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Theorem efgredleme 18156

Description: Lemma for efgred 18161. (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
efgredleme	⊢ (𝜑 → ((𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶)) ∧ (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶))))

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 efgredleme Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgredlemd.c	. . . . . 6 ⊢ (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	efgsf 18142	. . . . . . . 8 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
9	8	fdmi 6052	. . . . . . . . 9 ⊢ dom 𝑆 = {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}
10	9	feq2i 6037	. . . . . . . 8 ⊢ (𝑆:dom 𝑆⟶𝑊 ↔ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊)
11	8, 10	mpbir 221	. . . . . . 7 ⊢ 𝑆:dom 𝑆⟶𝑊
12	11	ffvelrni 6358	. . . . . 6 ⊢ (𝐶 ∈ dom 𝑆 → (𝑆‘𝐶) ∈ 𝑊)
13	1, 12	syl 17	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐶) ∈ 𝑊)
14		efgredlemb.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (0...(#‘(𝐵‘𝐿))))
15		elfzuz 12338	. . . . . . 7 ⊢ (𝑄 ∈ (0...(#‘(𝐵‘𝐿))) → 𝑄 ∈ (ℤ_≥‘0))
16	14, 15	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (ℤ_≥‘0))
17		efgredlemd.sc	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘𝐶) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)))
18	17	fveq2d 6195	. . . . . . . . 9 ⊢ (𝜑 → (#‘(𝑆‘𝐶)) = (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩))))
19		fviss 6256	. . . . . . . . . . . . 13 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
20	2, 19	eqsstri 3635	. . . . . . . . . . . 12 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
21		efgredlem.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
22		efgredlem.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
23		efgredlem.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)
24		efgredlem.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))
25		efgredlem.5	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
26		efgredlemb.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (((#‘𝐴) − 1) − 1)
27		efgredlemb.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (((#‘𝐵) − 1) − 1)
28	2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 26, 27	efgredlemf 18154	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊))
29	28	simprd 479	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
30	20, 29	sseldi 3601	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜))
31		swrdcl 13419	. . . . . . . . . . 11 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜))
32	30, 31	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜))
33	28	simpld 475	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
34	20, 33	sseldi 3601	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜))
35		swrdcl 13419	. . . . . . . . . . 11 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
36	34, 35	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
37		ccatlen 13360	. . . . . . . . . 10 ⊢ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜)) → (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩))) = ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩))))
38	32, 36, 37	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩))) = ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩))))
39		swrd0len 13422	. . . . . . . . . . . 12 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ 𝑄 ∈ (0...(#‘(𝐵‘𝐿)))) → (#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) = 𝑄)
40	30, 14, 39	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) = 𝑄)
41		2nn0 11309	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
42		uzaddcl 11744	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ (ℤ_≥‘0) ∧ 2 ∈ ℕ₀) → (𝑄 + 2) ∈ (ℤ_≥‘0))
43	16, 41, 42	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 2) ∈ (ℤ_≥‘0))
44		efgredlemb.p	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ (0...(#‘(𝐴‘𝐾))))
45		elfzuz3 12339	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (0...(#‘(𝐴‘𝐾))) → (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘𝑃))
46	44, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘𝑃))
47		efgredlemd.9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘(𝑄 + 2)))
48		uztrn 11704	. . . . . . . . . . . . . 14 ⊢ (((#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘𝑃) ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) → (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘(𝑄 + 2)))
49	46, 47, 48	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘(𝑄 + 2)))
50		elfzuzb 12336	. . . . . . . . . . . . 13 ⊢ ((𝑄 + 2) ∈ (0...(#‘(𝐴‘𝐾))) ↔ ((𝑄 + 2) ∈ (ℤ_≥‘0) ∧ (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘(𝑄 + 2))))
51	43, 49, 50	sylanbrc 698	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 2) ∈ (0...(#‘(𝐴‘𝐾))))
52		lencl 13324	. . . . . . . . . . . . . . 15 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → (#‘(𝐴‘𝐾)) ∈ ℕ₀)
53	34, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ ℕ₀)
54		nn0uz 11722	. . . . . . . . . . . . . 14 ⊢ ℕ₀ = (ℤ_≥‘0)
55	53, 54	syl6eleq 2711	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘0))
56		eluzfz2 12349	. . . . . . . . . . . . 13 ⊢ ((#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘0) → (#‘(𝐴‘𝐾)) ∈ (0...(#‘(𝐴‘𝐾))))
57	55, 56	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ (0...(#‘(𝐴‘𝐾))))
