Theorem efgsval2

Description: Value of the auxiliary function S defining a sequence of extensions. (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	efgsval2	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ∈ dom 𝑆 ) → ( 𝑆 ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) = 𝐵 )

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	1 2 3 4 5 6	efgsval	⊢ ( ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ∈ dom 𝑆 → ( 𝑆 ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) = ( ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ‘ ( ( ♯ ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) − 1 ) ) )
8		s1cl	⊢ ( 𝐵 ∈ 𝑊 → ⟨“ 𝐵 ”⟩ ∈ Word 𝑊 )
9		ccatlen	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ ⟨“ 𝐵 ”⟩ ∈ Word 𝑊 ) → ( ♯ ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) = ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) ) )
10	8 9	sylan2	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( ♯ ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) = ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) ) )
11		s1len	⊢ ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) = 1
12	11	oveq2i	⊢ ( ( ♯ ‘ 𝐴 ) + ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) ) = ( ( ♯ ‘ 𝐴 ) + 1 )
13	10 12	eqtrdi	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( ♯ ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) = ( ( ♯ ‘ 𝐴 ) + 1 ) )
14	13	oveq1d	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( ( ♯ ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) − 1 ) = ( ( ( ♯ ‘ 𝐴 ) + 1 ) − 1 ) )
15		lencl	⊢ ( 𝐴 ∈ Word 𝑊 → ( ♯ ‘ 𝐴 ) ∈ ℕ0 )
16	15	nn0cnd	⊢ ( 𝐴 ∈ Word 𝑊 → ( ♯ ‘ 𝐴 ) ∈ ℂ )
17		ax-1cn	⊢ 1 ∈ ℂ
18		pncan	⊢ ( ( ( ♯ ‘ 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( ♯ ‘ 𝐴 ) + 1 ) − 1 ) = ( ♯ ‘ 𝐴 ) )
19	16 17 18	sylancl	⊢ ( 𝐴 ∈ Word 𝑊 → ( ( ( ♯ ‘ 𝐴 ) + 1 ) − 1 ) = ( ♯ ‘ 𝐴 ) )
20	16	addlidd	⊢ ( 𝐴 ∈ Word 𝑊 → ( 0 + ( ♯ ‘ 𝐴 ) ) = ( ♯ ‘ 𝐴 ) )
21	19 20	eqtr4d	⊢ ( 𝐴 ∈ Word 𝑊 → ( ( ( ♯ ‘ 𝐴 ) + 1 ) − 1 ) = ( 0 + ( ♯ ‘ 𝐴 ) ) )
22	21	adantr	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( ( ( ♯ ‘ 𝐴 ) + 1 ) − 1 ) = ( 0 + ( ♯ ‘ 𝐴 ) ) )
23	14 22	eqtrd	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( ( ♯ ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) − 1 ) = ( 0 + ( ♯ ‘ 𝐴 ) ) )
24	23	fveq2d	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ‘ ( ( ♯ ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) − 1 ) ) = ( ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ‘ ( 0 + ( ♯ ‘ 𝐴 ) ) ) )
25		simpl	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → 𝐴 ∈ Word 𝑊 )
26	8	adantl	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ⟨“ 𝐵 ”⟩ ∈ Word 𝑊 )
27		1nn	⊢ 1 ∈ ℕ
28	11 27	eqeltri	⊢ ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) ∈ ℕ
29		lbfzo0	⊢ ( 0 ∈ ( 0 ..^ ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) ) ↔ ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) ∈ ℕ )
30	28 29	mpbir	⊢ 0 ∈ ( 0 ..^ ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) )
31	30	a1i	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → 0 ∈ ( 0 ..^ ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) ) )
32		ccatval3	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ ⟨“ 𝐵 ”⟩ ∈ Word 𝑊 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ ⟨“ 𝐵 ”⟩ ) ) ) → ( ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ‘ ( 0 + ( ♯ ‘ 𝐴 ) ) ) = ( ⟨“ 𝐵 ”⟩ ‘ 0 ) )
33	25 26 31 32	syl3anc	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ‘ ( 0 + ( ♯ ‘ 𝐴 ) ) ) = ( ⟨“ 𝐵 ”⟩ ‘ 0 ) )
34		s1fv	⊢ ( 𝐵 ∈ 𝑊 → ( ⟨“ 𝐵 ”⟩ ‘ 0 ) = 𝐵 )
35	34	adantl	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨“ 𝐵 ”⟩ ‘ 0 ) = 𝐵 )
36	24 33 35	3eqtrd	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ‘ ( ( ♯ ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) − 1 ) ) = 𝐵 )
37	7 36	sylan9eqr	⊢ ( ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ) ∧ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ∈ dom 𝑆 ) → ( 𝑆 ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) = 𝐵 )
38	37	3impa	⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ∈ dom 𝑆 ) → ( 𝑆 ‘ ( 𝐴 ++ ⟨“ 𝐵 ”⟩ ) ) = 𝐵 )