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Theorem efgredlemd 17980

Description: The reduced word that forms the base of the sequence in efgsval 17967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (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)))
efgredlem.1	⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
efgredlem.2	⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
efgredlem.3	⊢ (𝜑 → 𝐵 ∈ dom 𝑆)
efgredlem.4	⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))
efgredlem.5	⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
efgredlemb.k	⊢ 𝐾 = (((#‘𝐴) − 1) − 1)
efgredlemb.l	⊢ 𝐿 = (((#‘𝐵) − 1) − 1)
efgredlemb.p	⊢ (𝜑 → 𝑃 ∈ (0...(#‘(𝐴‘𝐾))))
efgredlemb.q	⊢ (𝜑 → 𝑄 ∈ (0...(#‘(𝐵‘𝐿))))
efgredlemb.u	⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2_𝑜))
efgredlemb.v	⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_𝑜))
efgredlemb.6	⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
efgredlemb.7	⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
efgredlemb.8	⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))
efgredlemd.9	⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘(𝑄 + 2)))
efgredlemd.c	⊢ (𝜑 → 𝐶 ∈ dom 𝑆)
efgredlemd.sc	⊢ (𝜑 → (𝑆‘𝐶) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)))

**Assertion**
Ref	Expression
efgredlemd	⊢ (𝜑 → (𝐴‘0) = (𝐵‘0))

Distinct variable groups: 𝑎,𝑏,𝐴 𝑦,𝑎,𝑧,𝑏 𝐿,𝑎,𝑏 𝐾,𝑎,𝑏 𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑃 𝑚,𝑎,𝑛,𝑡,𝑣,𝑤,𝑥,𝑀,𝑏 𝑈,𝑛,𝑣,𝑤,𝑦,𝑧 𝑘,𝑎,𝑇,𝑏,𝑚,𝑡,𝑥 𝑛,𝑉,𝑣,𝑤,𝑦,𝑧 𝑄,𝑛,𝑡,𝑣,𝑤,𝑦,𝑧 𝑊,𝑎,𝑏 𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥 ∼ ,𝑎,𝑏,𝑚,𝑡,𝑥,𝑦,𝑧 𝐵,𝑎,𝑏 𝐶,𝑎,𝑏,𝑘,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧 𝑆,𝑎,𝑏 𝐼,𝑎,𝑏,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧 𝐷,𝑎,𝑏,𝑚,𝑡 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛,𝑎,𝑏) 𝐴(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐵(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑘,𝑛) 𝑃(𝑥,𝑘,𝑚,𝑎,𝑏) 𝑄(𝑥,𝑘,𝑚,𝑎,𝑏) ∼ (𝑤,𝑣,𝑘,𝑛) 𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝑇(𝑦,𝑧,𝑤,𝑣,𝑛) 𝑈(𝑥,𝑡,𝑘,𝑚,𝑎,𝑏) 𝐼(𝑘) 𝐾(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛) 𝑀(𝑦,𝑧,𝑘) 𝑉(𝑥,𝑡,𝑘,𝑚,𝑎,𝑏)

Proof of Theorem efgredlemd Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgredlemd.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ dom 𝑆)
2		efgval.w	. . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
3		efgval.r	. . . . . . . . 9 ⊢ ∼ = ( ~_FG ‘𝐼)
4		efgval2.m	. . . . . . . . 9 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
5		efgval2.t	. . . . . . . . 9 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
6		efgred.d	. . . . . . . . 9 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
7		efgred.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
8	2, 3, 4, 5, 6, 7	efgsdm 17966	. . . . . . . 8 ⊢ (𝐶 ∈ dom 𝑆 ↔ (𝐶 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐶‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐶))(𝐶‘𝑖) ∈ ran (𝑇‘(𝐶‘(𝑖 − 1)))))
9	8	simp1bi 1069	. . . . . . 7 ⊢ (𝐶 ∈ dom 𝑆 → 𝐶 ∈ (Word 𝑊 ∖ {∅}))
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (Word 𝑊 ∖ {∅}))
11	10	eldifad 3552	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Word 𝑊)
12		efgredlem.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
13	2, 3, 4, 5, 6, 7	efgsdm 17966	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1)))))
14	13	simp1bi 1069	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
15	12, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
16	15	eldifad 3552	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Word 𝑊)
