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Theorem efgredlemc 18158

Description: The reduced word that forms the base of the sequence in efgsval 18144 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))

**Assertion**
Ref	Expression
efgredlemc	⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘𝑄) → (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 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 efgredlemc Dummy variables 𝑐 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzp1 11721	. 2 ⊢ (𝑃 ∈ (ℤ_≥‘𝑄) → (𝑃 = 𝑄 ∨ 𝑃 ∈ (ℤ_≥‘(𝑄 + 1))))
2		efgredlemb.8	. . . . . 6 ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))
3		efgval.w	. . . . . . . . . . 11 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
4		fviss 6256	. . . . . . . . . . 11 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
5	3, 4	eqsstri 3635	. . . . . . . . . 10 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
6		efgredlem.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)
7		efgval.r	. . . . . . . . . . . . . . 15 ⊢ ∼ = ( ~_FG ‘𝐼)
8		efgval2.m	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
9		efgval2.t	. . . . . . . . . . . . . . 15 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
10		efgred.d	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))
11		efgred.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
12	3, 7, 8, 9, 10, 11	efgsdm 18143	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1)))))
13	12	simp1bi 1076	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
14	6, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
15		eldifi 3732	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Word 𝑊 ∖ {∅}) → 𝐴 ∈ Word 𝑊)
16		wrdf 13310	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(#‘𝐴))⟶𝑊)
17	14, 15, 16	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:(0..^(#‘𝐴))⟶𝑊)
18		fzossfz 12488	. . . . . . . . . . . . 13 ⊢ (0..^((#‘𝐴) − 1)) ⊆ (0...((#‘𝐴) − 1))
19		efgredlemb.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (((#‘𝐴) − 1) − 1)
20		efgredlem.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
21		efgredlem.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)
22		efgredlem.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))
23		efgredlem.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
24	3, 7, 8, 9, 10, 11, 20, 6, 21, 22, 23	efgredlema 18153	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((#‘𝐴) − 1) ∈ ℕ ∧ ((#‘𝐵) − 1) ∈ ℕ))
25	24	simpld 475	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝐴) − 1) ∈ ℕ)
26		fzo0end 12560	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝐴) − 1) ∈ ℕ → (((#‘𝐴) − 1) − 1) ∈ (0..^((#‘𝐴) − 1)))
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((#‘𝐴) − 1) − 1) ∈ (0..^((#‘𝐴) − 1)))
28	19, 27	syl5eqel 2705	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ (0..^((#‘𝐴) − 1)))
29	18, 28	sseldi 3601	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (0...((#‘𝐴) − 1)))
30		lencl 13324	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑊 → (#‘𝐴) ∈ ℕ₀)
31	14, 15, 30	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘𝐴) ∈ ℕ₀)
32	31	nn0zd 11480	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘𝐴) ∈ ℤ)
33		fzoval 12471	. . . . . . . . . . . . 13 ⊢ ((#‘𝐴) ∈ ℤ → (0..^(#‘𝐴)) = (0...((#‘𝐴) − 1)))
34	32, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^(#‘𝐴)) = (0...((#‘𝐴) − 1)))
35	29, 34	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (0..^(#‘𝐴)))
36	17, 35	ffvelrnd 6360	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
37	5, 36	sseldi 3601	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜))
38		efgredlemb.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (0...(#‘(𝐴‘𝐾))))
39		elfzuz 12338	. . . . . . . . . 10 ⊢ (𝑃 ∈ (0...(#‘(𝐴‘𝐾))) → 𝑃 ∈ (ℤ_≥‘0))
40		eluzfz1 12348	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑃))
41	38, 39, 40	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0...𝑃))
42		lencl 13324	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → (#‘(𝐴‘𝐾)) ∈ ℕ₀)
43	37, 42	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ ℕ₀)
44		nn0uz 11722	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
45	43, 44	syl6eleq 2711	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘0))
46		eluzfz2 12349	. . . . . . . . . 10 ⊢ ((#‘(𝐴‘𝐾)) ∈ (ℤ_≥‘0) → (#‘(𝐴‘𝐾)) ∈ (0...(#‘(𝐴‘𝐾))))
47	45, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ (0...(#‘(𝐴‘𝐾))))
48		ccatswrd 13456	. . . . . . . . 9 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ (0 ∈ (0...𝑃) ∧ 𝑃 ∈ (0...(#‘(𝐴‘𝐾))) ∧ (#‘(𝐴‘𝐾)) ∈ (0...(#‘(𝐴‘𝐾))))) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐴‘𝐾) substr ⟨0, (#‘(𝐴‘𝐾))⟩))
