efgredlemd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > efgredlemd		Structured version Visualization version GIF version

Theorem efgredlemd 18357

Description: The reduced word that forms the base of the sequence in efgsval 18344 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 18343	. . . . . . . 8 ⊢ (𝐶 ∈ dom 𝑆 ↔ (𝐶 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐶‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐶))(𝐶‘𝑖) ∈ ran (𝑇‘(𝐶‘(𝑖 − 1)))))
9	8	simp1bi 1140	. . . . . . 7 ⊢ (𝐶 ∈ dom 𝑆 → 𝐶 ∈ (Word 𝑊 ∖ {∅}))
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (Word 𝑊 ∖ {∅}))
11	10	eldifad 3727	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Word 𝑊)
12		efgredlem.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
13	2, 3, 4, 5, 6, 7	efgsdm 18343	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1)))))
14	13	simp1bi 1140	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
15	12, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
16	15	eldifad 3727	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Word 𝑊)
17		wrdf 13496	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(♯‘𝐴))⟶𝑊)
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴:(0..^(♯‘𝐴))⟶𝑊)
19		fzossfz 12682	. . . . . . . . 9 ⊢ (0..^((♯‘𝐴) − 1)) ⊆ (0...((♯‘𝐴) − 1))
20		lencl 13510	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℕ₀)
21	16, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
22	21	nn0zd 11672	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ)
23		fzoval 12665	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℤ → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1)))
24	22, 23	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1)))
25	19, 24	syl5sseqr 3795	. . . . . . . 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 18353	. . . . . . . . . . 11 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ))
32	31	simpld 477	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ)
33		fzo0end 12754	. . . . . . . . . 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 2843	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (0..^((♯‘𝐴) − 1)))
36	25, 35	sseldd 3745	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐴)))
37	18, 36	ffvelrnd 6523	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
38	37	s1cld 13573	. . . . 5 ⊢ (𝜑 → ⟨“(𝐴‘𝐾)”⟩ ∈ Word 𝑊)
39		eldifsn 4462	. . . . . . . 8 ⊢ (𝐶 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝐶 ∈ Word 𝑊 ∧ 𝐶 ≠ ∅))
40		lennncl 13511	. . . . . . . 8 ⊢ ((𝐶 ∈ Word 𝑊 ∧ 𝐶 ≠ ∅) → (♯‘𝐶) ∈ ℕ)
41	39, 40	sylbi 207	. . . . . . 7 ⊢ (𝐶 ∈ (Word 𝑊 ∖ {∅}) → (♯‘𝐶) ∈ ℕ)
42	10, 41	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝐶) ∈ ℕ)
43		lbfzo0 12702	. . . . . 6 ⊢ (0 ∈ (0..^(♯‘𝐶)) ↔ (♯‘𝐶) ∈ ℕ)
44	42, 43	sylibr 224	. . . . 5 ⊢ (𝜑 → 0 ∈ (0..^(♯‘𝐶)))
45		ccatval1 13549	. . . . 5 ⊢ ((𝐶 ∈ Word 𝑊 ∧ ⟨“(𝐴‘𝐾)”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝐶))) → ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0) = (𝐶‘0))
46	11, 38, 44, 45	syl3anc 1477	. . . 4 ⊢ (𝜑 → ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0) = (𝐶‘0))
47	2, 3, 4, 5, 6, 7	efgsdm 18343	. . . . . . . . . . 11 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1)))))
48	47	simp1bi 1140	. . . . . . . . . 10 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
49	28, 48	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
50	49	eldifad 3727	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Word 𝑊)
51		wrdf 13496	. . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(♯‘𝐵))⟶𝑊)
52	50, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵:(0..^(♯‘𝐵))⟶𝑊)
53		fzossfz 12682	. . . . . . . . 9 ⊢ (0..^((♯‘𝐵) − 1)) ⊆ (0...((♯‘𝐵) − 1))
54		lencl 13510	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Word 𝑊 → (♯‘𝐵) ∈ ℕ₀)
55	50, 54	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ₀)