58		swrdlen 13423	. . . . . . . . . . . 12 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ (𝑄 + 2) ∈ (0...(#‘(𝐴‘𝐾))) ∧ (#‘(𝐴‘𝐾)) ∈ (0...(#‘(𝐴‘𝐾)))) → (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = ((#‘(𝐴‘𝐾)) − (𝑄 + 2)))
59	34, 51, 57, 58	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = ((#‘(𝐴‘𝐾)) − (𝑄 + 2)))
60	40, 59	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩))) = (𝑄 + ((#‘(𝐴‘𝐾)) − (𝑄 + 2))))
61		elfzelz 12342	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ (0...(#‘(𝐵‘𝐿))) → 𝑄 ∈ ℤ)
62	14, 61	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ ℤ)
63	62	zcnd 11483	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℂ)
64	53	nn0cnd 11353	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ ℂ)
65		2z 11409	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
66		zaddcl 11417	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑄 + 2) ∈ ℤ)
67	62, 65, 66	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 2) ∈ ℤ)
68	67	zcnd 11483	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 + 2) ∈ ℂ)
69	63, 64, 68	addsubassd 10412	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + (#‘(𝐴‘𝐾))) − (𝑄 + 2)) = (𝑄 + ((#‘(𝐴‘𝐾)) − (𝑄 + 2))))
70		2cn 11091	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
71	70	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℂ)
72	63, 64, 71	pnpcand 10429	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + (#‘(𝐴‘𝐾))) − (𝑄 + 2)) = ((#‘(𝐴‘𝐾)) − 2))
73	60, 69, 72	3eqtr2d 2662	. . . . . . . . 9 ⊢ (𝜑 → ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩))) = ((#‘(𝐴‘𝐾)) − 2))
74	18, 38, 73	3eqtrd 2660	. . . . . . . 8 ⊢ (𝜑 → (#‘(𝑆‘𝐶)) = ((#‘(𝐴‘𝐾)) − 2))
75		elfzelz 12342	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (0...(#‘(𝐴‘𝐾))) → 𝑃 ∈ ℤ)
76	44, 75	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
77		zsubcl 11419	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑃 − 2) ∈ ℤ)
78	76, 65, 77	sylancl 694	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 − 2) ∈ ℤ)
79	65	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℤ)
80	76	zcnd 11483	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℂ)
81		npcan 10290	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝑃 − 2) + 2) = 𝑃)
82	80, 70, 81	sylancl 694	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 − 2) + 2) = 𝑃)
83	82	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ_≥‘((𝑃 − 2) + 2)) = (ℤ_≥‘𝑃))
84	46, 83	eleqtrrd 2704	. . . . . . . . 9 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘((𝑃 − 2) + 2)))
85		eluzsub 11717	. . . . . . . . 9 ⊢ (((𝑃 − 2) ∈ ℤ ∧ 2 ∈ ℤ ∧ (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘((𝑃 − 2) + 2))) → ((#‘(𝐴‘𝐾)) − 2) ∈ (ℤ_≥‘(𝑃 − 2)))
86	78, 79, 84, 85	syl3anc 1326	. . . . . . . 8 ⊢ (𝜑 → ((#‘(𝐴‘𝐾)) − 2) ∈ (ℤ_≥‘(𝑃 − 2)))
87	74, 86	eqeltrd 2701	. . . . . . 7 ⊢ (𝜑 → (#‘(𝑆‘𝐶)) ∈ (ℤ_≥‘(𝑃 − 2)))
88		eluzsub 11717	. . . . . . . 8 ⊢ ((𝑄 ∈ ℤ ∧ 2 ∈ ℤ ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) → (𝑃 − 2) ∈ (ℤ_≥‘𝑄))
89	62, 79, 47, 88	syl3anc 1326	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 2) ∈ (ℤ_≥‘𝑄))
90		uztrn 11704	. . . . . . 7 ⊢ (((#‘(𝑆‘𝐶)) ∈ (ℤ_≥‘(𝑃 − 2)) ∧ (𝑃 − 2) ∈ (ℤ_≥‘𝑄)) → (#‘(𝑆‘𝐶)) ∈ (ℤ_≥‘𝑄))
91	87, 89, 90	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (#‘(𝑆‘𝐶)) ∈ (ℤ_≥‘𝑄))
92		elfzuzb 12336	. . . . . 6 ⊢ (𝑄 ∈ (0...(#‘(𝑆‘𝐶))) ↔ (𝑄 ∈ (ℤ_≥‘0) ∧ (#‘(𝑆‘𝐶)) ∈ (ℤ_≥‘𝑄)))
93	16, 91, 92	sylanbrc 698	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (0...(#‘(𝑆‘𝐶))))
94		efgredlemb.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_𝑜))
95	2, 3, 4, 5	efgtval 18136	. . . . 5 ⊢ (((𝑆‘𝐶) ∈ 𝑊 ∧ 𝑄 ∈ (0...(#‘(𝑆‘𝐶))) ∧ 𝑉 ∈ (𝐼 × 2_𝑜)) → (𝑄(𝑇‘(𝑆‘𝐶))𝑉) = ((𝑆‘𝐶) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩))
96	13, 93, 94, 95	syl3anc 1326	. . . 4 ⊢ (𝜑 → (𝑄(𝑇‘(𝑆‘𝐶))𝑉) = ((𝑆‘𝐶) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩))
97		swrdcl 13419	. . . . . 6 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜))
98	34, 97	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜))
99		wrd0 13330	. . . . . 6 ⊢ ∅ ∈ Word (𝐼 × 2_𝑜)
100	99	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ Word (𝐼 × 2_𝑜))
101	4	efgmf 18126	. . . . . . . 8 ⊢ 𝑀:(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜)
102	101	ffvelrni 6358	. . . . . . 7 ⊢ (𝑉 ∈ (𝐼 × 2_𝑜) → (𝑀‘𝑉) ∈ (𝐼 × 2_𝑜))
103	94, 102	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝑉) ∈ (𝐼 × 2_𝑜))
104	94, 103	s2cld 13616	. . . . 5 ⊢ (𝜑 → ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜))
105		eluzfz1 12348	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑄))
106	16, 105	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑄))
107	62	zred 11482	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ ℝ)
108		nn0addge1 11339	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 ∈ ℝ ∧ 2 ∈ ℕ₀) → 𝑄 ≤ (𝑄 + 2))
109	107, 41, 108	sylancl 694	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 ≤ (𝑄 + 2))
110		eluz2 11693	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 + 2) ∈ (ℤ_≥‘𝑄) ↔ (𝑄 ∈ ℤ ∧ (𝑄 + 2) ∈ ℤ ∧ 𝑄 ≤ (𝑄 + 2)))
111	62, 67, 109, 110	syl3anbrc 1246	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 + 2) ∈ (ℤ_≥‘𝑄))
112		uztrn 11704	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℤ_≥‘(𝑄 + 2)) ∧ (𝑄 + 2) ∈ (ℤ_≥‘𝑄)) → 𝑃 ∈ (ℤ_≥‘𝑄))
113	47, 111, 112	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘𝑄))