17		wrdf 13165	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(#‘𝐴))⟶𝑊)
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴:(0..^(#‘𝐴))⟶𝑊)
19		fzossfz 12357	. . . . . . . . 9 ⊢ (0..^((#‘𝐴) − 1)) ⊆ (0...((#‘𝐴) − 1))
20		lencl 13179	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑊 → (#‘𝐴) ∈ ℕ₀)
21	16, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝐴) ∈ ℕ₀)
22	21	nn0zd 11356	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝐴) ∈ ℤ)
23		fzoval 12340	. . . . . . . . . 10 ⊢ ((#‘𝐴) ∈ ℤ → (0..^(#‘𝐴)) = (0...((#‘𝐴) − 1)))
24	22, 23	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0..^(#‘𝐴)) = (0...((#‘𝐴) − 1)))
25	19, 24	syl5sseqr 3617	. . . . . . . 8 ⊢ (𝜑 → (0..^((#‘𝐴) − 1)) ⊆ (0..^(#‘𝐴)))
26		efgredlemb.k	. . . . . . . . 9 ⊢ 𝐾 = (((#‘𝐴) − 1) − 1)
27		efgredlem.1	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
28		efgredlem.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)
29		efgredlem.4	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))
30		efgredlem.5	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
31	2, 3, 4, 5, 6, 7, 27, 12, 28, 29, 30	efgredlema 17976	. . . . . . . . . . 11 ⊢ (𝜑 → (((#‘𝐴) − 1) ∈ ℕ ∧ ((#‘𝐵) − 1) ∈ ℕ))
32	31	simpld 474	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘𝐴) − 1) ∈ ℕ)
33		fzo0end 12426	. . . . . . . . . 10 ⊢ (((#‘𝐴) − 1) ∈ ℕ → (((#‘𝐴) − 1) − 1) ∈ (0..^((#‘𝐴) − 1)))
34	32, 33	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (((#‘𝐴) − 1) − 1) ∈ (0..^((#‘𝐴) − 1)))
35	26, 34	syl5eqel 2692	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (0..^((#‘𝐴) − 1)))
36	25, 35	sseldd 3569	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (0..^(#‘𝐴)))
37	18, 36	ffvelrnd 6268	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
38	37	s1cld 13236	. . . . 5 ⊢ (𝜑 → ⟨“(𝐴‘𝐾)”⟩ ∈ Word 𝑊)
39		eldifsn 4260	. . . . . . . 8 ⊢ (𝐶 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝐶 ∈ Word 𝑊 ∧ 𝐶 ≠ ∅))
40		lennncl 13180	. . . . . . . 8 ⊢ ((𝐶 ∈ Word 𝑊 ∧ 𝐶 ≠ ∅) → (#‘𝐶) ∈ ℕ)
41	39, 40	sylbi 206	. . . . . . 7 ⊢ (𝐶 ∈ (Word 𝑊 ∖ {∅}) → (#‘𝐶) ∈ ℕ)
42	10, 41	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘𝐶) ∈ ℕ)
43		lbfzo0 12375	. . . . . 6 ⊢ (0 ∈ (0..^(#‘𝐶)) ↔ (#‘𝐶) ∈ ℕ)
44	42, 43	sylibr 223	. . . . 5 ⊢ (𝜑 → 0 ∈ (0..^(#‘𝐶)))
45		ccatval1 13214	. . . . 5 ⊢ ((𝐶 ∈ Word 𝑊 ∧ ⟨“(𝐴‘𝐾)”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(#‘𝐶))) → ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0) = (𝐶‘0))
46	11, 38, 44, 45	syl3anc 1318	. . . 4 ⊢ (𝜑 → ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0) = (𝐶‘0))
47	2, 3, 4, 5, 6, 7	efgsdm 17966	. . . . . . . . . . 11 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1)))))
48	47	simp1bi 1069	. . . . . . . . . 10 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
49	28, 48	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
50	49	eldifad 3552	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Word 𝑊)
51		wrdf 13165	. . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(#‘𝐵))⟶𝑊)
52	50, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵:(0..^(#‘𝐵))⟶𝑊)
53		fzossfz 12357	. . . . . . . . 9 ⊢ (0..^((#‘𝐵) − 1)) ⊆ (0...((#‘𝐵) − 1))
54		lencl 13179	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Word 𝑊 → (#‘𝐵) ∈ ℕ₀)
55	50, 54	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ₀)
56	55	nn0zd 11356	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝐵) ∈ ℤ)
57		fzoval 12340	. . . . . . . . . 10 ⊢ ((#‘𝐵) ∈ ℤ → (0..^(#‘𝐵)) = (0...((#‘𝐵) − 1)))
58	56, 57	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0..^(#‘𝐵)) = (0...((#‘𝐵) − 1)))
