Theorem eleclclwwlknlem1

Description: Lemma 1 for eleclclwwlkn . (Contributed by Alexander van der Vekens, 11-May-2018) (Revised by AV, 30-Apr-2021)

	Ref	Expression
Hypothesis	erclwwlkn1.w	⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 )
Assertion	eleclclwwlknlem1	⊢ ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) → ( ( 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑋 cyclShift 𝑛 ) ) )

Proof

Step	Hyp	Ref	Expression
1		erclwwlkn1.w	⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 )
2		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3	2	clwwlknbp	⊢ ( 𝑌 ∈ ( 𝑁 ClWWalksN 𝐺 ) → ( 𝑌 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑌 ) = 𝑁 ) )
4	3 1	eleq2s	⊢ ( 𝑌 ∈ 𝑊 → ( 𝑌 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑌 ) = 𝑁 ) )
5	4	adantl	⊢ ( ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) → ( 𝑌 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑌 ) = 𝑁 ) )
6	5	adantl	⊢ ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) → ( 𝑌 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑌 ) = 𝑁 ) )
7	6	adantr	⊢ ( ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) ∧ ( 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) ) → ( 𝑌 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑌 ) = 𝑁 ) )
8		simpl	⊢ ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) → 𝐾 ∈ ( 0 ... 𝑁 ) )
9	8	adantr	⊢ ( ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) ∧ ( 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) ) → 𝐾 ∈ ( 0 ... 𝑁 ) )
10		simpl	⊢ ( ( 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) → 𝑋 = ( 𝑌 cyclShift 𝐾 ) )
11	10	adantl	⊢ ( ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) ∧ ( 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) ) → 𝑋 = ( 𝑌 cyclShift 𝐾 ) )
12		simprr	⊢ ( ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) ∧ ( 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) ) → ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) )
13	9 11 12	3jca	⊢ ( ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) ∧ ( 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) ) → ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) )
14		2cshwcshw	⊢ ( ( 𝑌 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑌 ) = 𝑁 ) → ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑋 cyclShift 𝑛 ) ) )
15	7 13 14	sylc	⊢ ( ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) ∧ ( 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑋 cyclShift 𝑛 ) )
16	15	ex	⊢ ( ( 𝐾 ∈ ( 0 ... 𝑁 ) ∧ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ) → ( ( 𝑋 = ( 𝑌 cyclShift 𝐾 ) ∧ ∃ 𝑚 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑌 cyclShift 𝑚 ) ) → ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑍 = ( 𝑋 cyclShift 𝑛 ) ) )