114		elfzuzb 12336	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (0...𝑃) ↔ (𝑄 ∈ (ℤ_≥‘0) ∧ 𝑃 ∈ (ℤ_≥‘𝑄)))
115	16, 113, 114	sylanbrc 698	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ (0...𝑃))
116		ccatswrd 13456	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ (0 ∈ (0...𝑄) ∧ 𝑄 ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(#‘(𝐴‘𝐾))))) → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩)) = ((𝐴‘𝐾) substr ⟨0, 𝑃⟩))
117	34, 106, 115, 44, 116	syl13anc 1328	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩)) = ((𝐴‘𝐾) substr ⟨0, 𝑃⟩))
118	117	oveq1d 6665	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩)) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))))
119		swrdcl 13419	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
120	34, 119	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
121		efgredlemb.u	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2_𝑜))
122	101	ffvelrni 6358	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (𝐼 × 2_𝑜) → (𝑀‘𝑈) ∈ (𝐼 × 2_𝑜))
123	121, 122	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀‘𝑈) ∈ (𝐼 × 2_𝑜))
124	121, 123	s2cld 13616	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜))
125		swrdcl 13419	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
126	34, 125	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
127		ccatass 13371	. . . . . . . . . . . 12 ⊢ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜)) → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))))
128	120, 124, 126, 127	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))))
129		efgredlemb.6	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
130	2, 3, 4, 5	efgtval 18136	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘𝐾) ∈ 𝑊 ∧ 𝑃 ∈ (0...(#‘(𝐴‘𝐾))) ∧ 𝑈 ∈ (𝐼 × 2_𝑜)) → (𝑃(𝑇‘(𝐴‘𝐾))𝑈) = ((𝐴‘𝐾) splice ⟨𝑃, 𝑃, ⟨“𝑈(𝑀‘𝑈)”⟩⟩))
131	33, 44, 121, 130	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃(𝑇‘(𝐴‘𝐾))𝑈) = ((𝐴‘𝐾) splice ⟨𝑃, 𝑃, ⟨“𝑈(𝑀‘𝑈)”⟩⟩))
132		splval 13502	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘𝐾) ∈ 𝑊 ∧ (𝑃 ∈ (0...(#‘(𝐴‘𝐾))) ∧ 𝑃 ∈ (0...(#‘(𝐴‘𝐾))) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜))) → ((𝐴‘𝐾) splice ⟨𝑃, 𝑃, ⟨“𝑈(𝑀‘𝑈)”⟩⟩) = ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))
133	33, 44, 44, 124, 132	syl13anc 1328	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴‘𝐾) splice ⟨𝑃, 𝑃, ⟨“𝑈(𝑀‘𝑈)”⟩⟩) = ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))
134	129, 131, 133	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐴) = ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))
135		efgredlemb.7	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
136	2, 3, 4, 5	efgtval 18136	. . . . . . . . . . . . . 14 ⊢ (((𝐵‘𝐿) ∈ 𝑊 ∧ 𝑄 ∈ (0...(#‘(𝐵‘𝐿))) ∧ 𝑉 ∈ (𝐼 × 2_𝑜)) → (𝑄(𝑇‘(𝐵‘𝐿))𝑉) = ((𝐵‘𝐿) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩))
137	29, 14, 94, 136	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄(𝑇‘(𝐵‘𝐿))𝑉) = ((𝐵‘𝐿) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩))
138		splval 13502	. . . . . . . . . . . . . 14 ⊢ (((𝐵‘𝐿) ∈ 𝑊 ∧ (𝑄 ∈ (0...(#‘(𝐵‘𝐿))) ∧ 𝑄 ∈ (0...(#‘(𝐵‘𝐿))) ∧ ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜))) → ((𝐵‘𝐿) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
139	29, 14, 14, 104, 138	syl13anc 1328	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵‘𝐿) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
140	135, 137, 139	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐵) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
141	24, 134, 140	3eqtr3d 2664	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
142	118, 128, 141	3eqtr2d 2662	. . . . . . . . . 10 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩)) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
143		swrdcl 13419	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
144	34, 143	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
145		ccatcl 13359	. . . . . . . . . . . 12 ⊢ ((⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜)) → (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜))
146	124, 126, 145	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜))
147		ccatass 13371	. . . . . . . . . . 11 ⊢ ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜)) → ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩)) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))))
148	98, 144, 146, 147	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩)) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))))
149		swrdcl 13419	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
150	30, 149	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
151		ccatass 13371	. . . . . . . . . . 11 ⊢ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜)) → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
152	32, 104, 150, 151	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
153	142, 148, 152	3eqtr3d 2664	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
154		ccatcl 13359	. . . . . . . . . . 11 ⊢ ((((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜)) → (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) ∈ Word (𝐼 × 2_𝑜))
155	144, 146, 154	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) ∈ Word (𝐼 × 2_𝑜))
156		ccatcl 13359	. . . . . . . . . . 11 ⊢ ((⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜)) → (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) ∈ Word (𝐼 × 2_𝑜))