59	53, 58	syl5sseqr 3617	. . . . . . . 8 ⊢ (𝜑 → (0..^((#‘𝐵) − 1)) ⊆ (0..^(#‘𝐵)))
60		efgredlemb.l	. . . . . . . . 9 ⊢ 𝐿 = (((#‘𝐵) − 1) − 1)
61	31	simprd 478	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘𝐵) − 1) ∈ ℕ)
62		fzo0end 12426	. . . . . . . . . 10 ⊢ (((#‘𝐵) − 1) ∈ ℕ → (((#‘𝐵) − 1) − 1) ∈ (0..^((#‘𝐵) − 1)))
63	61, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (((#‘𝐵) − 1) − 1) ∈ (0..^((#‘𝐵) − 1)))
64	60, 63	syl5eqel 2692	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (0..^((#‘𝐵) − 1)))
65	59, 64	sseldd 3569	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (0..^(#‘𝐵)))
66	52, 65	ffvelrnd 6268	. . . . . 6 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
67	66	s1cld 13236	. . . . 5 ⊢ (𝜑 → ⟨“(𝐵‘𝐿)”⟩ ∈ Word 𝑊)
68		ccatval1 13214	. . . . 5 ⊢ ((𝐶 ∈ Word 𝑊 ∧ ⟨“(𝐵‘𝐿)”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(#‘𝐶))) → ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0) = (𝐶‘0))
69	11, 67, 44, 68	syl3anc 1318	. . . 4 ⊢ (𝜑 → ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0) = (𝐶‘0))
70	46, 69	eqtr4d 2647	. . 3 ⊢ (𝜑 → ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0))
71		fviss 6166	. . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
72	2, 71	eqsstri 3598	. . . . . . . . 9 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
73	72, 37	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜))
74		lencl 13179	. . . . . . . 8 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → (#‘(𝐴‘𝐾)) ∈ ℕ₀)
75	73, 74	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ ℕ₀)
76	75	nn0red 11229	. . . . . 6 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ ℝ)
77		2rp 11713	. . . . . 6 ⊢ 2 ∈ ℝ⁺
78		ltaddrp 11743	. . . . . 6 ⊢ (((#‘(𝐴‘𝐾)) ∈ ℝ ∧ 2 ∈ ℝ⁺) → (#‘(𝐴‘𝐾)) < ((#‘(𝐴‘𝐾)) + 2))
79	76, 77, 78	sylancl 693	. . . . 5 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) < ((#‘(𝐴‘𝐾)) + 2))
80	21	nn0red 11229	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝐴) ∈ ℝ)
81	80	lem1d 10836	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘𝐴) − 1) ≤ (#‘𝐴))
82		fznn 12278	. . . . . . . . . . 11 ⊢ ((#‘𝐴) ∈ ℤ → (((#‘𝐴) − 1) ∈ (1...(#‘𝐴)) ↔ (((#‘𝐴) − 1) ∈ ℕ ∧ ((#‘𝐴) − 1) ≤ (#‘𝐴))))
83	22, 82	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((#‘𝐴) − 1) ∈ (1...(#‘𝐴)) ↔ (((#‘𝐴) − 1) ∈ ℕ ∧ ((#‘𝐴) − 1) ≤ (#‘𝐴))))
84	32, 81, 83	mpbir2and 959	. . . . . . . . 9 ⊢ (𝜑 → ((#‘𝐴) − 1) ∈ (1...(#‘𝐴)))
85	2, 3, 4, 5, 6, 7	efgsres 17974	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑆 ∧ ((#‘𝐴) − 1) ∈ (1...(#‘𝐴))) → (𝐴 ↾ (0..^((#‘𝐴) − 1))) ∈ dom 𝑆)
86	12, 84, 85	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↾ (0..^((#‘𝐴) − 1))) ∈ dom 𝑆)
87	2, 3, 4, 5, 6, 7	efgsval 17967	. . . . . . . 8 ⊢ ((𝐴 ↾ (0..^((#‘𝐴) − 1))) ∈ dom 𝑆 → (𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘((#‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) − 1)))
88	86, 87	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘((#‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) − 1)))
89		1eluzge0 11608	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ (ℤ_≥‘0)
90		fzss1 12251	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (ℤ_≥‘0) → (1...(#‘𝐴)) ⊆ (0...(#‘𝐴)))
91	89, 90	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (1...(#‘𝐴)) ⊆ (0...(#‘𝐴))
92	91, 84	sseldi 3566	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((#‘𝐴) − 1) ∈ (0...(#‘𝐴)))
93		swrd0val 13273	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑊 ∧ ((#‘𝐴) − 1) ∈ (0...(#‘𝐴))) → (𝐴 substr ⟨0, ((#‘𝐴) − 1)⟩) = (𝐴 ↾ (0..^((#‘𝐴) − 1))))
