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Theorem wlkp1 28928
Description: Append one path segment (edge) 𝐸 from vertex (π‘ƒβ€˜π‘) to a vertex 𝐢 to a walk ⟨𝐹, π‘ƒβŸ© to become a walk ⟨𝐻, π‘„βŸ© of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. Formerly proven directly for Eulerian paths (for pseudographs), see eupthp1 29459. (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 = (Vtxβ€˜πΊ)
wlkp1.i 𝐼 = (iEdgβ€˜πΊ)
wlkp1.f (πœ‘ β†’ Fun 𝐼)
wlkp1.a (πœ‘ β†’ 𝐼 ∈ Fin)
wlkp1.b (πœ‘ β†’ 𝐡 ∈ π‘Š)
wlkp1.c (πœ‘ β†’ 𝐢 ∈ 𝑉)
wlkp1.d (πœ‘ β†’ Β¬ 𝐡 ∈ dom 𝐼)
wlkp1.w (πœ‘ β†’ 𝐹(Walksβ€˜πΊ)𝑃)
wlkp1.n 𝑁 = (β™―β€˜πΉ)
wlkp1.e (πœ‘ β†’ 𝐸 ∈ (Edgβ€˜πΊ))
wlkp1.x (πœ‘ β†’ {(π‘ƒβ€˜π‘), 𝐢} βŠ† 𝐸)
wlkp1.u (πœ‘ β†’ (iEdgβ€˜π‘†) = (𝐼 βˆͺ {⟨𝐡, 𝐸⟩}))
wlkp1.h 𝐻 = (𝐹 βˆͺ {βŸ¨π‘, 𝐡⟩})
wlkp1.q 𝑄 = (𝑃 βˆͺ {⟨(𝑁 + 1), 𝐢⟩})
wlkp1.s (πœ‘ β†’ (Vtxβ€˜π‘†) = 𝑉)
wlkp1.l ((πœ‘ ∧ 𝐢 = (π‘ƒβ€˜π‘)) β†’ 𝐸 = {𝐢})
Assertion
Ref Expression
wlkp1 (πœ‘ β†’ 𝐻(Walksβ€˜π‘†)𝑄)
Proof of Theorem wlkp1
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 wlkp1.w . . . . . 6 (πœ‘ β†’ 𝐹(Walksβ€˜πΊ)𝑃)
2 wlkp1.i . . . . . . 7 𝐼 = (iEdgβ€˜πΊ)
32wlkf 28861 . . . . . 6 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝐹 ∈ Word dom 𝐼)
4 wrdf 14466 . . . . . . 7 (𝐹 ∈ Word dom 𝐼 β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐼)
5 wlkp1.n . . . . . . . . . 10 𝑁 = (β™―β€˜πΉ)
65eqcomi 2742 . . . . . . . . 9 (β™―β€˜πΉ) = 𝑁
76oveq2i 7417 . . . . . . . 8 (0..^(β™―β€˜πΉ)) = (0..^𝑁)
87feq2i 6707 . . . . . . 7 (𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐼 ↔ 𝐹:(0..^𝑁)⟢dom 𝐼)
94, 8sylib 217 . . . . . 6 (𝐹 ∈ Word dom 𝐼 β†’ 𝐹:(0..^𝑁)⟢dom 𝐼)
101, 3, 93syl 18 . . . . 5 (πœ‘ β†’ 𝐹:(0..^𝑁)⟢dom 𝐼)
115fvexi 6903 . . . . . . 7 𝑁 ∈ V
1211a1i 11 . . . . . 6 (πœ‘ β†’ 𝑁 ∈ V)
13 wlkp1.b . . . . . . . 8 (πœ‘ β†’ 𝐡 ∈ π‘Š)
14 snidg 4662 . . . . . . . 8 (𝐡 ∈ π‘Š β†’ 𝐡 ∈ {𝐡})
1513, 14syl 17 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ {𝐡})
16 wlkp1.e . . . . . . . 8 (πœ‘ β†’ 𝐸 ∈ (Edgβ€˜πΊ))
17 dmsnopg 6210 . . . . . . . 8 (𝐸 ∈ (Edgβ€˜πΊ) β†’ dom {⟨𝐡, 𝐸⟩} = {𝐡})
1816, 17syl 17 . . . . . . 7 (πœ‘ β†’ dom {⟨𝐡, 𝐸⟩} = {𝐡})
1915, 18eleqtrrd 2837 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ dom {⟨𝐡, 𝐸⟩})