157	104, 150, 156	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) ∈ Word (𝐼 × 2_𝑜))
158		uztrn 11704	. . . . . . . . . . . . . 14 ⊢ (((#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘𝑃) ∧ 𝑃 ∈ (ℤ_≥‘𝑄)) → (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘𝑄))
159	46, 113, 158	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘𝑄))
160		elfzuzb 12336	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ (0...(#‘(𝐴‘𝐾))) ↔ (𝑄 ∈ (ℤ_≥‘0) ∧ (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘𝑄)))
161	16, 159, 160	sylanbrc 698	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (0...(#‘(𝐴‘𝐾))))
162		swrd0len 13422	. . . . . . . . . . . 12 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ 𝑄 ∈ (0...(#‘(𝐴‘𝐾)))) → (#‘((𝐴‘𝐾) substr ⟨0, 𝑄⟩)) = 𝑄)
163	34, 161, 162	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘((𝐴‘𝐾) substr ⟨0, 𝑄⟩)) = 𝑄)
164	163, 40	eqtr4d 2659	. . . . . . . . . 10 ⊢ (𝜑 → (#‘((𝐴‘𝐾) substr ⟨0, 𝑄⟩)) = (#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)))
165		ccatopth 13470	. . . . . . . . . 10 ⊢ (((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) ∈ Word (𝐼 × 2_𝑜)) ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) ∈ Word (𝐼 × 2_𝑜)) ∧ (#‘((𝐴‘𝐾) substr ⟨0, 𝑄⟩)) = (#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩))) → ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))) ↔ (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) = ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∧ (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))))
166	98, 155, 32, 157, 164, 165	syl221anc 1337	. . . . . . . . 9 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))) ↔ (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) = ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∧ (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))))
167	153, 166	mpbid 222	. . . . . . . 8 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) = ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∧ (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
168	167	simpld 475	. . . . . . 7 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨0, 𝑄⟩) = ((𝐵‘𝐿) substr ⟨0, 𝑄⟩))
169	168	oveq1d 6665	. . . . . 6 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)))
170		ccatrid 13370	. . . . . . . 8 ⊢ (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ∅) = ((𝐴‘𝐾) substr ⟨0, 𝑄⟩))
171	98, 170	syl 17	. . . . . . 7 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ∅) = ((𝐴‘𝐾) substr ⟨0, 𝑄⟩))
172	171	oveq1d 6665	. . . . . 6 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ∅) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)))
173	169, 172, 17	3eqtr4rd 2667	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐶) = ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ∅) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)))
174	163	eqcomd 2628	. . . . 5 ⊢ (𝜑 → 𝑄 = (#‘((𝐴‘𝐾) substr ⟨0, 𝑄⟩)))
175		hash0 13158	. . . . . . 7 ⊢ (#‘∅) = 0
176	175	oveq2i 6661	. . . . . 6 ⊢ (𝑄 + (#‘∅)) = (𝑄 + 0)
177	63	addid1d 10236	. . . . . 6 ⊢ (𝜑 → (𝑄 + 0) = 𝑄)
178	176, 177	syl5req 2669	. . . . 5 ⊢ (𝜑 → 𝑄 = (𝑄 + (#‘∅)))
179	98, 100, 36, 104, 173, 174, 178	splval2 13508	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐶) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩) = ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)))
180		elfzuzb 12336	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (0...(𝑄 + 2)) ↔ (𝑄 ∈ (ℤ_≥‘0) ∧ (𝑄 + 2) ∈ (ℤ_≥‘𝑄)))
181	16, 111, 180	sylanbrc 698	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ (0...(𝑄 + 2)))
182		elfzuzb 12336	. . . . . . . . . . . . . 14 ⊢ ((𝑄 + 2) ∈ (0...𝑃) ↔ ((𝑄 + 2) ∈ (ℤ_≥‘0) ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))))
183	43, 47, 182	sylanbrc 698	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 2) ∈ (0...𝑃))
184		ccatswrd 13456	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ (𝑄 ∈ (0...(𝑄 + 2)) ∧ (𝑄 + 2) ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(#‘(𝐴‘𝐾))))) → (((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩)) = ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩))
185	34, 181, 183, 44, 184	syl13anc 1328	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩)) = ((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩))
186	185	oveq1d 6665	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩)) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))))
187		swrdcl 13419	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ∈ Word (𝐼 × 2_𝑜))
188	34, 187	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ∈ Word (𝐼 × 2_𝑜))
189		swrdcl 13419	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
190	34, 189	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
191		ccatass 13371	. . . . . . . . . . . 12 ⊢ ((((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜)) → ((((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩)) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))))
192	188, 190, 146, 191	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩)) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))))
193	167	simprd 479	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨𝑄, 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