49	37, 41, 38, 47, 48	syl13anc 1328	. . . . . . . 8 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐴‘𝐾) substr ⟨0, (#‘(𝐴‘𝐾))⟩))
50		swrdid 13428	. . . . . . . . 9 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨0, (#‘(𝐴‘𝐾))⟩) = (𝐴‘𝐾))
51	37, 50	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨0, (#‘(𝐴‘𝐾))⟩) = (𝐴‘𝐾))
52	49, 51	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (𝐴‘𝐾))
53	3, 7, 8, 9, 10, 11	efgsdm 18143	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1)))))
54	53	simp1bi 1076	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
55	21, 54	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
56		eldifi 3732	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (Word 𝑊 ∖ {∅}) → 𝐵 ∈ Word 𝑊)
57		wrdf 13310	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(#‘𝐵))⟶𝑊)
58	55, 56, 57	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵:(0..^(#‘𝐵))⟶𝑊)
59		fzossfz 12488	. . . . . . . . . . . . 13 ⊢ (0..^((#‘𝐵) − 1)) ⊆ (0...((#‘𝐵) − 1))
60		efgredlemb.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (((#‘𝐵) − 1) − 1)
61	24	simprd 479	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝐵) − 1) ∈ ℕ)
62		fzo0end 12560	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝐵) − 1) ∈ ℕ → (((#‘𝐵) − 1) − 1) ∈ (0..^((#‘𝐵) − 1)))
63	61, 62	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((#‘𝐵) − 1) − 1) ∈ (0..^((#‘𝐵) − 1)))
64	60, 63	syl5eqel 2705	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ (0..^((#‘𝐵) − 1)))
65	59, 64	sseldi 3601	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ (0...((#‘𝐵) − 1)))
66		lencl 13324	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ Word 𝑊 → (#‘𝐵) ∈ ℕ₀)
67	55, 56, 66	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ₀)
68	67	nn0zd 11480	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘𝐵) ∈ ℤ)
69		fzoval 12471	. . . . . . . . . . . . 13 ⊢ ((#‘𝐵) ∈ ℤ → (0..^(#‘𝐵)) = (0...((#‘𝐵) − 1)))
70	68, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^(#‘𝐵)) = (0...((#‘𝐵) − 1)))
71	65, 70	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (0..^(#‘𝐵)))
72	58, 71	ffvelrnd 6360	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
73	5, 72	sseldi 3601	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜))
74		efgredlemb.q	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (0...(#‘(𝐵‘𝐿))))
75		elfzuz 12338	. . . . . . . . . 10 ⊢ (𝑄 ∈ (0...(#‘(𝐵‘𝐿))) → 𝑄 ∈ (ℤ_≥‘0))
76		eluzfz1 12348	. . . . . . . . . 10 ⊢ (𝑄 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑄))
77	74, 75, 76	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0...𝑄))
78		lencl 13324	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → (#‘(𝐵‘𝐿)) ∈ ℕ₀)
79	73, 78	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ ℕ₀)
80	79, 44	syl6eleq 2711	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘0))
81		eluzfz2 12349	. . . . . . . . . 10 ⊢ ((#‘(𝐵‘𝐿)) ∈ (ℤ_≥‘0) → (#‘(𝐵‘𝐿)) ∈ (0...(#‘(𝐵‘𝐿))))
82	80, 81	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ (0...(#‘(𝐵‘𝐿))))
83		ccatswrd 13456	. . . . . . . . 9 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ (0 ∈ (0...𝑄) ∧ 𝑄 ∈ (0...(#‘(𝐵‘𝐿))) ∧ (#‘(𝐵‘𝐿)) ∈ (0...(#‘(𝐵‘𝐿))))) → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) substr ⟨0, (#‘(𝐵‘𝐿))⟩))
84	73, 77, 74, 82, 83	syl13anc 1328	. . . . . . . 8 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) substr ⟨0, (#‘(𝐵‘𝐿))⟩))
85		swrdid 13428	. . . . . . . . 9 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨0, (#‘(𝐵‘𝐿))⟩) = (𝐵‘𝐿))
86	73, 85	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨0, (#‘(𝐵‘𝐿))⟩) = (𝐵‘𝐿))
87	84, 86	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) = (𝐵‘𝐿))
88	52, 87	eqeq12d 2637	. . . . . 6 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)) ↔ (𝐴‘𝐾) = (𝐵‘𝐿)))
89	2, 88	mtbird 315	. . . . 5 ⊢ (𝜑 → ¬ (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
90		efgredlemb.6	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
91		efgredlemb.u	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2_𝑜))
92	3, 7, 8, 9	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)”⟩⟩))
93	36, 38, 91, 92	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃(𝑇‘(𝐴‘𝐾))𝑈) = ((𝐴‘𝐾) splice ⟨𝑃, 𝑃, ⟨“𝑈(𝑀‘𝑈)”⟩⟩))
94	8	efgmf 18126	. . . . . . . . . . . . . . . . 17 ⊢ 𝑀:(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜)
95	94	ffvelrni 6358	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (𝐼 × 2_𝑜) → (𝑀‘𝑈) ∈ (𝐼 × 2_𝑜))
96	91, 95	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀‘𝑈) ∈ (𝐼 × 2_𝑜))
97	91, 96	s2cld 13616	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜))
