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Theorem clWlkcompim 40987
 Description: Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
Hypotheses
Ref Expression
isclWlke.v 𝑉 = (Vtx‘𝐺)
isclWlke.i 𝐼 = (iEdg‘𝐺)
clWlkcomp.1 𝐹 = (1st𝑊)
clWlkcomp.2 𝑃 = (2nd𝑊)
Assertion
Ref Expression
clWlkcompim (𝑊 ∈ (ClWalkS‘𝐺) → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
Distinct variable groups:   𝑘,𝐹   𝑘,𝐺   𝑃,𝑘
Allowed substitution hints:   𝐼(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem clWlkcompim
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6131 . . . 4 (𝑊 ∈ (ClWalkS‘𝐺) → 𝐺 ∈ V)
2 clwlkS 40978 . . . . . 6 (𝐺 ∈ V → (ClWalkS‘𝐺) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))})
32eleq2d 2673 . . . . 5 (𝐺 ∈ V → (𝑊 ∈ (ClWalkS‘𝐺) ↔ 𝑊 ∈ {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))}))
4 elopaelxp 5114 . . . . . . 7 (𝑊 ∈ {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))} → 𝑊 ∈ (V × V))
54anim2i 591 . . . . . 6 ((𝐺 ∈ V ∧ 𝑊 ∈ {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))}) → (𝐺 ∈ V ∧ 𝑊 ∈ (V × V)))
65ex 449 . . . . 5 (𝐺 ∈ V → (𝑊 ∈ {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))} → (𝐺 ∈ V ∧ 𝑊 ∈ (V × V))))
73, 6sylbid 229 . . . 4 (𝐺 ∈ V → (𝑊 ∈ (ClWalkS‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ (V × V))))
81, 7mpcom 37 . . 3 (𝑊 ∈ (ClWalkS‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ (V × V)))
9 isclWlke.v . . . 4 𝑉 = (Vtx‘𝐺)
10 isclWlke.i . . . 4 𝐼 = (iEdg‘𝐺)
11 clWlkcomp.1 . . . 4 𝐹 = (1st𝑊)
12 clWlkcomp.2 . . . 4 𝑃 = (2nd𝑊)
139, 10, 11, 12clWlkcomp 40986 . . 3 ((𝐺 ∈ V ∧ 𝑊 ∈ (V × V)) → (𝑊 ∈ (ClWalkS‘𝐺) ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))))
148, 13syl 17 . 2 (𝑊 ∈ (ClWalkS‘𝐺) → (𝑊 ∈ (ClWalkS‘𝐺) ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))))
1514ibi 255 1 (𝑊 ∈ (ClWalkS‘𝐺) → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  if-wif 1006   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  {copab 4642   × cxp 5036  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  Vtxcvtx 25673  iEdgciedg 25674  1Walksc1wlks 40796  ClWalkScclwlks 40976 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-ifp 1007  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-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-1wlks 40800  df-clwlks 40977 This theorem is referenced by:  upgrclwlkcompim  40988  clwlksfclwwlk2wrd  41265  clwlksfclwwlk1hash  41267  clwlksf1clwwlklem0  41271
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