194	186, 192, 193	3eqtr3d 2664	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))) = (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
195		ccatcl 13359	. . . . . . . . . . . 12 ⊢ ((((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜)) → (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) ∈ Word (𝐼 × 2_𝑜))
196	190, 146, 195	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) ∈ Word (𝐼 × 2_𝑜))
197		swrdlen 13423	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ 𝑄 ∈ (0...(𝑄 + 2)) ∧ (𝑄 + 2) ∈ (0...(#‘(𝐴‘𝐾)))) → (#‘((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩)) = ((𝑄 + 2) − 𝑄))
198	34, 181, 51, 197	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩)) = ((𝑄 + 2) − 𝑄))
199		pncan2 10288	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝑄 + 2) − 𝑄) = 2)
200	63, 70, 199	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄 + 2) − 𝑄) = 2)
201	198, 200	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩)) = 2)
202		s2len 13634	. . . . . . . . . . . 12 ⊢ (#‘⟨“𝑉(𝑀‘𝑉)”⟩) = 2
203	201, 202	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩)) = (#‘⟨“𝑉(𝑀‘𝑉)”⟩))
204		ccatopth 13470	. . . . . . . . . . 11 ⊢ (((((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ∈ Word (𝐼 × 2_𝑜) ∧ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) ∈ Word (𝐼 × 2_𝑜)) ∧ (⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜)) ∧ (#‘((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩)) = (#‘⟨“𝑉(𝑀‘𝑉)”⟩)) → ((((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))) = (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) ↔ (((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) = ⟨“𝑉(𝑀‘𝑉)”⟩ ∧ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
205	188, 196, 104, 150, 203, 204	syl221anc 1337	. . . . . . . . . 10 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) ++ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))) = (⟨“𝑉(𝑀‘𝑉)”⟩ ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) ↔ (((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) = ⟨“𝑉(𝑀‘𝑉)”⟩ ∧ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
206	194, 205	mpbid 222	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) = ⟨“𝑉(𝑀‘𝑉)”⟩ ∧ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
207	206	simpld 475	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩) = ⟨“𝑉(𝑀‘𝑉)”⟩)
208	207	oveq2d 6666	. . . . . . 7 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩)) = (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩))
209		ccatswrd 13456	. . . . . . . 8 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ (0 ∈ (0...𝑄) ∧ 𝑄 ∈ (0...(𝑄 + 2)) ∧ (𝑄 + 2) ∈ (0...(#‘(𝐴‘𝐾))))) → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩)) = ((𝐴‘𝐾) substr ⟨0, (𝑄 + 2)⟩))
210	34, 106, 181, 51, 209	syl13anc 1328	. . . . . . 7 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨𝑄, (𝑄 + 2)⟩)) = ((𝐴‘𝐾) substr ⟨0, (𝑄 + 2)⟩))
211	208, 210	eqtr3d 2658	. . . . . 6 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) = ((𝐴‘𝐾) substr ⟨0, (𝑄 + 2)⟩))
212	211	oveq1d 6665	. . . . 5 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (((𝐴‘𝐾) substr ⟨0, (𝑄 + 2)⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)))
213		eluzfz1 12348	. . . . . . 7 ⊢ ((𝑄 + 2) ∈ (ℤ_≥‘0) → 0 ∈ (0...(𝑄 + 2)))
214	43, 213	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...(𝑄 + 2)))
215		ccatswrd 13456	. . . . . 6 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ (0 ∈ (0...(𝑄 + 2)) ∧ (𝑄 + 2) ∈ (0...(#‘(𝐴‘𝐾))) ∧ (#‘(𝐴‘𝐾)) ∈ (0...(#‘(𝐴‘𝐾))))) → (((𝐴‘𝐾) substr ⟨0, (𝑄 + 2)⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = ((𝐴‘𝐾) substr ⟨0, (#‘(𝐴‘𝐾))⟩))
216	34, 214, 51, 57, 215	syl13anc 1328	. . . . 5 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, (𝑄 + 2)⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = ((𝐴‘𝐾) substr ⟨0, (#‘(𝐴‘𝐾))⟩))
217		swrdid 13428	. . . . . 6 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨0, (#‘(𝐴‘𝐾))⟩) = (𝐴‘𝐾))
218	34, 217	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨0, (#‘(𝐴‘𝐾))⟩) = (𝐴‘𝐾))
219	212, 216, 218	3eqtrd 2660	. . . 4 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝐴‘𝐾))
220	96, 179, 219	3eqtrd 2660	. . 3 ⊢ (𝜑 → (𝑄(𝑇‘(𝑆‘𝐶))𝑉) = (𝐴‘𝐾))
221	2, 3, 4, 5	efgtf 18135	. . . . . . 7 ⊢ ((𝑆‘𝐶) ∈ 𝑊 → ((𝑇‘(𝑆‘𝐶)) = (𝑎 ∈ (0...(#‘(𝑆‘𝐶))), 𝑖 ∈ (𝐼 × 2_𝑜) ↦ ((𝑆‘𝐶) splice ⟨𝑎, 𝑎, ⟨“𝑖(𝑀‘𝑖)”⟩⟩)) ∧ (𝑇‘(𝑆‘𝐶)):((0...(#‘(𝑆‘𝐶))) × (𝐼 × 2_𝑜))⟶𝑊))
222	13, 221	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑇‘(𝑆‘𝐶)) = (𝑎 ∈ (0...(#‘(𝑆‘𝐶))), 𝑖 ∈ (𝐼 × 2_𝑜) ↦ ((𝑆‘𝐶) splice ⟨𝑎, 𝑎, ⟨“𝑖(𝑀‘𝑖)”⟩⟩)) ∧ (𝑇‘(𝑆‘𝐶)):((0...(#‘(𝑆‘𝐶))) × (𝐼 × 2_𝑜))⟶𝑊))
223	222	simprd 479	. . . . 5 ⊢ (𝜑 → (𝑇‘(𝑆‘𝐶)):((0...(#‘(𝑆‘𝐶))) × (𝐼 × 2_𝑜))⟶𝑊)
224		ffn 6045	. . . . 5 ⊢ ((𝑇‘(𝑆‘𝐶)):((0...(#‘(𝑆‘𝐶))) × (𝐼 × 2_𝑜))⟶𝑊 → (𝑇‘(𝑆‘𝐶)) Fn ((0...(#‘(𝑆‘𝐶))) × (𝐼 × 2_𝑜)))
225	223, 224	syl 17	. . . 4 ⊢ (𝜑 → (𝑇‘(𝑆‘𝐶)) Fn ((0...(#‘(𝑆‘𝐶))) × (𝐼 × 2_𝑜)))
226		fnovrn 6809	. . . 4 ⊢ (((𝑇‘(𝑆‘𝐶)) Fn ((0...(#‘(𝑆‘𝐶))) × (𝐼 × 2_𝑜)) ∧ 𝑄 ∈ (0...(#‘(𝑆‘𝐶))) ∧ 𝑉 ∈ (𝐼 × 2_𝑜)) → (𝑄(𝑇‘(𝑆‘𝐶))𝑉) ∈ ran (𝑇‘(𝑆‘𝐶)))
227	225, 93, 94, 226	syl3anc 1326	. . 3 ⊢ (𝜑 → (𝑄(𝑇‘(𝑆‘𝐶))𝑉) ∈ ran (𝑇‘(𝑆‘𝐶)))
228	220, 227	eqeltrrd 2702	. 2 ⊢ (𝜑 → (𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶)))
229		uztrn 11704	. . . . . . 7 ⊢ (((𝑃 − 2) ∈ (ℤ_≥‘𝑄) ∧ 𝑄 ∈ (ℤ_≥‘0)) → (𝑃 − 2) ∈ (ℤ_≥‘0))
230	89, 16, 229	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝑃 − 2) ∈ (ℤ_≥‘0))