98		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))⟩)))
99	36, 38, 38, 97, 98	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))⟩)))
100	90, 93, 99	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐴) = ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))
101		efgredlemb.7	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
102		efgredlemb.v	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_𝑜))
103	3, 7, 8, 9	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)”⟩⟩))
104	72, 74, 102, 103	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄(𝑇‘(𝐵‘𝐿))𝑉) = ((𝐵‘𝐿) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩))
105	94	ffvelrni 6358	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ (𝐼 × 2_𝑜) → (𝑀‘𝑉) ∈ (𝐼 × 2_𝑜))
106	102, 105	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀‘𝑉) ∈ (𝐼 × 2_𝑜))
107	102, 106	s2cld 13616	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜))
108		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))⟩)))
109	72, 74, 74, 107, 108	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))⟩)))
110	101, 104, 109	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐵) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
111	22, 100, 110	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))⟩)))
112	111	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((((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))⟩)))
113		swrdcl 13419	. . . . . . . . . . . . . 14 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
114	37, 113	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
115	114	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
116	97	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜))
117		ccatcl 13359	. . . . . . . . . . . 12 ⊢ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜)) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ∈ Word (𝐼 × 2_𝑜))
118	115, 116, 117	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ∈ Word (𝐼 × 2_𝑜))
119		swrdcl 13419	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
120	37, 119	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
121	120	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
122		swrdcl 13419	. . . . . . . . . . . . . 14 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜))
123	73, 122	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜))
124	123	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜))
125	107	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜))
126		ccatcl 13359	. . . . . . . . . . . 12 ⊢ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜)) → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_𝑜))
127	124, 125, 126	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_𝑜))
128		swrdcl 13419	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) → ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
129	73, 128	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
130	129	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
131		swrd0len 13422	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) ∧ 𝑃 ∈ (0...(#‘(𝐴‘𝐾)))) → (#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) = 𝑃)
132	37, 38, 131	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) = 𝑃)
133		swrd0len 13422	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_𝑜) ∧ 𝑄 ∈ (0...(#‘(𝐵‘𝐿)))) → (#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) = 𝑄)
134	73, 74, 133	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) = 𝑄)
135	132, 134	eqeq12d 2637	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) = (#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) ↔ 𝑃 = 𝑄))
136	135	biimpar 502	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) = (#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)))
137		s2len 13634	. . . . . . . . . . . . . . 15 ⊢ (#‘⟨“𝑈(𝑀‘𝑈)”⟩) = 2
138		s2len 13634	. . . . . . . . . . . . . . 15 ⊢ (#‘⟨“𝑉(𝑀‘𝑉)”⟩) = 2
139	137, 138	eqtr4i 2647	. . . . . . . . . . . . . 14 ⊢ (#‘⟨“𝑈(𝑀‘𝑈)”⟩) = (#‘⟨“𝑉(𝑀‘𝑉)”⟩)
140	139	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (#‘⟨“𝑈(𝑀‘𝑈)”⟩) = (#‘⟨“𝑉(𝑀‘𝑉)”⟩))
141	136, 140	oveq12d 6668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) + (#‘⟨“𝑈(𝑀‘𝑈)”⟩)) = ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘⟨“𝑉(𝑀‘𝑉)”⟩)))
142		ccatlen 13360	. . . . . . . . . . . . 13 ⊢ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜)) → (#‘(((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩)) = ((#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) + (#‘⟨“𝑈(𝑀‘𝑈)”⟩)))
143	115, 116, 142	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (#‘(((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩)) = ((#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) + (#‘⟨“𝑈(𝑀‘𝑈)”⟩)))
