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Metamath Proof Explorer

Theorem isclwwlk

Description: Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018) (Revised by AV, 24-Apr-2021)

Ref Expression
Hypotheses clwwlk.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
clwwlk.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion isclwwlk ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ↔ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )

Proof

Step Hyp Ref Expression
1 clwwlk.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 clwwlk.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
3 neeq1 ⊢ ( 𝑤 = 𝑊 → ( 𝑤 ≠ ∅ ↔ 𝑊 ≠ ∅ ) )
4 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ♯ ‘ 𝑤 ) = ( ♯ ‘ 𝑊 ) )
5 4 oveq1d ⊢ ( 𝑤 = 𝑊 → ( ( ♯ ‘ 𝑤 ) − 1 ) = ( ( ♯ ‘ 𝑊 ) − 1 ) )
6 5 oveq2d ⊢ ( 𝑤 = 𝑊 → ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) = ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
7 fveq1 ⊢ ( 𝑤 = 𝑊 → ( 𝑤 ‘ 𝑖 ) = ( 𝑊 ‘ 𝑖 ) )
8 fveq1 ⊢ ( 𝑤 = 𝑊 → ( 𝑤 ‘ ( 𝑖 + 1 ) ) = ( 𝑊 ‘ ( 𝑖 + 1 ) ) )
9 7 8 preq12d ⊢ ( 𝑤 = 𝑊 → { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } = { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } )
10 9 eleq1d ⊢ ( 𝑤 = 𝑊 → ( { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ↔ { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) )
11 6 10 raleqbidv ⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ↔ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ) )
12 fveq2 ⊢ ( 𝑤 = 𝑊 → ( lastS ‘ 𝑤 ) = ( lastS ‘ 𝑊 ) )
13 fveq1 ⊢ ( 𝑤 = 𝑊 → ( 𝑤 ‘ 0 ) = ( 𝑊 ‘ 0 ) )
14 12 13 preq12d ⊢ ( 𝑤 = 𝑊 → { ( lastS ‘ 𝑤 ) , ( 𝑤 ‘ 0 ) } = { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } )
15 14 eleq1d ⊢ ( 𝑤 = 𝑊 → ( { ( lastS ‘ 𝑤 ) , ( 𝑤 ‘ 0 ) } ∈ 𝐸 ↔ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )
16 3 11 15 3anbi123d ⊢ ( 𝑤 = 𝑊 → ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑤 ) , ( 𝑤 ‘ 0 ) } ∈ 𝐸 ) ↔ ( 𝑊 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
17 16 elrab ⊢ ( 𝑊 ∈ { 𝑤 ∈ Word 𝑉 ∣ ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑤 ) , ( 𝑤 ‘ 0 ) } ∈ 𝐸 ) } ↔ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
18 1 2 clwwlk ⊢ ( ClWWalks ‘ 𝐺 ) = { 𝑤 ∈ Word 𝑉 ∣ ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑤 ) , ( 𝑤 ‘ 0 ) } ∈ 𝐸 ) }
19 18 eleq2i ⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ↔ 𝑊 ∈ { 𝑤 ∈ Word 𝑉 ∣ ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑤 ) , ( 𝑤 ‘ 0 ) } ∈ 𝐸 ) } )
20 3anass ⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ↔ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
21 anass ⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) ↔ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 ≠ ∅ ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) ) )
22 3anass ⊢ ( ( 𝑊 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ↔ ( 𝑊 ≠ ∅ ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
23 22 bicomi ⊢ ( ( 𝑊 ≠ ∅ ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) ↔ ( 𝑊 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )
24 23 anbi2i ⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 ≠ ∅ ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) ) ↔ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
25 20 21 24 3bitri ⊢ ( ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ↔ ( 𝑊 ∈ Word 𝑉 ∧ ( 𝑊 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) ) )
26 17 19 25 3bitr4i ⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ↔ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ 𝐸 ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ 𝐸 ) )