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Theorem wlklenvclwlk 26366
 Description: The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.)
Assertion
Ref Expression
wlklenvclwlk ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸) → (#‘𝐹) = (#‘𝑊)))

Proof of Theorem wlklenvclwlk
StepHypRef Expression
1 df-br 4584 . . 3 (𝐹(𝑉 Walks 𝐸)(𝑊 ++ ⟨“(𝑊‘0)”⟩) ↔ ⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸))
2 wlkbprop 26051 . . . 4 (𝐹(𝑉 Walks 𝐸)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ V)))
3 wlklenvp1 26365 . . . . . . 7 (𝐹(𝑉 Walks 𝐸)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → (#‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) = ((#‘𝐹) + 1))
4 wrdsymb1 13197 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (𝑊‘0) ∈ 𝑉)
54s1cld 13236 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → ⟨“(𝑊‘0)”⟩ ∈ Word 𝑉)
6 ccatlen 13213 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉 ∧ ⟨“(𝑊‘0)”⟩ ∈ Word 𝑉) → (#‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) = ((#‘𝑊) + (#‘⟨“(𝑊‘0)”⟩)))
75, 6syldan 486 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (#‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) = ((#‘𝑊) + (#‘⟨“(𝑊‘0)”⟩)))
8 s1len 13238 . . . . . . . . . . . . 13 (#‘⟨“(𝑊‘0)”⟩) = 1
98a1i 11 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (#‘⟨“(𝑊‘0)”⟩) = 1)
109oveq2d 6565 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → ((#‘𝑊) + (#‘⟨“(𝑊‘0)”⟩)) = ((#‘𝑊) + 1))
117, 10eqtrd 2644 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (#‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) = ((#‘𝑊) + 1))
1211eqeq1d 2612 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → ((#‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) = ((#‘𝐹) + 1) ↔ ((#‘𝑊) + 1) = ((#‘𝐹) + 1)))
13 lencl 13179 . . . . . . . . . . 11 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
14 eqcom 2617 . . . . . . . . . . . . . 14 (((#‘𝑊) + 1) = ((#‘𝐹) + 1) ↔ ((#‘𝐹) + 1) = ((#‘𝑊) + 1))
15 nn0cn 11179 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℂ)
1615adantl 481 . . . . . . . . . . . . . . . 16 (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0) → (#‘𝐹) ∈ ℂ)
17 nn0cn 11179 . . . . . . . . . . . . . . . . 17 ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℂ)
1817adantr 480 . . . . . . . . . . . . . . . 16 (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0) → (#‘𝑊) ∈ ℂ)
19 1cnd 9935 . . . . . . . . . . . . . . . 16 (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0) → 1 ∈ ℂ)
2016, 18, 19addcan2d 10119 . . . . . . . . . . . . . . 15 (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0) → (((#‘𝐹) + 1) = ((#‘𝑊) + 1) ↔ (#‘𝐹) = (#‘𝑊)))
2120biimpd 218 . . . . . . . . . . . . . 14 (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0) → (((#‘𝐹) + 1) = ((#‘𝑊) + 1) → (#‘𝐹) = (#‘𝑊)))
2214, 21syl5bi 231 . . . . . . . . . . . . 13 (((#‘𝑊) ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ0) → (((#‘𝑊) + 1) = ((#‘𝐹) + 1) → (#‘𝐹) = (#‘𝑊)))
2322ex 449 . . . . . . . . . . . 12 ((#‘𝑊) ∈ ℕ0 → ((#‘𝐹) ∈ ℕ0 → (((#‘𝑊) + 1) = ((#‘𝐹) + 1) → (#‘𝐹) = (#‘𝑊))))
2423com23 84 . . . . . . . . . . 11 ((#‘𝑊) ∈ ℕ0 → (((#‘𝑊) + 1) = ((#‘𝐹) + 1) → ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) = (#‘𝑊))))
2513, 24syl 17 . . . . . . . . . 10 (𝑊 ∈ Word 𝑉 → (((#‘𝑊) + 1) = ((#‘𝐹) + 1) → ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) = (#‘𝑊))))
2625adantr 480 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (((#‘𝑊) + 1) = ((#‘𝐹) + 1) → ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) = (#‘𝑊))))
2712, 26sylbid 229 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → ((#‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) = ((#‘𝐹) + 1) → ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) = (#‘𝑊))))
2827com3l 87 . . . . . . 7 ((#‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) = ((#‘𝐹) + 1) → ((#‘𝐹) ∈ ℕ0 → ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (#‘𝐹) = (#‘𝑊))))
293, 28syl 17 . . . . . 6 (𝐹(𝑉 Walks 𝐸)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → ((#‘𝐹) ∈ ℕ0 → ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (#‘𝐹) = (#‘𝑊))))
3029com12 32 . . . . 5 ((#‘𝐹) ∈ ℕ0 → (𝐹(𝑉 Walks 𝐸)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (#‘𝐹) = (#‘𝑊))))
31303ad2ant1 1075 . . . 4 (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ V)) → (𝐹(𝑉 Walks 𝐸)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (#‘𝐹) = (#‘𝑊))))
322, 31mpcom 37 . . 3 (𝐹(𝑉 Walks 𝐸)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (#‘𝐹) = (#‘𝑊)))
331, 32sylbir 224 . 2 (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸) → ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (#‘𝐹) = (#‘𝑊)))
3433com12 32 1 ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸) → (#‘𝐹) = (#‘𝑊)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954  ℕ0cn0 11169  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149   Walks cwalk 26026 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-concat 13156  df-s1 13157  df-wlk 26036 This theorem is referenced by:  clwlkfoclwwlk  26372
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