144		ccatlen 13360	. . . . . . . . . . . . 13 ⊢ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜)) → (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩)) = ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘⟨“𝑉(𝑀‘𝑉)”⟩)))
145	124, 125, 144	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩)) = ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘⟨“𝑉(𝑀‘𝑉)”⟩)))
146	141, 143, 145	3eqtr4d 2666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (#‘(((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩)) = (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩)))
147		ccatopth 13470	. . . . . . . . . . 11 ⊢ ((((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜)) ∧ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜)) ∧ (#‘(((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩)) = (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩))) → (((((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))⟩)) ↔ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
148	118, 121, 127, 130, 146, 147	syl221anc 1337	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((((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))⟩)) ↔ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
149	112, 148	mpbid 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
150	149	simpld 475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩))
151		ccatopth 13470	. . . . . . . . 9 ⊢ (((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜)) ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜)) ∧ (#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) = (#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩))) → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ↔ (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) = ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ = ⟨“𝑉(𝑀‘𝑉)”⟩)))
152	115, 116, 124, 125, 136, 151	syl221anc 1337	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ↔ (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) = ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ = ⟨“𝑉(𝑀‘𝑉)”⟩)))
153	150, 152	mpbid 222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) = ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ = ⟨“𝑉(𝑀‘𝑉)”⟩))
154	153	simpld 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐴‘𝐾) substr ⟨0, 𝑃⟩) = ((𝐵‘𝐿) substr ⟨0, 𝑄⟩))
155	149	simprd 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))
156	154, 155	oveq12d 6668	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
157	89, 156	mtand 691	. . . 4 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
158	157	pm2.21d 118	. . 3 ⊢ (𝜑 → (𝑃 = 𝑄 → (𝐴‘0) = (𝐵‘0)))
159		uzp1 11721	. . . 4 ⊢ (𝑃 ∈ (ℤ_≥‘(𝑄 + 1)) → (𝑃 = (𝑄 + 1) ∨ 𝑃 ∈ (ℤ_≥‘((𝑄 + 1) + 1))))
160	91	s1cld 13383	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_𝑜))
161		ccatcl 13359	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_𝑜)) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_𝑜))
162	114, 160, 161	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_𝑜))
163	96	s1cld 13383	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ⟨“(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜))
164		ccatass 13371	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“(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))⟩))))
165	162, 163, 120, 164	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (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))⟩))))
166		ccatass 13371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜)) → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) = (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ (⟨“𝑈”⟩ ++ ⟨“(𝑀‘𝑈)”⟩)))
167	114, 160, 163, 166	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) = (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ (⟨“𝑈”⟩ ++ ⟨“(𝑀‘𝑈)”⟩)))
168		df-s2 13593	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨“𝑈(𝑀‘𝑈)”⟩ = (⟨“𝑈”⟩ ++ ⟨“(𝑀‘𝑈)”⟩)
169	168	oveq2i 6661	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ (⟨“𝑈”⟩ ++ ⟨“(𝑀‘𝑈)”⟩))
170	167, 169	syl6eqr 2674	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) = (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈(𝑀‘𝑈)”⟩))
171	170	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (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))⟩)))
172	102	s1cld 13383	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_𝑜))
173	106	s1cld 13383	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜))
174		ccatass 13371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜)) → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉”⟩ ++ ⟨“(𝑀‘𝑉)”⟩)))