94	16, 92, 93	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 substr ⟨0, ((#‘𝐴) − 1)⟩) = (𝐴 ↾ (0..^((#‘𝐴) − 1))))
95	94	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝐴 substr ⟨0, ((#‘𝐴) − 1)⟩)) = (#‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))))
96		swrd0len 13274	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑊 ∧ ((#‘𝐴) − 1) ∈ (0...(#‘𝐴))) → (#‘(𝐴 substr ⟨0, ((#‘𝐴) − 1)⟩)) = ((#‘𝐴) − 1))
97	16, 92, 96	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝐴 substr ⟨0, ((#‘𝐴) − 1)⟩)) = ((#‘𝐴) − 1))
98	95, 97	eqtr3d 2646	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = ((#‘𝐴) − 1))
99	98	oveq1d 6564	. . . . . . . . 9 ⊢ (𝜑 → ((#‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) − 1) = (((#‘𝐴) − 1) − 1))
100	99, 26	syl6eqr 2662	. . . . . . . 8 ⊢ (𝜑 → ((#‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) − 1) = 𝐾)
101	100	fveq2d 6107	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘((#‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) − 1)) = ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘𝐾))
102		fvres 6117	. . . . . . . 8 ⊢ (𝐾 ∈ (0..^((#‘𝐴) − 1)) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘𝐾) = (𝐴‘𝐾))
103	35, 102	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘𝐾) = (𝐴‘𝐾))
104	88, 101, 103	3eqtrd 2648	. . . . . 6 ⊢ (𝜑 → (𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝐴‘𝐾))
105	104	fveq2d 6107	. . . . 5 ⊢ (𝜑 → (#‘(𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1))))) = (#‘(𝐴‘𝐾)))
106	2, 3, 4, 5, 6, 7	efgsdmi 17968	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑆 ∧ ((#‘𝐴) − 1) ∈ ℕ) → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((#‘𝐴) − 1) − 1))))
107	12, 32, 106	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((#‘𝐴) − 1) − 1))))
108	26	fveq2i 6106	. . . . . . . . 9 ⊢ (𝐴‘𝐾) = (𝐴‘(((#‘𝐴) − 1) − 1))
109	108	fveq2i 6106	. . . . . . . 8 ⊢ (𝑇‘(𝐴‘𝐾)) = (𝑇‘(𝐴‘(((#‘𝐴) − 1) − 1)))
110	109	rneqi 5273	. . . . . . 7 ⊢ ran (𝑇‘(𝐴‘𝐾)) = ran (𝑇‘(𝐴‘(((#‘𝐴) − 1) − 1)))
111	107, 110	syl6eleqr 2699	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾)))
112	2, 3, 4, 5	efgtlen 17962	. . . . . 6 ⊢ (((𝐴‘𝐾) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) → (#‘(𝑆‘𝐴)) = ((#‘(𝐴‘𝐾)) + 2))
113	37, 111, 112	syl2anc 691	. . . . 5 ⊢ (𝜑 → (#‘(𝑆‘𝐴)) = ((#‘(𝐴‘𝐾)) + 2))
114	79, 105, 113	3brtr4d 4615	. . . 4 ⊢ (𝜑 → (#‘(𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1))))) < (#‘(𝑆‘𝐴)))
115		efgredlemb.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (0...(#‘(𝐴‘𝐾))))
116		efgredlemb.q	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ (0...(#‘(𝐵‘𝐿))))
117		efgredlemb.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2_𝑜))
118		efgredlemb.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_𝑜))
119		efgredlemb.6	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
120		efgredlemb.7	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
121		efgredlemb.8	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))
122		efgredlemd.9	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘(𝑄 + 2)))
123		efgredlemd.sc	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘𝐶) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)))
124	2, 3, 4, 5, 6, 7, 27, 12, 28, 29, 30, 26, 60, 115, 116, 117, 118, 119, 120, 121, 122, 1, 123	efgredleme 17979	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶)) ∧ (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶))))
125	124	simpld 474	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶)))
