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Theorem efgredlemc 18598

Description: The reduced word that forms the base of the sequence in efgsval 18584 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.)

**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 12128	. 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 6611	. . . . . . . . . . 11 ⊢ ( I ‘Word (𝐼 × 2_o)) ⊆ Word (𝐼 × 2_o)
5	3, 4	eqsstri 3924	. . . . . . . . . 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 18583	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1)))))
13	12	simp1bi 1138	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
14	6, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅}))
15		eldifi 4026	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (Word 𝑊 ∖ {∅}) → 𝐴 ∈ Word 𝑊)
16		wrdf 13712	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(♯‘𝐴))⟶𝑊)
17	14, 15, 16	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:(0..^(♯‘𝐴))⟶𝑊)
18		fzossfz 12906	. . . . . . . . . . . . 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 18593	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ))
25	24	simpld 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ)
26		fzo0end 12979	. . . . . . . . . . . . . . 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 2886	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ (0..^((♯‘𝐴) − 1)))
29	18, 28	sseldi 3889	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ (0...((♯‘𝐴) − 1)))
30		lencl 13729	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℕ₀)
31	14, 15, 30	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ₀)
32	31	nn0zd 11935	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ)
33		fzoval 12889	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℤ → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1)))
34	32, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1)))
35	29, 34	eleqtrrd 2885	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐴)))
36	17, 35	ffvelrnd 6720	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊)
37	5, 36	sseldi 3889	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2_o))
38		efgredlemb.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))
39		lencl 13729	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
40	37, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℕ₀)
41		nn0uz 12129	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
42	40, 41	syl6eleq 2892	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ (ℤ_≥‘0))
43		eluzfz2 12765	. . . . . . . . . 10 ⊢ ((♯‘(𝐴‘𝐾)) ∈ (ℤ_≥‘0) → (♯‘(𝐴‘𝐾)) ∈ (0...(♯‘(𝐴‘𝐾))))
44	42, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ (0...(♯‘(𝐴‘𝐾))))
45		ccatpfx 13899	. . . . . . . . 9 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) ∧ 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))) ∧ (♯‘(𝐴‘𝐾)) ∈ (0...(♯‘(𝐴‘𝐾)))) → (((𝐴‘𝐾) prefix 𝑃) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((𝐴‘𝐾) prefix (♯‘(𝐴‘𝐾))))
46	37, 38, 44, 45	syl3anc 1364	. . . . . . . 8 ⊢ (𝜑 → (((𝐴‘𝐾) prefix 𝑃) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((𝐴‘𝐾) prefix (♯‘(𝐴‘𝐾))))
47		pfxid 13882	. . . . . . . . 9 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) → ((𝐴‘𝐾) prefix (♯‘(𝐴‘𝐾))) = (𝐴‘𝐾))
48	37, 47	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝐾) prefix (♯‘(𝐴‘𝐾))) = (𝐴‘𝐾))
49	46, 48	eqtrd 2830	. . . . . . 7 ⊢ (𝜑 → (((𝐴‘𝐾) prefix 𝑃) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = (𝐴‘𝐾))
50	3, 7, 8, 9, 10, 11	efgsdm 18583	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1)))))
51	50	simp1bi 1138	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
52	21, 51	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅}))
53		eldifi 4026	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (Word 𝑊 ∖ {∅}) → 𝐵 ∈ Word 𝑊)
54		wrdf 13712	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(♯‘𝐵))⟶𝑊)
55	52, 53, 54	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵:(0..^(♯‘𝐵))⟶𝑊)
56		fzossfz 12906	. . . . . . . . . . . . 13 ⊢ (0..^((♯‘𝐵) − 1)) ⊆ (0...((♯‘𝐵) − 1))
57		efgredlemb.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)
58	24	simprd 496	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝐵) − 1) ∈ ℕ)
59		fzo0end 12979	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐵) − 1) ∈ ℕ → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1)))
60	58, 59	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1)))