175	123, 172, 173, 174	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉”⟩ ++ ⟨“(𝑀‘𝑉)”⟩)))
176		df-s2 13593	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨“𝑉(𝑀‘𝑉)”⟩ = (⟨“𝑉”⟩ ++ ⟨“(𝑀‘𝑉)”⟩)
177	176	oveq2i 6661	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉”⟩ ++ ⟨“(𝑀‘𝑉)”⟩))
178	175, 177	syl6eqr 2674	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉(𝑀‘𝑉)”⟩))
179	178	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (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))⟩)))
180	111, 171, 179	3eqtr4d 2666	. . . . . . . . . . . . . . 15 ⊢ (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))⟩)))
181	165, 180	eqtr3d 2658	. . . . . . . . . . . . . 14 ⊢ (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))⟩)))
182	181	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((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))⟩)))
183	162	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_𝑜))
184	163	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜))
185	120	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
186		ccatcl 13359	. . . . . . . . . . . . . . 15 ⊢ ((⟨“(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜)) → (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜))
187	184, 185, 186	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜))
188		ccatcl 13359	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_𝑜)) → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_𝑜))
189	123, 172, 188	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_𝑜))
190	189	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_𝑜))
191	173	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜))
192		ccatcl 13359	. . . . . . . . . . . . . . 15 ⊢ (((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜)) → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_𝑜))
193	190, 191, 192	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_𝑜))
194	129	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜))
195		ccatlen 13360	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_𝑜)) → (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩)) = ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘⟨“𝑉”⟩)))
196	123, 172, 195	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩)) = ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘⟨“𝑉”⟩)))
197		s1len 13385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (#‘⟨“𝑉”⟩) = 1
198	197	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (#‘⟨“𝑉”⟩) = 1)
199	134, 198	oveq12d 6668	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((#‘((𝐵‘𝐿) substr ⟨0, 𝑄⟩)) + (#‘⟨“𝑉”⟩)) = (𝑄 + 1))
200	196, 199	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩)) = (𝑄 + 1))
201	132, 200	eqeq12d 2637	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) = (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩)) ↔ 𝑃 = (𝑄 + 1)))
202	201	biimpar 502	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) = (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩)))
203		s1len 13385	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘⟨“𝑈”⟩) = 1
204		s1len 13385	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘⟨“(𝑀‘𝑉)”⟩) = 1
205	203, 204	eqtr4i 2647	. . . . . . . . . . . . . . . . 17 ⊢ (#‘⟨“𝑈”⟩) = (#‘⟨“(𝑀‘𝑉)”⟩)
206	205	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (#‘⟨“𝑈”⟩) = (#‘⟨“(𝑀‘𝑉)”⟩))
207	202, 206	oveq12d 6668	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) + (#‘⟨“𝑈”⟩)) = ((#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩)) + (#‘⟨“(𝑀‘𝑉)”⟩)))
208	114	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜))
209	160	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_𝑜))
210		ccatlen 13360	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_𝑜)) → (#‘(((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩)) = ((#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) + (#‘⟨“𝑈”⟩)))
211	208, 209, 210	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (#‘(((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩)) = ((#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) + (#‘⟨“𝑈”⟩)))
212		ccatlen 13360	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜)) → (#‘((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩)) = ((#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩)) + (#‘⟨“(𝑀‘𝑉)”⟩)))
213	190, 191, 212	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (#‘((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩)) = ((#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩)) + (#‘⟨“(𝑀‘𝑉)”⟩)))
214	207, 211, 213	3eqtr4d 2666	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (#‘(((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩)) = (#‘((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩)))