126	2, 3, 4, 5, 6, 7	efgsp1 17973	. . . . . . 7 ⊢ ((𝐶 ∈ dom 𝑆 ∧ (𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶))) → (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) ∈ dom 𝑆)
127	1, 125, 126	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) ∈ dom 𝑆)
128	2, 3, 4, 5, 6, 7	efgsval2 17969	. . . . . 6 ⊢ ((𝐶 ∈ Word 𝑊 ∧ (𝐴‘𝐾) ∈ 𝑊 ∧ (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) ∈ dom 𝑆) → (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)) = (𝐴‘𝐾))
129	11, 37, 127, 128	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)) = (𝐴‘𝐾))
130	104, 129	eqtr4d 2647	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)))
131		fveq2 6103	. . . . . . . . 9 ⊢ (𝑎 = (𝐴 ↾ (0..^((#‘𝐴) − 1))) → (𝑆‘𝑎) = (𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))))
132	131	fveq2d 6107	. . . . . . . 8 ⊢ (𝑎 = (𝐴 ↾ (0..^((#‘𝐴) − 1))) → (#‘(𝑆‘𝑎)) = (#‘(𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1))))))
133	132	breq1d 4593	. . . . . . 7 ⊢ (𝑎 = (𝐴 ↾ (0..^((#‘𝐴) − 1))) → ((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) ↔ (#‘(𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1))))) < (#‘(𝑆‘𝐴))))
134	131	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑎 = (𝐴 ↾ (0..^((#‘𝐴) − 1))) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ (𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘𝑏)))
135		fveq1 6102	. . . . . . . . 9 ⊢ (𝑎 = (𝐴 ↾ (0..^((#‘𝐴) − 1))) → (𝑎‘0) = ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0))
136	135	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑎 = (𝐴 ↾ (0..^((#‘𝐴) − 1))) → ((𝑎‘0) = (𝑏‘0) ↔ ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = (𝑏‘0)))
137	134, 136	imbi12d 333	. . . . . . 7 ⊢ (𝑎 = (𝐴 ↾ (0..^((#‘𝐴) − 1))) → (((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)) ↔ ((𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘𝑏) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = (𝑏‘0))))
138	133, 137	imbi12d 333	. . . . . 6 ⊢ (𝑎 = (𝐴 ↾ (0..^((#‘𝐴) − 1))) → (((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((#‘(𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘𝑏) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = (𝑏‘0)))))
139		fveq2 6103	. . . . . . . . 9 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → (𝑆‘𝑏) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)))
140	139	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → ((𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘𝑏) ↔ (𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩))))
141		fveq1 6102	. . . . . . . . 9 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → (𝑏‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0))
142	141	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → (((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = (𝑏‘0) ↔ ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0)))
143	140, 142	imbi12d 333	. . . . . . 7 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → (((𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘𝑏) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = (𝑏‘0)) ↔ ((𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0))))
144	143	imbi2d 329	. . . . . 6 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → (((#‘(𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘𝑏) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = (𝑏‘0))) ↔ ((#‘(𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0)))))
