Theorem efgsf

Description: Value of the auxiliary function S defining a sequence of extensions starting at some irreducible word. (Contributed by Mario Carneiro, 1-Oct-2015)

	Ref	Expression
Hypotheses	efgval.w	⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) )
	efgval.r	⊢ ∼ = ( ~FG ‘ 𝐼 )
	efgval2.m	⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ ⟨ 𝑦 , ( 1o ∖ 𝑧 ) ⟩ )
	efgval2.t	⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice ⟨ 𝑛 , 𝑛 , ⟨“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”⟩ ⟩ ) ) )
	efgred.d	⊢ 𝐷 = ( 𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) )
	efgred.s	⊢ 𝑆 = ( 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ↦ ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) )
Assertion	efgsf	⊢ 𝑆 : { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ⟶ 𝑊

Proof

Step	Hyp	Ref	Expression
1		efgval.w	⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) )
2		efgval.r	⊢ ∼ = ( ~FG ‘ 𝐼 )
3		efgval2.m	⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ ⟨ 𝑦 , ( 1o ∖ 𝑧 ) ⟩ )
4		efgval2.t	⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice ⟨ 𝑛 , 𝑛 , ⟨“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”⟩ ⟩ ) ) )
5		efgred.d	⊢ 𝐷 = ( 𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) )
6		efgred.s	⊢ 𝑆 = ( 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ↦ ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) )
7		id	⊢ ( 𝑚 = 𝑡 → 𝑚 = 𝑡 )
8		fveq2	⊢ ( 𝑚 = 𝑡 → ( ♯ ‘ 𝑚 ) = ( ♯ ‘ 𝑡 ) )
9	8	oveq1d	⊢ ( 𝑚 = 𝑡 → ( ( ♯ ‘ 𝑚 ) − 1 ) = ( ( ♯ ‘ 𝑡 ) − 1 ) )
10	7 9	fveq12d	⊢ ( 𝑚 = 𝑡 → ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) = ( 𝑡 ‘ ( ( ♯ ‘ 𝑡 ) − 1 ) ) )
11	10	eleq1d	⊢ ( 𝑚 = 𝑡 → ( ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) ∈ 𝑊 ↔ ( 𝑡 ‘ ( ( ♯ ‘ 𝑡 ) − 1 ) ) ∈ 𝑊 ) )
12	11	ralrab2	⊢ ( ∀ 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) ∈ 𝑊 ↔ ∀ 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ( ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) → ( 𝑡 ‘ ( ( ♯ ‘ 𝑡 ) − 1 ) ) ∈ 𝑊 ) )
13		eldifi	⊢ ( 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) → 𝑡 ∈ Word 𝑊 )
14		wrdf	⊢ ( 𝑡 ∈ Word 𝑊 → 𝑡 : ( 0 ..^ ( ♯ ‘ 𝑡 ) ) ⟶ 𝑊 )
15	13 14	syl	⊢ ( 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) → 𝑡 : ( 0 ..^ ( ♯ ‘ 𝑡 ) ) ⟶ 𝑊 )
16		eldifsn	⊢ ( 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ↔ ( 𝑡 ∈ Word 𝑊 ∧ 𝑡 ≠ ∅ ) )
17		lennncl	⊢ ( ( 𝑡 ∈ Word 𝑊 ∧ 𝑡 ≠ ∅ ) → ( ♯ ‘ 𝑡 ) ∈ ℕ )
18	16 17	sylbi	⊢ ( 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) → ( ♯ ‘ 𝑡 ) ∈ ℕ )
19		fzo0end	⊢ ( ( ♯ ‘ 𝑡 ) ∈ ℕ → ( ( ♯ ‘ 𝑡 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑡 ) ) )
20	18 19	syl	⊢ ( 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) → ( ( ♯ ‘ 𝑡 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑡 ) ) )
21	15 20	ffvelcdmd	⊢ ( 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) → ( 𝑡 ‘ ( ( ♯ ‘ 𝑡 ) − 1 ) ) ∈ 𝑊 )
22	21	a1d	⊢ ( 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) → ( ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) → ( 𝑡 ‘ ( ( ♯ ‘ 𝑡 ) − 1 ) ) ∈ 𝑊 ) )
23	12 22	mprgbir	⊢ ∀ 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) ∈ 𝑊
24	6	fmpt	⊢ ( ∀ 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) ∈ 𝑊 ↔ 𝑆 : { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ⟶ 𝑊 )
25	23 24	mpbi	⊢ 𝑆 : { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ⟶ 𝑊