2012, 19fsnd 6874 . . . . 5 (πœ‘ β†’ {βŸ¨π‘, 𝐡⟩}:{𝑁}⟢dom {⟨𝐡, 𝐸⟩})
21 fzodisjsn 13667 . . . . . 6 ((0..^𝑁) ∩ {𝑁}) = βˆ…
2221a1i 11 . . . . 5 (πœ‘ β†’ ((0..^𝑁) ∩ {𝑁}) = βˆ…)
23 fun 6751 . . . . 5 (((𝐹:(0..^𝑁)⟢dom 𝐼 ∧ {βŸ¨π‘, 𝐡⟩}:{𝑁}⟢dom {⟨𝐡, 𝐸⟩}) ∧ ((0..^𝑁) ∩ {𝑁}) = βˆ…) β†’ (𝐹 βˆͺ {βŸ¨π‘, 𝐡⟩}):((0..^𝑁) βˆͺ {𝑁})⟢(dom 𝐼 βˆͺ dom {⟨𝐡, 𝐸⟩}))
2410, 20, 22, 23syl21anc 837 . . . 4 (πœ‘ β†’ (𝐹 βˆͺ {βŸ¨π‘, 𝐡⟩}):((0..^𝑁) βˆͺ {𝑁})⟢(dom 𝐼 βˆͺ dom {⟨𝐡, 𝐸⟩}))
25 wlkp1.h . . . . . 6 𝐻 = (𝐹 βˆͺ {βŸ¨π‘, 𝐡⟩})
2625a1i 11 . . . . 5 (πœ‘ β†’ 𝐻 = (𝐹 βˆͺ {βŸ¨π‘, 𝐡⟩}))
27 wlkp1.v . . . . . . . 8 𝑉 = (Vtxβ€˜πΊ)
28 wlkp1.f . . . . . . . 8 (πœ‘ β†’ Fun 𝐼)
29 wlkp1.a . . . . . . . 8 (πœ‘ β†’ 𝐼 ∈ Fin)
30 wlkp1.c . . . . . . . 8 (πœ‘ β†’ 𝐢 ∈ 𝑉)
31 wlkp1.d . . . . . . . 8 (πœ‘ β†’ Β¬ 𝐡 ∈ dom 𝐼)
32 wlkp1.x . . . . . . . 8 (πœ‘ β†’ {(π‘ƒβ€˜π‘), 𝐢} βŠ† 𝐸)
33 wlkp1.u . . . . . . . 8 (πœ‘ β†’ (iEdgβ€˜π‘†) = (𝐼 βˆͺ {⟨𝐡, 𝐸⟩}))
3427, 2, 28, 29, 13, 30, 31, 1, 5, 16, 32, 33, 25wlkp1lem2 28921 . . . . . . 7 (πœ‘ β†’ (β™―β€˜π») = (𝑁 + 1))
3534oveq2d 7422 . . . . . 6 (πœ‘ β†’ (0..^(β™―β€˜π»)) = (0..^(𝑁 + 1)))
36 wlkcl 28862 . . . . . . . 8 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ∈ β„•0)
37 eleq1 2822 . . . . . . . . . . 11 ((β™―β€˜πΉ) = 𝑁 β†’ ((β™―β€˜πΉ) ∈ β„•0 ↔ 𝑁 ∈ β„•0))
3837eqcoms 2741 . . . . . . . . . 10 (𝑁 = (β™―β€˜πΉ) β†’ ((β™―β€˜πΉ) ∈ β„•0 ↔ 𝑁 ∈ β„•0))
39 elnn0uz 12864 . . . . . . . . . . 11 (𝑁 ∈ β„•0 ↔ 𝑁 ∈ (β„€β‰₯β€˜0))
4039biimpi 215 . . . . . . . . . 10 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
4138, 40syl6bi 253 . . . . . . . . 9 (𝑁 = (β™―β€˜πΉ) β†’ ((β™―β€˜πΉ) ∈ β„•0 β†’ 𝑁 ∈ (β„€β‰₯β€˜0)))
425, 41ax-mp 5 . . . . . . . 8 ((β™―β€˜πΉ) ∈ β„•0 β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
431, 36, 423syl 18 . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜0))
44 fzosplitsn 13737 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜0) β†’ (0..^(𝑁 + 1)) = ((0..^𝑁) βˆͺ {𝑁}))
4543, 44syl 17 . . . . . 6 (πœ‘ β†’ (0..^(𝑁 + 1)) = ((0..^𝑁) βˆͺ {𝑁}))
4635, 45eqtrd 2773 . . . . 5 (πœ‘ β†’ (0..^(β™―β€˜π»)) = ((0..^𝑁) βˆͺ {𝑁}))
4733dmeqd 5904 . . . . . 6 (πœ‘ β†’ dom (iEdgβ€˜π‘†) = dom (𝐼 βˆͺ {⟨𝐡, 𝐸⟩}))
48 dmun 5909 . . . . . 6 dom (𝐼 βˆͺ {⟨𝐡, 𝐸⟩}) = (dom 𝐼 βˆͺ dom {⟨𝐡, 𝐸⟩})