145	138, 144	rspc2va 3294	. . . . 5 ⊢ ((((𝐴 ↾ (0..^((#‘𝐴) − 1))) ∈ dom 𝑆 ∧ (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) ∈ dom 𝑆) ∧ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) → ((#‘(𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0))))
146	86, 127, 27, 145	syl21anc 1317	. . . 4 ⊢ (𝜑 → ((#‘(𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐴 ↾ (0..^((#‘𝐴) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0))))
147	114, 130, 146	mp2d 47	. . 3 ⊢ (𝜑 → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0))
148	72, 66	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜))
149		lencl 13179	. . . . . . . 8 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → (#‘(𝐵‘𝐿)) ∈ ℕ₀)
150	148, 149	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ ℕ₀)
151	150	nn0red 11229	. . . . . 6 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ ℝ)
152		ltaddrp 11743	. . . . . 6 ⊢ (((#‘(𝐵‘𝐿)) ∈ ℝ ∧ 2 ∈ ℝ⁺) → (#‘(𝐵‘𝐿)) < ((#‘(𝐵‘𝐿)) + 2))
153	151, 77, 152	sylancl 693	. . . . 5 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) < ((#‘(𝐵‘𝐿)) + 2))
154	55	nn0red 11229	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝐵) ∈ ℝ)
155	154	lem1d 10836	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘𝐵) − 1) ≤ (#‘𝐵))
156		fznn 12278	. . . . . . . . . . 11 ⊢ ((#‘𝐵) ∈ ℤ → (((#‘𝐵) − 1) ∈ (1...(#‘𝐵)) ↔ (((#‘𝐵) − 1) ∈ ℕ ∧ ((#‘𝐵) − 1) ≤ (#‘𝐵))))
157	56, 156	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((#‘𝐵) − 1) ∈ (1...(#‘𝐵)) ↔ (((#‘𝐵) − 1) ∈ ℕ ∧ ((#‘𝐵) − 1) ≤ (#‘𝐵))))
158	61, 155, 157	mpbir2and 959	. . . . . . . . 9 ⊢ (𝜑 → ((#‘𝐵) − 1) ∈ (1...(#‘𝐵)))
159	2, 3, 4, 5, 6, 7	efgsres 17974	. . . . . . . . 9 ⊢ ((𝐵 ∈ dom 𝑆 ∧ ((#‘𝐵) − 1) ∈ (1...(#‘𝐵))) → (𝐵 ↾ (0..^((#‘𝐵) − 1))) ∈ dom 𝑆)
160	28, 158, 159	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ↾ (0..^((#‘𝐵) − 1))) ∈ dom 𝑆)
161	2, 3, 4, 5, 6, 7	efgsval 17967	. . . . . . . 8 ⊢ ((𝐵 ↾ (0..^((#‘𝐵) − 1))) ∈ dom 𝑆 → (𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘((#‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) − 1)))
162	160, 161	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘((#‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) − 1)))
163		fzss1 12251	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (ℤ_≥‘0) → (1...(#‘𝐵)) ⊆ (0...(#‘𝐵)))
164	89, 163	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (1...(#‘𝐵)) ⊆ (0...(#‘𝐵))
165	164, 158	sseldi 3566	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((#‘𝐵) − 1) ∈ (0...(#‘𝐵)))
166		swrd0val 13273	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ Word 𝑊 ∧ ((#‘𝐵) − 1) ∈ (0...(#‘𝐵))) → (𝐵 substr ⟨0, ((#‘𝐵) − 1)⟩) = (𝐵 ↾ (0..^((#‘𝐵) − 1))))
167	50, 165, 166	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 substr ⟨0, ((#‘𝐵) − 1)⟩) = (𝐵 ↾ (0..^((#‘𝐵) − 1))))
168	167	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝐵 substr ⟨0, ((#‘𝐵) − 1)⟩)) = (#‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))))
169		swrd0len 13274	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Word 𝑊 ∧ ((#‘𝐵) − 1) ∈ (0...(#‘𝐵))) → (#‘(𝐵 substr ⟨0, ((#‘𝐵) − 1)⟩)) = ((#‘𝐵) − 1))
170	50, 165, 169	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝐵 substr ⟨0, ((#‘𝐵) − 1)⟩)) = ((#‘𝐵) − 1))
171	168, 170	eqtr3d 2646	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = ((#‘𝐵) − 1))
172	171	oveq1d 6564	. . . . . . . . 9 ⊢ (𝜑 → ((#‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) − 1) = (((#‘𝐵) − 1) − 1))
173	172, 60	syl6eqr 2662	. . . . . . . 8 ⊢ (𝜑 → ((#‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) − 1) = 𝐿)
