨“(𝑊‘0)”⟩ = ⟨“𝑣”⟩)
17 eqeq1 2614 . . . . . . . . . . . . 13 (⟨“(𝑊‘0)”⟩ = 𝑊 → (⟨“(𝑊‘0)”⟩ = ⟨“𝑣”⟩ ↔ 𝑊 = ⟨“𝑣”⟩))
1817eqcoms 2618 . . . . . . . . . . . 12 (𝑊 = ⟨“(𝑊‘0)”⟩ → (⟨“(𝑊‘0)”⟩ = ⟨“𝑣”⟩ ↔ 𝑊 = ⟨“𝑣”⟩))
1918rexbidv 3034 . . . . . . . . . . 11 (𝑊 = ⟨“(𝑊‘0)”⟩ → (∃𝑣𝑉 ⟨“(𝑊‘0)”⟩ = ⟨“𝑣”⟩ ↔ ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩))
2016, 19syl5ib 233 . . . . . . . . . 10 (𝑊 = ⟨“(𝑊‘0)”⟩ → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩))
2112, 20mpcom 37 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩)
2221expcom 450 . . . . . . . 8 ((#‘𝑊) = 1 → (𝑊 ∈ Word 𝑉 → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩))
2311, 22sylbi 206 . . . . . . 7 ((#‘𝑊) = (0 + 1) → (𝑊 ∈ Word 𝑉 → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩))
2423adantl 481 . . . . . 6 ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)) → (𝑊 ∈ Word 𝑉 → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩))
2524com12 32 . . . . 5 (𝑊 ∈ Word 𝑉 → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩))
2625adantl 481 . . . 4 ((0 ∈ ℕ0𝑊 ∈ Word 𝑉) → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩))
2726adantl 481 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0𝑊 ∈ Word 𝑉)) → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (0 + 1)) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩))
289, 27sylbid 229 . 2 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (0 ∈ ℕ0𝑊 ∈ Word 𝑉)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩))
291, 28mpcom 37 1 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169  #chash 12979  Word cword 13146  ⟨“cs1 13149   WWalks cwwlk 26205   WWalksN cwwlkn 26206 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-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-s1 13157  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by: (None)
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