4947, 48eqtrdi 2789 . . . . 5 (πœ‘ β†’ dom (iEdgβ€˜π‘†) = (dom 𝐼 βˆͺ dom {⟨𝐡, 𝐸⟩}))
5026, 46, 49feq123d 6704 . . . 4 (πœ‘ β†’ (𝐻:(0..^(β™―β€˜π»))⟢dom (iEdgβ€˜π‘†) ↔ (𝐹 βˆͺ {βŸ¨π‘, 𝐡⟩}):((0..^𝑁) βˆͺ {𝑁})⟢(dom 𝐼 βˆͺ dom {⟨𝐡, 𝐸⟩})))
5124, 50mpbird 257 . . 3 (πœ‘ β†’ 𝐻:(0..^(β™―β€˜π»))⟢dom (iEdgβ€˜π‘†))
52 iswrdb 14467 . . 3 (𝐻 ∈ Word dom (iEdgβ€˜π‘†) ↔ 𝐻:(0..^(β™―β€˜π»))⟢dom (iEdgβ€˜π‘†))
5351, 52sylibr 233 . 2 (πœ‘ β†’ 𝐻 ∈ Word dom (iEdgβ€˜π‘†))
5427wlkp 28863 . . . . . . 7 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
551, 54syl 17 . . . . . 6 (πœ‘ β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
565oveq2i 7417 . . . . . . 7 (0...𝑁) = (0...(β™―β€˜πΉ))
5756feq2i 6707 . . . . . 6 (𝑃:(0...𝑁)βŸΆπ‘‰ ↔ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
5855, 57sylibr 233 . . . . 5 (πœ‘ β†’ 𝑃:(0...𝑁)βŸΆπ‘‰)
59 ovexd 7441 . . . . . 6 (πœ‘ β†’ (𝑁 + 1) ∈ V)
6059, 30fsnd 6874 . . . . 5 (πœ‘ β†’ {⟨(𝑁 + 1), 𝐢⟩}:{(𝑁 + 1)}βŸΆπ‘‰)
61 fzp1disj 13557 . . . . . 6 ((0...𝑁) ∩ {(𝑁 + 1)}) = βˆ…
6261a1i 11 . . . . 5 (πœ‘ β†’ ((0...𝑁) ∩ {(𝑁 + 1)}) = βˆ…)
63 fun 6751 . . . . 5 (((𝑃:(0...𝑁)βŸΆπ‘‰ ∧ {⟨(𝑁 + 1), 𝐢⟩}:{(𝑁 + 1)}βŸΆπ‘‰) ∧ ((0...𝑁) ∩ {(𝑁 + 1)}) = βˆ…) β†’ (𝑃 βˆͺ {⟨(𝑁 + 1), 𝐢⟩}):((0...𝑁) βˆͺ {(𝑁 + 1)})⟢(𝑉 βˆͺ 𝑉))
6458, 60, 62, 63syl21anc 837 . . . 4 (πœ‘ β†’ (𝑃 βˆͺ {⟨(𝑁 + 1), 𝐢⟩}):((0...𝑁) βˆͺ {(𝑁 + 1)})⟢(𝑉 βˆͺ 𝑉))
65 fzsuc 13545 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜0) β†’ (0...(𝑁 + 1)) = ((0...𝑁) βˆͺ {(𝑁 + 1)}))
6643, 65syl 17 . . . . 5 (πœ‘ β†’ (0...(𝑁 + 1)) = ((0...𝑁) βˆͺ {(𝑁 + 1)}))
67 unidm 4152 . . . . . . 7 (𝑉 βˆͺ 𝑉) = 𝑉
6867eqcomi 2742 . . . . . 6 𝑉 = (𝑉 βˆͺ 𝑉)
6968a1i 11 . . . . 5 (πœ‘ β†’ 𝑉 = (𝑉 βˆͺ 𝑉))
7066, 69feq23d 6710 . . . 4 (πœ‘ β†’ ((𝑃 βˆͺ {⟨(𝑁 + 1), 𝐢⟩}):(0...(𝑁 + 1))βŸΆπ‘‰ ↔ (𝑃 βˆͺ {⟨(𝑁 + 1), 𝐢⟩}):((0...𝑁) βˆͺ {(𝑁 + 1)})⟢(𝑉 βˆͺ 𝑉)))
7164, 70mpbird 257 . . 3 (πœ‘ β†’ (𝑃 βˆͺ {⟨(𝑁 + 1), 𝐢⟩}):(0...(𝑁 + 1))βŸΆπ‘‰)
72 wlkp1.q . . . . 5 𝑄 = (𝑃 βˆͺ {⟨(𝑁 + 1), 𝐢⟩})
7372a1i 11 . . . 4 (πœ‘ β†’ 𝑄 = (𝑃 βˆͺ {⟨(𝑁 + 1), 𝐢⟩}))
7434oveq2d 7422 . . . 4 (πœ‘ β†’ (0...(β™―β€˜π»)) = (0...(𝑁 + 1)))
75 wlkp1.s . . . 4 (πœ‘ β†’ (Vtxβ€˜π‘†) = 𝑉)
7673, 74, 75feq123d 6704 . . 3 (πœ‘ β†’ (𝑄:(0...(β™―β€˜π»))⟢(Vtxβ€˜π‘†) ↔ (𝑃 βˆͺ {⟨(𝑁 + 1), 𝐢⟩}):(0...(𝑁 + 1))βŸΆπ‘‰))