174	173	fveq2d 6107	. . . . . . 7 ⊢ (𝜑 → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘((#‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) − 1)) = ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘𝐿))
175		fvres 6117	. . . . . . . 8 ⊢ (𝐿 ∈ (0..^((#‘𝐵) − 1)) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘𝐿) = (𝐵‘𝐿))
176	64, 175	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘𝐿) = (𝐵‘𝐿))
177	162, 174, 176	3eqtrd 2648	. . . . . 6 ⊢ (𝜑 → (𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝐵‘𝐿))
178	177	fveq2d 6107	. . . . 5 ⊢ (𝜑 → (#‘(𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1))))) = (#‘(𝐵‘𝐿)))
179	2, 3, 4, 5, 6, 7	efgsdmi 17968	. . . . . . . . 9 ⊢ ((𝐵 ∈ dom 𝑆 ∧ ((#‘𝐵) − 1) ∈ ℕ) → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1))))
180	28, 61, 179	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1))))
181	29, 180	eqeltrd 2688	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1))))
182	60	fveq2i 6106	. . . . . . . . 9 ⊢ (𝐵‘𝐿) = (𝐵‘(((#‘𝐵) − 1) − 1))
183	182	fveq2i 6106	. . . . . . . 8 ⊢ (𝑇‘(𝐵‘𝐿)) = (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1)))
184	183	rneqi 5273	. . . . . . 7 ⊢ ran (𝑇‘(𝐵‘𝐿)) = ran (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1)))
185	181, 184	syl6eleqr 2699	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿)))
186	2, 3, 4, 5	efgtlen 17962	. . . . . 6 ⊢ (((𝐵‘𝐿) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) → (#‘(𝑆‘𝐴)) = ((#‘(𝐵‘𝐿)) + 2))
187	66, 185, 186	syl2anc 691	. . . . 5 ⊢ (𝜑 → (#‘(𝑆‘𝐴)) = ((#‘(𝐵‘𝐿)) + 2))
188	153, 178, 187	3brtr4d 4615	. . . 4 ⊢ (𝜑 → (#‘(𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1))))) < (#‘(𝑆‘𝐴)))
189	124	simprd 478	. . . . . . 7 ⊢ (𝜑 → (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶)))
190	2, 3, 4, 5, 6, 7	efgsp1 17973	. . . . . . 7 ⊢ ((𝐶 ∈ dom 𝑆 ∧ (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶))) → (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) ∈ dom 𝑆)
191	1, 189, 190	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) ∈ dom 𝑆)
192	2, 3, 4, 5, 6, 7	efgsval2 17969	. . . . . 6 ⊢ ((𝐶 ∈ Word 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊 ∧ (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) ∈ dom 𝑆) → (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)) = (𝐵‘𝐿))
193	11, 66, 191, 192	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)) = (𝐵‘𝐿))
194	177, 193	eqtr4d 2647	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)))
195		fveq2 6103	. . . . . . . . 9 ⊢ (𝑎 = (𝐵 ↾ (0..^((#‘𝐵) − 1))) → (𝑆‘𝑎) = (𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))))
196	195	fveq2d 6107	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ↾ (0..^((#‘𝐵) − 1))) → (#‘(𝑆‘𝑎)) = (#‘(𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1))))))
197	196	breq1d 4593	. . . . . . 7 ⊢ (𝑎 = (𝐵 ↾ (0..^((#‘𝐵) − 1))) → ((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) ↔ (#‘(𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1))))) < (#‘(𝑆‘𝐴))))
198	195	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ↾ (0..^((#‘𝐵) − 1))) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ (𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘𝑏)))
199		fveq1 6102	. . . . . . . . 9 ⊢ (𝑎 = (𝐵 ↾ (0..^((#‘𝐵) − 1))) → (𝑎‘0) = ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0))
200	199	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ↾ (0..^((#‘𝐵) − 1))) → ((𝑎‘0) = (𝑏‘0) ↔ ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = (𝑏‘0)))
