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Theorem wlkp1 29714
Description: Append one path segment (edge) 𝐸 from vertex (𝑃𝑁) to a vertex 𝐶 to a walk 𝐹, 𝑃 to become a walk 𝐻, 𝑄 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. Formerly proven directly for Eulerian paths (for pseudographs), see eupthp1 30245. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Proof shortened by AV, 18-Apr-2021.) (Revised by AV, 8-Apr-2024.)
Hypotheses
Ref Expression
wlkp1.v 𝑉 = (Vtx‘𝐺)
wlkp1.i 𝐼 = (iEdg‘𝐺)
wlkp1.f (𝜑 → Fun 𝐼)
wlkp1.a (𝜑𝐼 ∈ Fin)
wlkp1.b (𝜑𝐵𝑊)
wlkp1.c (𝜑𝐶𝑉)
wlkp1.d (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
wlkp1.w (𝜑𝐹(Walks‘𝐺)𝑃)
wlkp1.n 𝑁 = (♯‘𝐹)
wlkp1.e (𝜑𝐸 ∈ (Edg‘𝐺))
wlkp1.x (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)
wlkp1.u (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
wlkp1.h 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
wlkp1.q 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
wlkp1.s (𝜑 → (Vtx‘𝑆) = 𝑉)
wlkp1.l ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})
Assertion
Ref Expression
wlkp1 (𝜑𝐻(Walks‘𝑆)𝑄)

Proof of Theorem wlkp1
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 wlkp1.w . . . . . 6 (𝜑𝐹(Walks‘𝐺)𝑃)
2 wlkp1.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
32wlkf 29647 . . . . . 6 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
4 wrdf 14554 . . . . . . 7 (𝐹 ∈ Word dom 𝐼𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
5 wlkp1.n . . . . . . . . . 10 𝑁 = (♯‘𝐹)
65eqcomi 2744 . . . . . . . . 9 (♯‘𝐹) = 𝑁
76oveq2i 7442 . . . . . . . 8 (0..^(♯‘𝐹)) = (0..^𝑁)
87feq2i 6729 . . . . . . 7 (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼𝐹:(0..^𝑁)⟶dom 𝐼)
94, 8sylib 218 . . . . . 6 (𝐹 ∈ Word dom 𝐼𝐹:(0..^𝑁)⟶dom 𝐼)
101, 3, 93syl 18 . . . . 5 (𝜑𝐹:(0..^𝑁)⟶dom 𝐼)
115fvexi 6921 . . . . . . 7 𝑁 ∈ V
1211a1i 11 . . . . . 6 (𝜑𝑁 ∈ V)
13 wlkp1.b . . . . . . . 8 (𝜑𝐵𝑊)
14 snidg 4665 . . . . . . . 8 (𝐵𝑊𝐵 ∈ {𝐵})
1513, 14syl 17 . . . . . . 7 (𝜑𝐵 ∈ {𝐵})
16 wlkp1.e . . . . . . . 8 (𝜑𝐸 ∈ (Edg‘𝐺))
17 dmsnopg 6235 . . . . . . . 8 (𝐸 ∈ (Edg‘𝐺) → dom {⟨𝐵, 𝐸⟩} = {𝐵})
1816, 17syl 17 . . . . . . 7 (𝜑 → dom {⟨𝐵, 𝐸⟩} = {𝐵})
1915, 18eleqtrrd 2842 . . . . . 6 (𝜑𝐵 ∈ dom {⟨𝐵, 𝐸⟩})
2012, 19fsnd 6892 . . . . 5 (𝜑 → {⟨𝑁, 𝐵⟩}:{𝑁}⟶dom {⟨𝐵, 𝐸⟩})
21 fzodisjsn 13734 . . . . . 6 ((0..^𝑁) ∩ {𝑁}) = ∅
2221a1i 11 . . . . 5 (𝜑 → ((0..^𝑁) ∩ {𝑁}) = ∅)
23 fun 6771 . . . . 5 (((𝐹:(0..^𝑁)⟶dom 𝐼 ∧ {⟨𝑁, 𝐵⟩}:{𝑁}⟶dom {⟨𝐵, 𝐸⟩}) ∧ ((0..^𝑁) ∩ {𝑁}) = ∅) → (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
2410, 20, 22, 23syl21anc 838 . . . 4 (𝜑 → (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
25 wlkp1.h . . . . . 6 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
2625a1i 11 . . . . 5 (𝜑𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩}))
27 wlkp1.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
28 wlkp1.f . . . . . . . 8 (𝜑 → Fun 𝐼)
29 wlkp1.a . . . . . . . 8 (𝜑𝐼 ∈ Fin)
30 wlkp1.c . . . . . . . 8 (𝜑𝐶𝑉)
31 wlkp1.d . . . . . . . 8 (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
32 wlkp1.x . . . . . . . 8 (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)
33 wlkp1.u . . . . . . . 8 (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
3427, 2, 28, 29, 13, 30, 31, 1, 5, 16, 32, 33, 25wlkp1lem2 29707 . . . . . . 7 (𝜑 → (♯‘𝐻) = (𝑁 + 1))
3534oveq2d 7447 . . . . . 6 (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
36 wlkcl 29648 . . . . . . . 8 (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
37 eleq1 2827 . . . . . . . . . . 11 ((♯‘𝐹) = 𝑁 → ((♯‘𝐹) ∈ ℕ0𝑁 ∈ ℕ0))
3837eqcoms 2743 . . . . . . . . . 10 (𝑁 = (♯‘𝐹) → ((♯‘𝐹) ∈ ℕ0𝑁 ∈ ℕ0))
39 elnn0uz 12921 . . . . . . . . . . 11 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
4039biimpi 216 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
4138, 40biimtrdi 253 . . . . . . . . 9 (𝑁 = (♯‘𝐹) → ((♯‘𝐹) ∈ ℕ0𝑁 ∈ (ℤ‘0)))
425, 41ax-mp 5 . . . . . . . 8 ((♯‘𝐹) ∈ ℕ0𝑁 ∈ (ℤ‘0))
431, 36, 423syl 18 . . . . . . 7 (𝜑𝑁 ∈ (ℤ‘0))
44 fzosplitsn 13811 . . . . . . 7 (𝑁 ∈ (ℤ‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
4543, 44syl 17 . . . . . 6 (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
