Theorem efgsdm

Description: Elementhood in the domain of S , the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-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	efgsdm	⊢ ( 𝐹 ∈ dom 𝑆 ↔ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) )

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		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 0 ) = ( 𝐹 ‘ 0 ) )
8	7	eleq1d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ↔ ( 𝐹 ‘ 0 ) ∈ 𝐷 ) )
9		fveq2	⊢ ( 𝑓 = 𝐹 → ( ♯ ‘ 𝑓 ) = ( ♯ ‘ 𝐹 ) )
10	9	oveq2d	⊢ ( 𝑓 = 𝐹 → ( 1 ..^ ( ♯ ‘ 𝑓 ) ) = ( 1 ..^ ( ♯ ‘ 𝐹 ) ) )
11		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑖 ) )
12		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑖 − 1 ) ) = ( 𝐹 ‘ ( 𝑖 − 1 ) ) )
13	12	fveq2d	⊢ ( 𝑓 = 𝐹 → ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) = ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) )
14	13	rneqd	⊢ ( 𝑓 = 𝐹 → ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) = ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) )
15	11 14	eleq12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ↔ ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) )
16	10 15	raleqbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) )
17	8 16	anbi12d	⊢ ( 𝑓 = 𝐹 → ( ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) ↔ ( ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) ) )
18	1 2 3 4 5 6	efgsf	⊢ 𝑆 : { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ⟶ 𝑊
19	18	fdmi	⊢ dom 𝑆 = { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) }
20		fveq1	⊢ ( 𝑡 = 𝑓 → ( 𝑡 ‘ 0 ) = ( 𝑓 ‘ 0 ) )
21	20	eleq1d	⊢ ( 𝑡 = 𝑓 → ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ↔ ( 𝑓 ‘ 0 ) ∈ 𝐷 ) )
22		fveq2	⊢ ( 𝑘 = 𝑖 → ( 𝑡 ‘ 𝑘 ) = ( 𝑡 ‘ 𝑖 ) )
23		fvoveq1	⊢ ( 𝑘 = 𝑖 → ( 𝑡 ‘ ( 𝑘 − 1 ) ) = ( 𝑡 ‘ ( 𝑖 − 1 ) ) )
24	23	fveq2d	⊢ ( 𝑘 = 𝑖 → ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) = ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) )
25	24	rneqd	⊢ ( 𝑘 = 𝑖 → ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) = ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) )
26	22 25	eleq12d	⊢ ( 𝑘 = 𝑖 → ( ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ↔ ( 𝑡 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) ) )
27	26	cbvralvw	⊢ ( ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) )
28		fveq2	⊢ ( 𝑡 = 𝑓 → ( ♯ ‘ 𝑡 ) = ( ♯ ‘ 𝑓 ) )
29	28	oveq2d	⊢ ( 𝑡 = 𝑓 → ( 1 ..^ ( ♯ ‘ 𝑡 ) ) = ( 1 ..^ ( ♯ ‘ 𝑓 ) ) )
30		fveq1	⊢ ( 𝑡 = 𝑓 → ( 𝑡 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑖 ) )
31		fveq1	⊢ ( 𝑡 = 𝑓 → ( 𝑡 ‘ ( 𝑖 − 1 ) ) = ( 𝑓 ‘ ( 𝑖 − 1 ) ) )
32	31	fveq2d	⊢ ( 𝑡 = 𝑓 → ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) = ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) )
33	32	rneqd	⊢ ( 𝑡 = 𝑓 → ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) = ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) )
34	30 33	eleq12d	⊢ ( 𝑡 = 𝑓 → ( ( 𝑡 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) ↔ ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) )
35	29 34	raleqbidv	⊢ ( 𝑡 = 𝑓 → ( ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑖 − 1 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) )
36	27 35	syl5bb	⊢ ( 𝑡 = 𝑓 → ( ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ↔ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) )
37	21 36	anbi12d	⊢ ( 𝑡 = 𝑓 → ( ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) ↔ ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) ) )
38	37	cbvrabv	⊢ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } = { 𝑓 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) }
39	19 38	eqtri	⊢ dom 𝑆 = { 𝑓 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑓 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ( 𝑓 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑓 ‘ ( 𝑖 − 1 ) ) ) ) }
40	17 39	elrab2	⊢ ( 𝐹 ∈ dom 𝑆 ↔ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) ) )
41		3anass	⊢ ( ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) ↔ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) ) )
42	40 41	bitr4i	⊢ ( 𝐹 ∈ dom 𝑆 ↔ ( 𝐹 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐹 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐹 ‘ ( 𝑖 − 1 ) ) ) ) )