231		elfzuzb 12336	. . . . . 6 ⊢ ((𝑃 − 2) ∈ (0...(#‘(𝑆‘𝐶))) ↔ ((𝑃 − 2) ∈ (ℤ_≥‘0) ∧ (#‘(𝑆‘𝐶)) ∈ (ℤ_≥‘(𝑃 − 2))))
232	230, 87, 231	sylanbrc 698	. . . . 5 ⊢ (𝜑 → (𝑃 − 2) ∈ (0...(#‘(𝑆‘𝐶))))
233	2, 3, 4, 5	efgtval 18136	. . . . 5 ⊢ (((𝑆‘𝐶) ∈ 𝑊 ∧ (𝑃 − 2) ∈ (0...(#‘(𝑆‘𝐶))) ∧ 𝑈 ∈ (𝐼 × 2_𝑜)) → ((𝑃 − 2)(𝑇‘(𝑆‘𝐶))𝑈) = ((𝑆‘𝐶) splice ⟨(𝑃 − 2), (𝑃 − 2), ⟨“𝑈(𝑀‘𝑈)”⟩⟩))
234	13, 232, 121, 233	syl3anc 1326	. . . 4 ⊢ (𝜑 → ((𝑃 − 2)(𝑇‘(𝑆‘𝐶))𝑈) = ((𝑆‘𝐶) splice ⟨(𝑃 − 2), (𝑃 − 2), ⟨“𝑈(𝑀‘𝑈)”⟩⟩))
235		swrdcl 13419	. . . . . 6 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ∈ Word (𝐼 × 2_𝑜))
236	30, 235	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ∈ Word (𝐼 × 2_𝑜))
237		swrdcl 13419	. . . . . 6 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
238	30, 237	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
239		ccatswrd 13456	. . . . . . . . . . 11 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝑄 + 2) ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(#‘(𝐴‘𝐾))) ∧ (#‘(𝐴‘𝐾)) ∈ (0...(#‘(𝐴‘𝐾))))) → (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩))
240	34, 183, 44, 57, 239	syl13anc 1328	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩))
241	206	simprd 479	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))
242		elfzuzb 12336	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ (0...(𝑃 − 2)) ↔ (𝑄 ∈ (ℤ_≥‘0) ∧ (𝑃 − 2) ∈ (ℤ_≥‘𝑄)))
243	16, 89, 242	sylanbrc 698	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ (0...(𝑃 − 2)))
244	2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 26, 27, 44, 14, 121, 94, 129, 135	efgredlemg 18155	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) = (#‘(𝐵‘𝐿)))
245	244, 46	eqeltrrd 2702	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘𝑃))
246		0le2 11111	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 2
247	246	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ≤ 2)
248	76	zred 11482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑃 ∈ ℝ)
249		2re 11090	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
250		subge02 10544	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ ℝ ∧ 2 ∈ ℝ) → (0 ≤ 2 ↔ (𝑃 − 2) ≤ 𝑃))
251	248, 249, 250	sylancl 694	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0 ≤ 2 ↔ (𝑃 − 2) ≤ 𝑃))
252	247, 251	mpbid 222	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑃 − 2) ≤ 𝑃)
253		eluz2 11693	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℤ_≥‘(𝑃 − 2)) ↔ ((𝑃 − 2) ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ (𝑃 − 2) ≤ 𝑃))
254	78, 76, 252, 253	syl3anbrc 1246	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘(𝑃 − 2)))
255		uztrn 11704	. . . . . . . . . . . . . . . . 17 ⊢ (((#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘𝑃) ∧ 𝑃 ∈ (ℤ_≥‘(𝑃 − 2))) → (#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘(𝑃 − 2)))
256	245, 254, 255	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘(𝑃 − 2)))
257		elfzuzb 12336	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 − 2) ∈ (0...(#‘(𝐵‘𝐿))) ↔ ((𝑃 − 2) ∈ (ℤ_≥‘0) ∧ (#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘(𝑃 − 2))))
258	230, 256, 257	sylanbrc 698	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃 − 2) ∈ (0...(#‘(𝐵‘𝐿))))
259		lencl 13324	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → (#‘(𝐵‘𝐿)) ∈ ℕ₀)
260	30, 259	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ ℕ₀)
261	260, 54	syl6eleq 2711	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘0))
262		eluzfz2 12349	. . . . . . . . . . . . . . . 16 ⊢ ((#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘0) → (#‘(𝐵‘𝐿)) ∈ (0...(#‘(𝐵‘𝐿))))
263	261, 262	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ (0...(#‘(𝐵‘𝐿))))
264		ccatswrd 13456	. . . . . . . . . . . . . . 15 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ (𝑄 ∈ (0...(𝑃 − 2)) ∧ (𝑃 − 2) ∈ (0...(#‘(𝐵‘𝐿))) ∧ (#‘(𝐵‘𝐿)) ∈ (0...(#‘(𝐵‘𝐿))))) → (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))
265	30, 243, 258, 263, 264	syl13anc 1328	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))
266	241, 265	eqtr4d 2659	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩)))
267		swrdcl 13419	. . . . . . . . . . . . . . 15 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ∈ Word (𝐼 × 2_𝑜))
268	30, 267	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ∈ Word (𝐼 × 2_𝑜))
269		swrdcl 13419	. . . . . . . . . . . . . . 15 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
270	30, 269	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
271		swrdlen 13423	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ (𝑄 + 2) ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(#‘(𝐴‘𝐾)))) → (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩)) = (𝑃 − (𝑄 + 2)))
272	34, 183, 44, 271	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩)) = (𝑃 − (𝑄 + 2)))
273		swrdlen 13423	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ 𝑄 ∈ (0...(𝑃 − 2)) ∧ (𝑃 − 2) ∈ (0...(#‘(𝐵‘𝐿)))) → (#‘((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)) = ((𝑃 − 2) − 𝑄))
274	30, 243, 258, 273	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (#‘((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)) = ((𝑃 − 2) − 𝑄))
275	80, 63, 71	sub32d 10424	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑃 − 𝑄) − 2) = ((𝑃 − 2) − 𝑄))
276	80, 63, 71	subsub4d 10423	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑃 − 𝑄) − 2) = (𝑃 − (𝑄 + 2)))