56	55	nn0zd 11672	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
57		fzoval 12665	. . . . . . . . . 10 ⊢ ((♯‘𝐵) ∈ ℤ → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1)))
58	56, 57	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1)))
59	53, 58	syl5sseqr 3795	. . . . . . . 8 ⊢ (𝜑 → (0..^((♯‘𝐵) − 1)) ⊆ (0..^(♯‘𝐵)))
60		efgredlemb.l	. . . . . . . . 9 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)
61	31	simprd 482	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝐵) − 1) ∈ ℕ)
62		fzo0end 12754	. . . . . . . . . 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 2843	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (0..^((♯‘𝐵) − 1)))
65	59, 64	sseldd 3745	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (0..^(♯‘𝐵)))
66	52, 65	ffvelrnd 6523	. . . . . 6 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
67	66	s1cld 13573	. . . . 5 ⊢ (𝜑 → ⟨“(𝐵‘𝐿)”⟩ ∈ Word 𝑊)
68		ccatval1 13549	. . . . 5 ⊢ ((𝐶 ∈ Word 𝑊 ∧ ⟨“(𝐵‘𝐿)”⟩ ∈ Word 𝑊 ∧ 0 ∈ (0..^(♯‘𝐶))) → ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0) = (𝐶‘0))
69	11, 67, 44, 68	syl3anc 1477	. . . 4 ⊢ (𝜑 → ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0) = (𝐶‘0))
70	46, 69	eqtr4d 2797	. . 3 ⊢ (𝜑 → ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0))
71		fviss 6418	. . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
72	2, 71	eqsstri 3776	. . . . . . . . 9 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
73	72, 37	sseldi 3742	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜))
74		lencl 13510	. . . . . . . 8 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
75	73, 74	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
76	75	nn0red 11544	. . . . . 6 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℝ)
77		2rp 12030	. . . . . 6 ⊢ 2 ∈ ℝ⁺
78		ltaddrp 12060	. . . . . 6 ⊢ (((♯‘(𝐴‘𝐾)) ∈ ℝ ∧ 2 ∈ ℝ⁺) → (♯‘(𝐴‘𝐾)) < ((♯‘(𝐴‘𝐾)) + 2))
79	76, 77, 78	sylancl 697	. . . . 5 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) < ((♯‘(𝐴‘𝐾)) + 2))
80	21	nn0red 11544	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℝ)
81	80	lem1d 11149	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝐴) − 1) ≤ (♯‘𝐴))
82		fznn 12601	. . . . . . . . . . 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 995	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ (1...(♯‘𝐴)))
85	2, 3, 4, 5, 6, 7	efgsres 18351	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑆 ∧ ((♯‘𝐴) − 1) ∈ (1...(♯‘𝐴))) → (𝐴 ↾ (0..^((♯‘𝐴) − 1))) ∈ dom 𝑆)
86	12, 84, 85	syl2anc 696	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↾ (0..^((♯‘𝐴) − 1))) ∈ dom 𝑆)
87	2, 3, 4, 5, 6, 7	efgsval 18344	. . . . . . . 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		fz1ssfz0 12629	. . . . . . . . . . . . . 14 ⊢ (1...(♯‘𝐴)) ⊆ (0...(♯‘𝐴))
90	89, 84	sseldi 3742	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ (0...(♯‘𝐴)))
91		swrd0val 13620	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑊 ∧ ((♯‘𝐴) − 1) ∈ (0...(♯‘𝐴))) → (𝐴 substr ⟨0, ((♯‘𝐴) − 1)⟩) = (𝐴 ↾ (0..^((♯‘𝐴) − 1))))
92	16, 90, 91	syl2anc 696	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 substr ⟨0, ((♯‘𝐴) − 1)⟩) = (𝐴 ↾ (0..^((♯‘𝐴) − 1))))
93	92	fveq2d 6356	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐴 substr ⟨0, ((♯‘𝐴) − 1)⟩)) = (♯‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))))
94		swrd0len 13621	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑊 ∧ ((♯‘𝐴) − 1) ∈ (0...(♯‘𝐴))) → (♯‘(𝐴 substr ⟨0, ((♯‘𝐴) − 1)⟩)) = ((♯‘𝐴) − 1))
95	16, 90, 94	syl2anc 696	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐴 substr ⟨0, ((♯‘𝐴) − 1)⟩)) = ((♯‘𝐴) − 1))
96	93, 95	eqtr3d 2796	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))) = ((♯‘𝐴) − 1))
97	96	oveq1d 6828	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))) − 1) = (((♯‘𝐴) − 1) − 1))
98	97, 26	syl6eqr 2812	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))) − 1) = 𝐾)