215		ccatopth 13470	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_𝑜) ∧ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜)) ∧ (((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_𝑜)) ∧ (#‘(((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩)) = (#‘((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩))) → (((((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))⟩)) ↔ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∧ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
216	183, 187, 193, 194, 214, 215	syl221anc 1337	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((((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))⟩)) ↔ ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∧ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))))
217	182, 216	mpbid 222	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∧ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
218	217	simpld 475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩))
219		ccatopth 13470	. . . . . . . . . . . 12 ⊢ (((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_𝑜)) ∧ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_𝑜)) ∧ (#‘((𝐴‘𝐾) substr ⟨0, 𝑃⟩)) = (#‘(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩))) → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ↔ (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ∧ ⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩)))
220	208, 209, 190, 191, 202, 219	syl221anc 1337	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ↔ (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ∧ ⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩)))
221	218, 220	mpbid 222	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ∧ ⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩))
222	221	simpld 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐴‘𝐾) substr ⟨0, 𝑃⟩) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩))
223	222	oveq1d 6665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))
224	123	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜))
225	172	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_𝑜))
226		ccatass 13371	. . . . . . . . 9 ⊢ ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ∈ Word (𝐼 × 2_𝑜) ∧ ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_𝑜) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜)) → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))))
227	224, 225, 185, 226	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ⟨“𝑉”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))))
228	221	simprd 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩)
229		s111 13395	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ (𝐼 × 2_𝑜) ∧ (𝑀‘𝑉) ∈ (𝐼 × 2_𝑜)) → (⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩ ↔ 𝑈 = (𝑀‘𝑉)))
230	91, 106, 229	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩ ↔ 𝑈 = (𝑀‘𝑉)))
231	230	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩ ↔ 𝑈 = (𝑀‘𝑉)))
232	228, 231	mpbid 222	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → 𝑈 = (𝑀‘𝑉))
233	232	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (𝑀‘𝑈) = (𝑀‘(𝑀‘𝑉)))
234	8	efgmnvl 18127	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (𝐼 × 2_𝑜) → (𝑀‘(𝑀‘𝑉)) = 𝑉)
235	102, 234	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀‘(𝑀‘𝑉)) = 𝑉)
236	235	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (𝑀‘(𝑀‘𝑉)) = 𝑉)
237	233, 236	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (𝑀‘𝑈) = 𝑉)
238	237	s1eqd 13381	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“(𝑀‘𝑈)”⟩ = ⟨“𝑉”⟩)
239	238	oveq1d 6665	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)))
240	217	simprd 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))
241	239, 240	eqtr3d 2658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩))
242	241	oveq2d 6666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩))) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
243	223, 227, 242	3eqtrd 2660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) substr ⟨0, 𝑃⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (#‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (#‘(𝐵‘𝐿))⟩)))
244	89, 243	mtand 691	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 = (𝑄 + 1))
245	244	pm2.21d 118	. . . . 5 ⊢ (𝜑 → (𝑃 = (𝑄 + 1) → (𝐴‘0) = (𝐵‘0)))
246		elfzelz 12342	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ (0...(#‘(𝐵‘𝐿))) → 𝑄 ∈ ℤ)
247	74, 246	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℤ)
248	247	zcnd 11483	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℂ)