277	274, 275, 276	3eqtr2d 2662	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (#‘((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)) = (𝑃 − (𝑄 + 2)))
278	272, 277	eqtr4d 2659	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩)) = (#‘((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)))
279		ccatopth 13470	. . . . . . . . . . . . . 14 ⊢ (((((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜)) ∧ (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜)) ∧ (#‘((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩)) = (#‘((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩))) → ((((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩)) ↔ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ∧ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩))))
280	190, 146, 268, 270, 278, 279	syl221anc 1337	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩)) ↔ (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ∧ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩))))
281	266, 280	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ∧ (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩)))
282	281	simpld 475	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩))
283	281	simprd 479	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩))
284		elfzuzb 12336	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 − 2) ∈ (0...𝑃) ↔ ((𝑃 − 2) ∈ (ℤ_≥‘0) ∧ 𝑃 ∈ (ℤ_≥‘(𝑃 − 2))))
285	230, 254, 284	sylanbrc 698	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃 − 2) ∈ (0...𝑃))
286		elfzuz 12338	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (0...(#‘(𝐴‘𝐾))) → 𝑃 ∈ (ℤ_≥‘0))
287	44, 286	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘0))
288		elfzuzb 12336	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (0...(#‘(𝐵‘𝐿))) ↔ (𝑃 ∈ (ℤ_≥‘0) ∧ (#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘𝑃)))
289	287, 245, 288	sylanbrc 698	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ (0...(#‘(𝐵‘𝐿))))
290		ccatswrd 13456	. . . . . . . . . . . . . . 15 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝑃 − 2) ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(#‘(𝐵‘𝐿))) ∧ (#‘(𝐵‘𝐿)) ∈ (0...(#‘(𝐵‘𝐿))))) → (((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩))
291	30, 285, 289, 263, 290	syl13anc 1328	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), (#‘(𝐵‘𝐿))⟩))
292	283, 291	eqtr4d 2659	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
293		swrdcl 13419	. . . . . . . . . . . . . . 15 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
294	30, 293	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
295		s2len 13634	. . . . . . . . . . . . . . 15 ⊢ (#‘⟨“𝑈(𝑀‘𝑈)”⟩) = 2
296		swrdlen 13423	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ (𝑃 − 2) ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(#‘(𝐵‘𝐿)))) → (#‘((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩)) = (𝑃 − (𝑃 − 2)))
297	30, 285, 289, 296	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (#‘((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩)) = (𝑃 − (𝑃 − 2)))
298		nncan 10310	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℂ ∧ 2 ∈ ℂ) → (𝑃 − (𝑃 − 2)) = 2)
299	80, 70, 298	sylancl 694	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃 − (𝑃 − 2)) = 2)
300	297, 299	eqtr2d 2657	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 = (#‘((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩)))
301	295, 300	syl5eq 2668	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘⟨“𝑈(𝑀‘𝑈)”⟩) = (#‘((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩)))
302		ccatopth 13470	. . . . . . . . . . . . . 14 ⊢ (((⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜)) ∧ (((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜)) ∧ (#‘⟨“𝑈(𝑀‘𝑈)”⟩) = (#‘((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩))) → ((⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) ↔ (⟨“𝑈(𝑀‘𝑈)”⟩ = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩))))
303	124, 126, 294, 238, 301, 302	syl221anc 1337	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((⟨“𝑈(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) ↔ (⟨“𝑈(𝑀‘𝑈)”⟩ = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩))))
304	292, 303	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → (⟨“𝑈(𝑀‘𝑈)”⟩ = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
305	304	simprd 479	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩))
306	282, 305	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨(𝑄 + 2), 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
307	240, 306	eqtr3d 2658	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩) = (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
308	307	oveq2d 6666	. . . . . . . 8 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩))))
309		ccatass 13371	. . . . . . . . 9 ⊢ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜)) → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩))))
310	32, 268, 238, 309	syl3anc 1326	. . . . . . . 8 ⊢ (𝜑 → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩))))
311	308, 310	eqtr4d 2659	. . . . . . 7 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