61	57, 60	syl5eqel 2886	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ (0..^((♯‘𝐵) − 1)))
62	56, 61	sseldi 3889	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ (0...((♯‘𝐵) − 1)))
63		lencl 13729	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ Word 𝑊 → (♯‘𝐵) ∈ ℕ₀)
64	52, 53, 63	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ₀)
65	64	nn0zd 11935	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ)
66		fzoval 12889	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐵) ∈ ℤ → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1)))
67	65, 66	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1)))
68	62, 67	eleqtrrd 2885	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ (0..^(♯‘𝐵)))
69	55, 68	ffvelrnd 6720	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊)
70	5, 69	sseldi 3889	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2_o))
71		efgredlemb.q	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))
72		lencl 13729	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_o) → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
73	70, 72	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℕ₀)
74	73, 41	syl6eleq 2892	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ (ℤ_≥‘0))
75		eluzfz2 12765	. . . . . . . . . 10 ⊢ ((♯‘(𝐵‘𝐿)) ∈ (ℤ_≥‘0) → (♯‘(𝐵‘𝐿)) ∈ (0...(♯‘(𝐵‘𝐿))))
76	74, 75	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ (0...(♯‘(𝐵‘𝐿))))
77		ccatpfx 13899	. . . . . . . . 9 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_o) ∧ 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))) ∧ (♯‘(𝐵‘𝐿)) ∈ (0...(♯‘(𝐵‘𝐿)))) → (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) prefix (♯‘(𝐵‘𝐿))))
78	70, 71, 76, 77	syl3anc 1364	. . . . . . . 8 ⊢ (𝜑 → (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)) = ((𝐵‘𝐿) prefix (♯‘(𝐵‘𝐿))))
79		pfxid 13882	. . . . . . . . 9 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_o) → ((𝐵‘𝐿) prefix (♯‘(𝐵‘𝐿))) = (𝐵‘𝐿))
80	70, 79	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐵‘𝐿) prefix (♯‘(𝐵‘𝐿))) = (𝐵‘𝐿))
81	78, 80	eqtrd 2830	. . . . . . 7 ⊢ (𝜑 → (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)) = (𝐵‘𝐿))
82	49, 81	eqeq12d 2809	. . . . . 6 ⊢ (𝜑 → ((((𝐴‘𝐾) prefix 𝑃) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)) ↔ (𝐴‘𝐾) = (𝐵‘𝐿)))
83	2, 82	mtbird 326	. . . . 5 ⊢ (𝜑 → ¬ (((𝐴‘𝐾) prefix 𝑃) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
84		efgredlemb.6	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
85		efgredlemb.u	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2_o))
86	3, 7, 8, 9	efgtval 18576	. . . . . . . . . . . . . 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)”⟩⟩))
87	36, 38, 85, 86	syl3anc 1364	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃(𝑇‘(𝐴‘𝐾))𝑈) = ((𝐴‘𝐾) splice ⟨𝑃, 𝑃, ⟨“𝑈(𝑀‘𝑈)”⟩⟩))
88	8	efgmf 18566	. . . . . . . . . . . . . . . . 17 ⊢ 𝑀:(𝐼 × 2_o)⟶(𝐼 × 2_o)
89	88	ffvelrni 6718	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 ∈ (𝐼 × 2_o) → (𝑀‘𝑈) ∈ (𝐼 × 2_o))
90	85, 89	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀‘𝑈) ∈ (𝐼 × 2_o))
91	85, 90	s2cld 14069	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_o))
92		splval 13949	. . . . . . . . . . . . . 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) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)))
93	36, 38, 38, 91, 92	syl13anc 1365	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴‘𝐾) splice ⟨𝑃, 𝑃, ⟨“𝑈(𝑀‘𝑈)”⟩⟩) = ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)))
94	84, 87, 93	3eqtrd 2834	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐴) = ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)))
95		efgredlemb.7	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
96		efgredlemb.v	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_o))
97	3, 7, 8, 9	efgtval 18576	. . . . . . . . . . . . . 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)”⟩⟩))
98	69, 71, 96, 97	syl3anc 1364	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄(𝑇‘(𝐵‘𝐿))𝑉) = ((𝐵‘𝐿) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩))
99	88	ffvelrni 6718	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ (𝐼 × 2_o) → (𝑀‘𝑉) ∈ (𝐼 × 2_o))
100	96, 99	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀‘𝑉) ∈ (𝐼 × 2_o))
101	96, 100	s2cld 14069	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o))
102		splval 13949	. . . . . . . . . . . . . 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) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