99	98	fveq2d 6356	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘((♯‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))) − 1)) = ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘𝐾))
100		fvres 6368	. . . . . . . 8 ⊢ (𝐾 ∈ (0..^((♯‘𝐴) − 1)) → ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘𝐾) = (𝐴‘𝐾))
101	35, 100	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘𝐾) = (𝐴‘𝐾))
102	88, 99, 101	3eqtrd 2798	. . . . . 6 ⊢ (𝜑 → (𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))) = (𝐴‘𝐾))
103	102	fveq2d 6356	. . . . 5 ⊢ (𝜑 → (♯‘(𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1))))) = (♯‘(𝐴‘𝐾)))
104	2, 3, 4, 5, 6, 7	efgsdmi 18345	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝑆 ∧ ((♯‘𝐴) − 1) ∈ ℕ) → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))))
105	12, 32, 104	syl2anc 696	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))))
106	26	fveq2i 6355	. . . . . . . . 9 ⊢ (𝐴‘𝐾) = (𝐴‘(((♯‘𝐴) − 1) − 1))
107	106	fveq2i 6355	. . . . . . . 8 ⊢ (𝑇‘(𝐴‘𝐾)) = (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1)))
108	107	rneqi 5507	. . . . . . 7 ⊢ ran (𝑇‘(𝐴‘𝐾)) = ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1)))
109	105, 108	syl6eleqr 2850	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾)))
110	2, 3, 4, 5	efgtlen 18339	. . . . . 6 ⊢ (((𝐴‘𝐾) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2))
111	37, 109, 110	syl2anc 696	. . . . 5 ⊢ (𝜑 → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2))
112	79, 103, 111	3brtr4d 4836	. . . 4 ⊢ (𝜑 → (♯‘(𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1))))) < (♯‘(𝑆‘𝐴)))
113		efgredlemb.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))
114		efgredlemb.q	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))
115		efgredlemb.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2_𝑜))
116		efgredlemb.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_𝑜))
117		efgredlemb.6	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
118		efgredlemb.7	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
119		efgredlemb.8	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))
120		efgredlemd.9	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘(𝑄 + 2)))
121		efgredlemd.sc	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘𝐶) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)))
122	2, 3, 4, 5, 6, 7, 27, 12, 28, 29, 30, 26, 60, 113, 114, 115, 116, 117, 118, 119, 120, 1, 121	efgredleme 18356	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶)) ∧ (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶))))
123	122	simpld 477	. . . . . . 7 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶)))
124	2, 3, 4, 5, 6, 7	efgsp1 18350	. . . . . . 7 ⊢ ((𝐶 ∈ dom 𝑆 ∧ (𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶))) → (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) ∈ dom 𝑆)
125	1, 123, 124	syl2anc 696	. . . . . 6 ⊢ (𝜑 → (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) ∈ dom 𝑆)
126	2, 3, 4, 5, 6, 7	efgsval2 18346	. . . . . 6 ⊢ ((𝐶 ∈ Word 𝑊 ∧ (𝐴‘𝐾) ∈ 𝑊 ∧ (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) ∈ dom 𝑆) → (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)) = (𝐴‘𝐾))
127	11, 37, 125, 126	syl3anc 1477	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)) = (𝐴‘𝐾))
128	102, 127	eqtr4d 2797	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)))
129		fveq2 6352	. . . . . . . . 9 ⊢ (𝑎 = (𝐴 ↾ (0..^((♯‘𝐴) − 1))) → (𝑆‘𝑎) = (𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))))
130	129	fveq2d 6356	. . . . . . . 8 ⊢ (𝑎 = (𝐴 ↾ (0..^((♯‘𝐴) − 1))) → (♯‘(𝑆‘𝑎)) = (♯‘(𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1))))))
131	130	breq1d 4814	. . . . . . 7 ⊢ (𝑎 = (𝐴 ↾ (0..^((♯‘𝐴) − 1))) → ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) ↔ (♯‘(𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1))))) < (♯‘(𝑆‘𝐴))))