312		ccatswrd 13456	. . . . . . . . 9 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ (0 ∈ (0...𝑄) ∧ 𝑄 ∈ (0...(𝑃 − 2)) ∧ (𝑃 − 2) ∈ (0...(#‘(𝐵‘𝐿))))) → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)) = ((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩))
313	30, 106, 243, 258, 312	syl13anc 1328	. . . . . . . 8 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)) = ((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩))
314	313	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (𝑃 − 2)⟩)) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
315	17, 311, 314	3eqtrd 2660	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐶) = (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
316		ccatrid 13370	. . . . . . . 8 ⊢ (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ∈ Word (𝐼 × 2_𝑜) → (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ∅) = ((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩))
317	236, 316	syl 17	. . . . . . 7 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ∅) = ((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩))
318	317	oveq1d 6665	. . . . . 6 ⊢ (𝜑 → ((((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ∅) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
319	315, 318	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐶) = ((((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ∅) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
320		swrd0len 13422	. . . . . . 7 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ (𝑃 − 2) ∈ (0...(#‘(𝐵‘𝐿)))) → (#‘((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩)) = (𝑃 − 2))
321	30, 258, 320	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (#‘((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩)) = (𝑃 − 2))
322	321	eqcomd 2628	. . . . 5 ⊢ (𝜑 → (𝑃 − 2) = (#‘((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩)))
323	175	oveq2i 6661	. . . . . 6 ⊢ ((𝑃 − 2) + (#‘∅)) = ((𝑃 − 2) + 0)
324	78	zcnd 11483	. . . . . . 7 ⊢ (𝜑 → (𝑃 − 2) ∈ ℂ)
325	324	addid1d 10236	. . . . . 6 ⊢ (𝜑 → ((𝑃 − 2) + 0) = (𝑃 − 2))
326	323, 325	syl5req 2669	. . . . 5 ⊢ (𝜑 → (𝑃 − 2) = ((𝑃 − 2) + (#‘∅)))
327	236, 100, 238, 124, 319, 322, 326	splval2 13508	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐶) splice ⟨(𝑃 − 2), (𝑃 − 2), ⟨“𝑈(𝑀‘𝑈)”⟩⟩) = ((((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
328	304	simpld 475	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝑈(𝑀‘𝑈)”⟩ = ((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩))
329	328	oveq2d 6666	. . . . . . 7 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩)))
330		eluzfz1 12348	. . . . . . . . 9 ⊢ ((𝑃 − 2) ∈ (ℤ_≥‘0) → 0 ∈ (0...(𝑃 − 2)))
331	230, 330	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...(𝑃 − 2)))
332		ccatswrd 13456	. . . . . . . 8 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ (0 ∈ (0...(𝑃 − 2)) ∧ (𝑃 − 2) ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(#‘(𝐵‘𝐿))))) → (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩)) = ((𝐵‘𝐿) substr ⟨0, 𝑃⟩))
333	30, 331, 285, 289, 332	syl13anc 1328	. . . . . . 7 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ((𝐵‘𝐿) substr ⟨(𝑃 − 2), 𝑃⟩)) = ((𝐵‘𝐿) substr ⟨0, 𝑃⟩))
334	329, 333	eqtrd 2656	. . . . . 6 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = ((𝐵‘𝐿) substr ⟨0, 𝑃⟩))
335	334	oveq1d 6665	. . . . 5 ⊢ (𝜑 → ((((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑃⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)))
336		eluzfz1 12348	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑃))
337	287, 336	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑃))
338		ccatswrd 13456	. . . . . 6 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ (0 ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(#‘(𝐵‘𝐿))) ∧ (#‘(𝐵‘𝐿)) ∈ (0...(#‘(𝐵‘𝐿))))) → (((𝐵‘𝐿) substr ⟨0, 𝑃⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) substr ⟨0, (#‘(𝐵‘𝐿))⟩))
339	30, 337, 289, 263, 338	syl13anc 1328	. . . . 5 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, 𝑃⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) substr ⟨0, (#‘(𝐵‘𝐿))⟩))
340		swrdid 13428	. . . . . 6 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨0, (#‘(𝐵‘𝐿))⟩) = (𝐵‘𝐿))
341	30, 340	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨0, (#‘(𝐵‘𝐿))⟩) = (𝐵‘𝐿))
342	335, 339, 341	3eqtrd 2660	. . . 4 ⊢ (𝜑 → ((((𝐵‘𝐿) substr ⟨0, (𝑃 − 2)⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑃, (#‘(𝐵‘𝐿))⟩)) = (𝐵‘𝐿))
343	234, 327, 342	3eqtrd 2660	. . 3 ⊢ (𝜑 → ((𝑃 − 2)(𝑇‘(𝑆‘𝐶))𝑈) = (𝐵‘𝐿))
344		fnovrn 6809	. . . 4 ⊢ (((𝑇‘(𝑆‘𝐶)) Fn ((0...(#‘(𝑆‘𝐶))) × (𝐼 × 2_𝑜)) ∧ (𝑃 − 2) ∈ (0...(#‘(𝑆‘𝐶))) ∧ 𝑈 ∈ (𝐼 × 2_𝑜)) → ((𝑃 − 2)(𝑇‘(𝑆‘𝐶))𝑈) ∈ ran (𝑇‘(𝑆‘𝐶)))
345	225, 232, 121, 344	syl3anc 1326	. . 3 ⊢ (𝜑 → ((𝑃 − 2)(𝑇‘(𝑆‘𝐶))𝑈) ∈ ran (𝑇‘(𝑆‘𝐶)))
346	343, 345	eqeltrrd 2702	. 2 ⊢ (𝜑 → (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶)))
347	228, 346	jca 554	1 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶)) ∧ (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ∖ cdif 3571 ∅c0 3915 {csn 4177 ⟨cop 4183 ⟨cotp 4185 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 dom cdm 5114 ran crn 5115 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1_𝑜c1o 7553 2_𝑜c2o 7554 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 substr csubstr 13295 splice csplice 13296 ⟨“cs2 13586 ~_FG cefg 18119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-s2 13593

This theorem is referenced by: efgredlemd 18157

W3C validator