201	198, 200	imbi12d 333	. . . . . . 7 ⊢ (𝑎 = (𝐵 ↾ (0..^((#‘𝐵) − 1))) → (((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)) ↔ ((𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘𝑏) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = (𝑏‘0))))
202	197, 201	imbi12d 333	. . . . . 6 ⊢ (𝑎 = (𝐵 ↾ (0..^((#‘𝐵) − 1))) → (((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((#‘(𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘𝑏) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = (𝑏‘0)))))
203		fveq2 6103	. . . . . . . . 9 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → (𝑆‘𝑏) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)))
204	203	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → ((𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘𝑏) ↔ (𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩))))
205		fveq1 6102	. . . . . . . . 9 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → (𝑏‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0))
206	205	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → (((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = (𝑏‘0) ↔ ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0)))
207	204, 206	imbi12d 333	. . . . . . 7 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → (((𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘𝑏) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = (𝑏‘0)) ↔ ((𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0))))
208	207	imbi2d 329	. . . . . 6 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → (((#‘(𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘𝑏) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = (𝑏‘0))) ↔ ((#‘(𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0)))))
209	202, 208	rspc2va 3294	. . . . 5 ⊢ ((((𝐵 ↾ (0..^((#‘𝐵) − 1))) ∈ dom 𝑆 ∧ (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) ∈ dom 𝑆) ∧ ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) → ((#‘(𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0))))
210	160, 191, 27, 209	syl21anc 1317	. . . 4 ⊢ (𝜑 → ((#‘(𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1))))) < (#‘(𝑆‘𝐴)) → ((𝑆‘(𝐵 ↾ (0..^((#‘𝐵) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0))))
211	188, 194, 210	mp2d 47	. . 3 ⊢ (𝜑 → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0))
212	70, 147, 211	3eqtr4d 2654	. 2 ⊢ (𝜑 → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0))
213		lbfzo0 12375	. . . 4 ⊢ (0 ∈ (0..^((#‘𝐴) − 1)) ↔ ((#‘𝐴) − 1) ∈ ℕ)
214	32, 213	sylibr 223	. . 3 ⊢ (𝜑 → 0 ∈ (0..^((#‘𝐴) − 1)))
215		fvres 6117	. . 3 ⊢ (0 ∈ (0..^((#‘𝐴) − 1)) → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = (𝐴‘0))
216	214, 215	syl 17	. 2 ⊢ (𝜑 → ((𝐴 ↾ (0..^((#‘𝐴) − 1)))‘0) = (𝐴‘0))
217		lbfzo0 12375	. . . 4 ⊢ (0 ∈ (0..^((#‘𝐵) − 1)) ↔ ((#‘𝐵) − 1) ∈ ℕ)
218	61, 217	sylibr 223	. . 3 ⊢ (𝜑 → 0 ∈ (0..^((#‘𝐵) − 1)))
219		fvres 6117	. . 3 ⊢ (0 ∈ (0..^((#‘𝐵) − 1)) → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = (𝐵‘0))
220	218, 219	syl 17	. 2 ⊢ (𝜑 → ((𝐵 ↾ (0..^((#‘𝐵) − 1)))‘0) = (𝐵‘0))
221	212, 216, 220	3eqtr3d 2652	1 ⊢ (𝜑 → (𝐴‘0) = (𝐵‘0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 {csn 4125 ⟨cop 4131 ⟨cotp 4133 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 × cxp 5036 dom cdm 5038 ran crn 5039 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1_𝑜c1o 7440 2_𝑜c2o 7441 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 ⟨“cs1 13149 substr csubstr 13150 splice csplice 13151 ⟨“cs2 13437 ~_FG cefg 17942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-s2 13444

This theorem is referenced by: efgredlemc 17981

W3C validator