132	129	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑎 = (𝐴 ↾ (0..^((♯‘𝐴) − 1))) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ (𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))) = (𝑆‘𝑏)))
133		fveq1 6351	. . . . . . . . 9 ⊢ (𝑎 = (𝐴 ↾ (0..^((♯‘𝐴) − 1))) → (𝑎‘0) = ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘0))
134	133	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑎 = (𝐴 ↾ (0..^((♯‘𝐴) − 1))) → ((𝑎‘0) = (𝑏‘0) ↔ ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘0) = (𝑏‘0)))
135	132, 134	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))))
136	131, 135	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)))))
137		fveq2 6352	. . . . . . . . 9 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → (𝑆‘𝑏) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)))
138	137	eqeq2d 2770	. . . . . . . 8 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → ((𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))) = (𝑆‘𝑏) ↔ (𝑆‘(𝐴 ↾ (0..^((♯‘𝐴) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐴‘𝐾)”⟩))))
139		fveq1 6351	. . . . . . . . 9 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → (𝑏‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0))
140	139	eqeq2d 2770	. . . . . . . 8 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐴‘𝐾)”⟩) → (((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘0) = (𝑏‘0) ↔ ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0)))
141	138, 140	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))))
142	141	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)))))
143	136, 142	rspc2va 3462	. . . . 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))))
144	86, 125, 27, 143	syl21anc 1476	. . . 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))))
145	112, 128, 144	mp2d 49	. . 3 ⊢ (𝜑 → ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐴‘𝐾)”⟩)‘0))
146	72, 66	sseldi 3742	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜))
147		lencl 13510	. . . . . . . 8 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
148	146, 147	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
149	148	nn0red 11544	. . . . . 6 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℝ)
150		ltaddrp 12060	. . . . . 6 ⊢ (((♯‘(𝐵‘𝐿)) ∈ ℝ ∧ 2 ∈ ℝ⁺) → (♯‘(𝐵‘𝐿)) < ((♯‘(𝐵‘𝐿)) + 2))
151	149, 77, 150	sylancl 697	. . . . 5 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) < ((♯‘(𝐵‘𝐿)) + 2))
152	55	nn0red 11544	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐵) ∈ ℝ)
153	152	lem1d 11149	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝐵) − 1) ≤ (♯‘𝐵))
154		fznn 12601	. . . . . . . . . . 11 ⊢ ((♯‘𝐵) ∈ ℤ → (((♯‘𝐵) − 1) ∈ (1...(♯‘𝐵)) ↔ (((♯‘𝐵) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ≤ (♯‘𝐵))))
155	56, 154	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((♯‘𝐵) − 1) ∈ (1...(♯‘𝐵)) ↔ (((♯‘𝐵) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ≤ (♯‘𝐵))))
156	61, 153, 155	mpbir2and 995	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐵) − 1) ∈ (1...(♯‘𝐵)))
157	2, 3, 4, 5, 6, 7	efgsres 18351	. . . . . . . . 9 ⊢ ((𝐵 ∈ dom 𝑆 ∧ ((♯‘𝐵) − 1) ∈ (1...(♯‘𝐵))) → (𝐵 ↾ (0..^((♯‘𝐵) − 1))) ∈ dom 𝑆)
158	28, 156, 157	syl2anc 696	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ↾ (0..^((♯‘𝐵) − 1))) ∈ dom 𝑆)
159	2, 3, 4, 5, 6, 7	efgsval 18344	. . . . . . . 8 ⊢ ((𝐵 ↾ (0..^((♯‘𝐵) − 1))) ∈ dom 𝑆 → (𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) = ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘((♯‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) − 1)))
160	158, 159	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) = ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘((♯‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) − 1)))
161		fz1ssfz0 12629	. . . . . . . . . . . . . 14 ⊢ (1...(♯‘𝐵)) ⊆ (0...(♯‘𝐵))