249		1cnd 10056	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
250	248, 249, 249	addassd 10062	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + 1) = (𝑄 + (1 + 1)))
251		df-2 11079	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
252	251	oveq2i 6661	. . . . . . . . 9 ⊢ (𝑄 + 2) = (𝑄 + (1 + 1))
253	250, 252	syl6eqr 2674	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 1) + 1) = (𝑄 + 2))
254	253	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘((𝑄 + 1) + 1)) = (ℤ_≥‘(𝑄 + 2)))
255	254	eleq2d 2687	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘((𝑄 + 1) + 1)) ↔ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))))
256	3, 7, 8, 9, 10, 11	efgsfo 18152	. . . . . . . . . 10 ⊢ 𝑆:dom 𝑆–onto→𝑊
257		swrdcl 13419	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_𝑜) → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
258	37, 257	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_𝑜))
259		ccatcl 13359	. . . . . . . . . . . 12 ⊢ ((((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))⟩)) ∈ Word (𝐼 × 2_𝑜))
260	123, 258, 259	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_𝑜))
261	3	efgrcl 18128	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_𝑜)))
262	36, 261	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_𝑜)))
263	262	simprd 479	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2_𝑜))
264	260, 263	eleqtrrd 2704	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) ∈ 𝑊)
265		foelrn 6378	. . . . . . . . . 10 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) ∈ 𝑊) → ∃𝑐 ∈ dom 𝑆(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))
266	256, 264, 265	sylancr 695	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑐 ∈ dom 𝑆(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))
267	266	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) → ∃𝑐 ∈ dom 𝑆(((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))
268	20	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((#‘(𝑆‘𝑎)) < (#‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))
269	6	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝐴 ∈ dom 𝑆)
270	21	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝐵 ∈ dom 𝑆)
271	22	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝑆‘𝐴) = (𝑆‘𝐵))
272	23	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → ¬ (𝐴‘0) = (𝐵‘0))
273	38	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑃 ∈ (0...(#‘(𝐴‘𝐾))))
274	74	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑄 ∈ (0...(#‘(𝐵‘𝐿))))
275	91	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑈 ∈ (𝐼 × 2_𝑜))
276	102	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑉 ∈ (𝐼 × 2_𝑜))
277	90	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
278	101	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
279	2	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))
280		simplr 792	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑃 ∈ (ℤ_≥‘(𝑄 + 2)))
281		simprl 794	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑐 ∈ dom 𝑆)
282		simprr 796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))
283	282	eqcomd 2628	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝑆‘𝑐) = (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)))
284	3, 7, 8, 9, 10, 11, 268, 269, 270, 271, 272, 19, 60, 273, 274, 275, 276, 277, 278, 279, 280, 281, 283	efgredlemd 18157	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝐴‘0) = (𝐵‘0))
285	267, 284	rexlimddv 3035	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) → (𝐴‘0) = (𝐵‘0))
286	285	ex 450	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘(𝑄 + 2)) → (𝐴‘0) = (𝐵‘0)))
287	255, 286	sylbid 230	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘((𝑄 + 1) + 1)) → (𝐴‘0) = (𝐵‘0)))
288	245, 287	jaod 395	. . . 4 ⊢ (𝜑 → ((𝑃 = (𝑄 + 1) ∨ 𝑃 ∈ (ℤ_≥‘((𝑄 + 1) + 1))) → (𝐴‘0) = (𝐵‘0)))
289	159, 288	syl5 34	. . 3 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘(𝑄 + 1)) → (𝐴‘0) = (𝐵‘0)))
290	158, 289	jaod 395	. 2 ⊢ (𝜑 → ((𝑃 = 𝑄 ∨ 𝑃 ∈ (ℤ_≥‘(𝑄 + 1))) → (𝐴‘0) = (𝐵‘0)))
291	1, 290	syl5 34	1 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘𝑄) → (𝐴‘0) = (𝐵‘0)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∖ 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 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1_𝑜c1o 7553 2_𝑜c2o 7554 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 − cmin 10266 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 ⟨“cs1 13294 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-rp 11833 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: efgredlemb 18159

W3C validator