4635, 45eqtrd 2775 . . . . 5 (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
4733dmeqd 5919 . . . . . 6 (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
48 dmun 5924 . . . . . 6 dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})
4947, 48eqtrdi 2791 . . . . 5 (𝜑 → dom (iEdg‘𝑆) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
5026, 46, 49feq123d 6726 . . . 4 (𝜑 → (𝐻:(0..^(♯‘𝐻))⟶dom (iEdg‘𝑆) ↔ (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})))
5124, 50mpbird 257 . . 3 (𝜑𝐻:(0..^(♯‘𝐻))⟶dom (iEdg‘𝑆))
52 iswrdb 14555 . . 3 (𝐻 ∈ Word dom (iEdg‘𝑆) ↔ 𝐻:(0..^(♯‘𝐻))⟶dom (iEdg‘𝑆))
5351, 52sylibr 234 . 2 (𝜑𝐻 ∈ Word dom (iEdg‘𝑆))
5427wlkp 29649 . . . . . . 7 (𝐹(Walks‘𝐺)𝑃𝑃:(0...(♯‘𝐹))⟶𝑉)
551, 54syl 17 . . . . . 6 (𝜑𝑃:(0...(♯‘𝐹))⟶𝑉)
565oveq2i 7442 . . . . . . 7 (0...𝑁) = (0...(♯‘𝐹))
5756feq2i 6729 . . . . . 6 (𝑃:(0...𝑁)⟶𝑉𝑃:(0...(♯‘𝐹))⟶𝑉)
5855, 57sylibr 234 . . . . 5 (𝜑𝑃:(0...𝑁)⟶𝑉)
59 ovexd 7466 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ V)
6059, 30fsnd 6892 . . . . 5 (𝜑 → {⟨(𝑁 + 1), 𝐶⟩}:{(𝑁 + 1)}⟶𝑉)
61 fzp1disj 13620 . . . . . 6 ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅
6261a1i 11 . . . . 5 (𝜑 → ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅)
63 fun 6771 . . . . 5 (((𝑃:(0...𝑁)⟶𝑉 ∧ {⟨(𝑁 + 1), 𝐶⟩}:{(𝑁 + 1)}⟶𝑉) ∧ ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅) → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉𝑉))
6458, 60, 62, 63syl21anc 838 . . . 4 (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉𝑉))
65 fzsuc 13608 . . . . . 6 (𝑁 ∈ (ℤ‘0) → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)}))
6643, 65syl 17 . . . . 5 (𝜑 → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)}))
67 unidm 4167 . . . . . . 7 (𝑉𝑉) = 𝑉
6867eqcomi 2744 . . . . . 6 𝑉 = (𝑉𝑉)
6968a1i 11 . . . . 5 (𝜑𝑉 = (𝑉𝑉))
7066, 69feq23d 6732 . . . 4 (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉 ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉𝑉)))
7164, 70mpbird 257 . . 3 (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉)
72 wlkp1.q . . . . 5 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
7372a1i 11 . . . 4 (𝜑𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}))
7434oveq2d 7447 . . . 4 (𝜑 → (0...(♯‘𝐻)) = (0...(𝑁 + 1)))
75 wlkp1.s . . . 4 (𝜑 → (Vtx‘𝑆) = 𝑉)
7673, 74, 75feq123d 6726 . . 3 (𝜑 → (𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉))
7771, 76mpbird 257 . 2 (𝜑𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆))
78 wlkp1.l . . 3 ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})
7927, 2, 28, 29, 13, 30, 31, 1, 5, 16, 32, 33, 25, 72, 75, 78wlkp1lem8 29713 . 2 (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐻))if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))))
8027, 2, 28, 29, 13, 30, 31, 1, 5, 16, 32, 33, 25, 72, 75wlkp1lem4 29709 . . 3 (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
81 eqid 2735 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
82 eqid 2735 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
8381, 82iswlk 29643 . . 3 ((𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V) → (𝐻(Walks‘𝑆)𝑄 ↔ (𝐻 ∈ Word dom (iEdg‘𝑆) ∧ 𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐻))if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))))))
8480, 83syl 17 . 2 (𝜑 → (𝐻(Walks‘𝑆)𝑄 ↔ (𝐻 ∈ Word dom (iEdg‘𝑆) ∧ 𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(♯‘𝐻))if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))))))
8553, 77, 79, 84mpbir3and 1341 1 (𝜑𝐻(Walks‘𝑆)𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  if-wif 1062  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631  {cpr 4633  cop 4637   class class class wbr 5148  dom cdm 5689  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  Fincfn 8984  0cc0 11153  1c1 11154   + caddc 11156  0cn0 12524  cuz 12876  ...cfz 13544  ..^cfzo 13691  chash 14366  Word cword 14549  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079  Walkscwlks 29629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-wlks 29632
This theorem is referenced by:  eupthp1  30245
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