162	161, 156	sseldi 3742	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘𝐵) − 1) ∈ (0...(♯‘𝐵)))
163		swrd0val 13620	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ Word 𝑊 ∧ ((♯‘𝐵) − 1) ∈ (0...(♯‘𝐵))) → (𝐵 substr ⟨0, ((♯‘𝐵) − 1)⟩) = (𝐵 ↾ (0..^((♯‘𝐵) − 1))))
164	50, 162, 163	syl2anc 696	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 substr ⟨0, ((♯‘𝐵) − 1)⟩) = (𝐵 ↾ (0..^((♯‘𝐵) − 1))))
165	164	fveq2d 6356	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐵 substr ⟨0, ((♯‘𝐵) − 1)⟩)) = (♯‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))))
166		swrd0len 13621	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Word 𝑊 ∧ ((♯‘𝐵) − 1) ∈ (0...(♯‘𝐵))) → (♯‘(𝐵 substr ⟨0, ((♯‘𝐵) − 1)⟩)) = ((♯‘𝐵) − 1))
167	50, 162, 166	syl2anc 696	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐵 substr ⟨0, ((♯‘𝐵) − 1)⟩)) = ((♯‘𝐵) − 1))
168	165, 167	eqtr3d 2796	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) = ((♯‘𝐵) − 1))
169	168	oveq1d 6828	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) − 1) = (((♯‘𝐵) − 1) − 1))
170	169, 60	syl6eqr 2812	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) − 1) = 𝐿)
171	170	fveq2d 6356	. . . . . . 7 ⊢ (𝜑 → ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘((♯‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) − 1)) = ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘𝐿))
172		fvres 6368	. . . . . . . 8 ⊢ (𝐿 ∈ (0..^((♯‘𝐵) − 1)) → ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘𝐿) = (𝐵‘𝐿))
173	64, 172	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘𝐿) = (𝐵‘𝐿))
174	160, 171, 173	3eqtrd 2798	. . . . . 6 ⊢ (𝜑 → (𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) = (𝐵‘𝐿))
175	174	fveq2d 6356	. . . . 5 ⊢ (𝜑 → (♯‘(𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1))))) = (♯‘(𝐵‘𝐿)))
176	2, 3, 4, 5, 6, 7	efgsdmi 18345	. . . . . . . . 9 ⊢ ((𝐵 ∈ dom 𝑆 ∧ ((♯‘𝐵) − 1) ∈ ℕ) → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
177	28, 61, 176	syl2anc 696	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
178	29, 177	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))))
179	60	fveq2i 6355	. . . . . . . . 9 ⊢ (𝐵‘𝐿) = (𝐵‘(((♯‘𝐵) − 1) − 1))
180	179	fveq2i 6355	. . . . . . . 8 ⊢ (𝑇‘(𝐵‘𝐿)) = (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))
181	180	rneqi 5507	. . . . . . 7 ⊢ ran (𝑇‘(𝐵‘𝐿)) = ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))
182	178, 181	syl6eleqr 2850	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿)))
183	2, 3, 4, 5	efgtlen 18339	. . . . . 6 ⊢ (((𝐵‘𝐿) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐵‘𝐿)) + 2))
184	66, 182, 183	syl2anc 696	. . . . 5 ⊢ (𝜑 → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐵‘𝐿)) + 2))
185	151, 175, 184	3brtr4d 4836	. . . 4 ⊢ (𝜑 → (♯‘(𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1))))) < (♯‘(𝑆‘𝐴)))
186	122	simprd 482	. . . . . . 7 ⊢ (𝜑 → (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶)))
187	2, 3, 4, 5, 6, 7	efgsp1 18350	. . . . . . 7 ⊢ ((𝐶 ∈ dom 𝑆 ∧ (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶))) → (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) ∈ dom 𝑆)
188	1, 186, 187	syl2anc 696	. . . . . 6 ⊢ (𝜑 → (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) ∈ dom 𝑆)
189	2, 3, 4, 5, 6, 7	efgsval2 18346	. . . . . 6 ⊢ ((𝐶 ∈ Word 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊 ∧ (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) ∈ dom 𝑆) → (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)) = (𝐵‘𝐿))
190	11, 66, 188, 189	syl3anc 1477	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)) = (𝐵‘𝐿))
191	174, 190	eqtr4d 2797	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)))
192		fveq2 6352	. . . . . . . . 9 ⊢ (𝑎 = (𝐵 ↾ (0..^((♯‘𝐵) − 1))) → (𝑆‘𝑎) = (𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))))
193	192	fveq2d 6356	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ↾ (0..^((♯‘𝐵) − 1))) → (♯‘(𝑆‘𝑎)) = (♯‘(𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1))))))