7771, 76mpbird 257 . 2 (πœ‘ β†’ 𝑄:(0...(β™―β€˜π»))⟢(Vtxβ€˜π‘†))
78 wlkp1.l . . 3 ((πœ‘ ∧ 𝐢 = (π‘ƒβ€˜π‘)) β†’ 𝐸 = {𝐢})
7927, 2, 28, 29, 13, 30, 31, 1, 5, 16, 32, 33, 25, 72, 75, 78wlkp1lem8 28927 . 2 (πœ‘ β†’ βˆ€π‘˜ ∈ (0..^(β™―β€˜π»))if-((π‘„β€˜π‘˜) = (π‘„β€˜(π‘˜ + 1)), ((iEdgβ€˜π‘†)β€˜(π»β€˜π‘˜)) = {(π‘„β€˜π‘˜)}, {(π‘„β€˜π‘˜), (π‘„β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘†)β€˜(π»β€˜π‘˜))))
8027, 2, 28, 29, 13, 30, 31, 1, 5, 16, 32, 33, 25, 72, 75wlkp1lem4 28923 . . 3 (πœ‘ β†’ (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
81 eqid 2733 . . . 4 (Vtxβ€˜π‘†) = (Vtxβ€˜π‘†)
82 eqid 2733 . . . 4 (iEdgβ€˜π‘†) = (iEdgβ€˜π‘†)
8381, 82iswlk 28857 . . 3 ((𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V) β†’ (𝐻(Walksβ€˜π‘†)𝑄 ↔ (𝐻 ∈ Word dom (iEdgβ€˜π‘†) ∧ 𝑄:(0...(β™―β€˜π»))⟢(Vtxβ€˜π‘†) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π»))if-((π‘„β€˜π‘˜) = (π‘„β€˜(π‘˜ + 1)), ((iEdgβ€˜π‘†)β€˜(π»β€˜π‘˜)) = {(π‘„β€˜π‘˜)}, {(π‘„β€˜π‘˜), (π‘„β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘†)β€˜(π»β€˜π‘˜))))))
8480, 83syl 17 . 2 (πœ‘ β†’ (𝐻(Walksβ€˜π‘†)𝑄 ↔ (𝐻 ∈ Word dom (iEdgβ€˜π‘†) ∧ 𝑄:(0...(β™―β€˜π»))⟢(Vtxβ€˜π‘†) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜π»))if-((π‘„β€˜π‘˜) = (π‘„β€˜(π‘˜ + 1)), ((iEdgβ€˜π‘†)β€˜(π»β€˜π‘˜)) = {(π‘„β€˜π‘˜)}, {(π‘„β€˜π‘˜), (π‘„β€˜(π‘˜ + 1))} βŠ† ((iEdgβ€˜π‘†)β€˜(π»β€˜π‘˜))))))
8553, 77, 79, 84mpbir3and 1343 1 (πœ‘ β†’ 𝐻(Walksβ€˜π‘†)𝑄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397  if-wif 1062   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  {cpr 4630  βŸ¨cop 4634   class class class wbr 5148  dom cdm 5676  Fun wfun 6535  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406  Fincfn 8936  0cc0 11107  1c1 11108   + caddc 11110  β„•0cn0 12469  β„€β‰₯cuz 12819  ...cfz 13481  ..^cfzo 13624  β™―chash 14287  Word cword 14461  Vtxcvtx 28246  iEdgciedg 28247  Edgcedg 28297  Walkscwlks 28843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-oadd 8467  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-dju 9893  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-fzo 13625  df-hash 14288  df-word 14462  df-wlks 28846
This theorem is referenced by:  eupthp1  29459
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