194	193	breq1d 4814	. . . . . . 7 ⊢ (𝑎 = (𝐵 ↾ (0..^((♯‘𝐵) − 1))) → ((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) ↔ (♯‘(𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1))))) < (♯‘(𝑆‘𝐴))))
195	192	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ↾ (0..^((♯‘𝐵) − 1))) → ((𝑆‘𝑎) = (𝑆‘𝑏) ↔ (𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) = (𝑆‘𝑏)))
196		fveq1 6351	. . . . . . . . 9 ⊢ (𝑎 = (𝐵 ↾ (0..^((♯‘𝐵) − 1))) → (𝑎‘0) = ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘0))
197	196	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑎 = (𝐵 ↾ (0..^((♯‘𝐵) − 1))) → ((𝑎‘0) = (𝑏‘0) ↔ ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘0) = (𝑏‘0)))
198	195, 197	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))))
199	194, 198	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)))))
200		fveq2 6352	. . . . . . . . 9 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → (𝑆‘𝑏) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)))
201	200	eqeq2d 2770	. . . . . . . 8 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → ((𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) = (𝑆‘𝑏) ↔ (𝑆‘(𝐵 ↾ (0..^((♯‘𝐵) − 1)))) = (𝑆‘(𝐶 ++ ⟨“(𝐵‘𝐿)”⟩))))
202		fveq1 6351	. . . . . . . . 9 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → (𝑏‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0))
203	202	eqeq2d 2770	. . . . . . . 8 ⊢ (𝑏 = (𝐶 ++ ⟨“(𝐵‘𝐿)”⟩) → (((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘0) = (𝑏‘0) ↔ ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0)))
204	201, 203	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))))
205	204	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)))))
206	199, 205	rspc2va 3462	. . . . 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))))
207	158, 188, 27, 206	syl21anc 1476	. . . 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))))
208	185, 191, 207	mp2d 49	. . 3 ⊢ (𝜑 → ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘0) = ((𝐶 ++ ⟨“(𝐵‘𝐿)”⟩)‘0))
209	70, 145, 208	3eqtr4d 2804	. 2 ⊢ (𝜑 → ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘0) = ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘0))
210		lbfzo0 12702	. . . 4 ⊢ (0 ∈ (0..^((♯‘𝐴) − 1)) ↔ ((♯‘𝐴) − 1) ∈ ℕ)
211	32, 210	sylibr 224	. . 3 ⊢ (𝜑 → 0 ∈ (0..^((♯‘𝐴) − 1)))
212		fvres 6368	. . 3 ⊢ (0 ∈ (0..^((♯‘𝐴) − 1)) → ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘0) = (𝐴‘0))
213	211, 212	syl 17	. 2 ⊢ (𝜑 → ((𝐴 ↾ (0..^((♯‘𝐴) − 1)))‘0) = (𝐴‘0))
214		lbfzo0 12702	. . . 4 ⊢ (0 ∈ (0..^((♯‘𝐵) − 1)) ↔ ((♯‘𝐵) − 1) ∈ ℕ)
215	61, 214	sylibr 224	. . 3 ⊢ (𝜑 → 0 ∈ (0..^((♯‘𝐵) − 1)))
216		fvres 6368	. . 3 ⊢ (0 ∈ (0..^((♯‘𝐵) − 1)) → ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘0) = (𝐵‘0))
217	215, 216	syl 17	. 2 ⊢ (𝜑 → ((𝐵 ↾ (0..^((♯‘𝐵) − 1)))‘0) = (𝐵‘0))
218	209, 213, 217	3eqtr3d 2802	1 ⊢ (𝜑 → (𝐴‘0) = (𝐵‘0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∀wral 3050 {crab 3054 ∖ cdif 3712 ∅c0 4058 {csn 4321 ⟨cop 4327 ⟨cotp 4329 ∪ ciun 4672 class class class wbr 4804 ↦ cmpt 4881 I cid 5173 × cxp 5264 dom cdm 5266 ran crn 5267 ↾ cres 5268 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ↦ cmpt2 6815 1_𝑜c1o 7722 2_𝑜c2o 7723 ℝcr 10127 0cc0 10128 1c1 10129 + caddc 10131 < clt 10266 ≤ cle 10267 − cmin 10458 ℕcn 11212 2c2 11262 ℕ₀cn0 11484 ℤcz 11569 ℤ_≥cuz 11879 ℝ⁺crp 12025 ...cfz 12519 ..^cfzo 12659 ♯chash 13311 Word cword 13477 ++ cconcat 13479 ⟨“cs1 13480 substr csubstr 13481 splice csplice 13482 ⟨“cs2 13786 ~_FG cefg 18319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-ot 4330 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-fz 12520 df-fzo 12660 df-hash 13312 df-word 13485 df-concat 13487 df-s1 13488 df-substr 13489 df-splice 13490 df-s2 13793

This theorem is referenced by: efgredlemc 18358

W3C validator