103	69, 71, 71, 101, 102	syl13anc 1365	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵‘𝐿) splice ⟨𝑄, 𝑄, ⟨“𝑉(𝑀‘𝑉)”⟩⟩) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
104	95, 98, 103	3eqtrd 2834	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘𝐵) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
105	22, 94, 104	3eqtr3d 2838	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
106	105	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
107		pfxcl 13875	. . . . . . . . . . . . . 14 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) → ((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o))
108	37, 107	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o))
109	108	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o))
110	91	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_o))
111		ccatcl 13772	. . . . . . . . . . . 12 ⊢ ((((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_o)) → (((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ∈ Word (𝐼 × 2_o))
112	109, 110, 111	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ∈ Word (𝐼 × 2_o))
113		swrdcl 13843	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) → ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o))
114	37, 113	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o))
115	114	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o))
116		pfxcl 13875	. . . . . . . . . . . . . 14 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_o) → ((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o))
117	70, 116	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o))
118	117	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o))
119	101	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o))
120		ccatcl 13772	. . . . . . . . . . . 12 ⊢ ((((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o)) → (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_o))
121	118, 119, 120	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_o))
122		swrdcl 13843	. . . . . . . . . . . . 13 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2_o) → ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_o))
123	70, 122	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_o))
124	123	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_o))
125		pfxlen 13881	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) ∧ 𝑃 ∈ (0...(♯‘(𝐴‘𝐾)))) → (♯‘((𝐴‘𝐾) prefix 𝑃)) = 𝑃)
126	37, 38, 125	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘((𝐴‘𝐾) prefix 𝑃)) = 𝑃)
127		pfxlen 13881	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵‘𝐿) ∈ Word (𝐼 × 2_o) ∧ 𝑄 ∈ (0...(♯‘(𝐵‘𝐿)))) → (♯‘((𝐵‘𝐿) prefix 𝑄)) = 𝑄)
128	70, 71, 127	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘((𝐵‘𝐿) prefix 𝑄)) = 𝑄)
129	126, 128	eqeq12d 2809	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘((𝐴‘𝐾) prefix 𝑃)) = (♯‘((𝐵‘𝐿) prefix 𝑄)) ↔ 𝑃 = 𝑄))
130	129	biimpar 478	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (♯‘((𝐴‘𝐾) prefix 𝑃)) = (♯‘((𝐵‘𝐿) prefix 𝑄)))
131		s2len 14087	. . . . . . . . . . . . . . 15 ⊢ (♯‘⟨“𝑈(𝑀‘𝑈)”⟩) = 2
132		s2len 14087	. . . . . . . . . . . . . . 15 ⊢ (♯‘⟨“𝑉(𝑀‘𝑉)”⟩) = 2
133	131, 132	eqtr4i 2821	. . . . . . . . . . . . . 14 ⊢ (♯‘⟨“𝑈(𝑀‘𝑈)”⟩) = (♯‘⟨“𝑉(𝑀‘𝑉)”⟩)
134	133	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (♯‘⟨“𝑈(𝑀‘𝑈)”⟩) = (♯‘⟨“𝑉(𝑀‘𝑉)”⟩))
135	130, 134	oveq12d 7037	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((♯‘((𝐴‘𝐾) prefix 𝑃)) + (♯‘⟨“𝑈(𝑀‘𝑈)”⟩)) = ((♯‘((𝐵‘𝐿) prefix 𝑄)) + (♯‘⟨“𝑉(𝑀‘𝑉)”⟩)))
136		ccatlen 13773	. . . . . . . . . . . . 13 ⊢ ((((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_o)) → (♯‘(((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩)) = ((♯‘((𝐴‘𝐾) prefix 𝑃)) + (♯‘⟨“𝑈(𝑀‘𝑈)”⟩)))
137	109, 110, 136	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (♯‘(((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩)) = ((♯‘((𝐴‘𝐾) prefix 𝑃)) + (♯‘⟨“𝑈(𝑀‘𝑈)”⟩)))
138		ccatlen 13773	. . . . . . . . . . . . 13 ⊢ ((((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o)) → (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩)) = ((♯‘((𝐵‘𝐿) prefix 𝑄)) + (♯‘⟨“𝑉(𝑀‘𝑉)”⟩)))
139	118, 119, 138	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩)) = ((♯‘((𝐵‘𝐿) prefix 𝑄)) + (♯‘⟨“𝑉(𝑀‘𝑉)”⟩)))
140	135, 137, 139	3eqtr4d 2840	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (♯‘(((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩)) = (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩)))
141		ccatopth 13914	. . . . . . . . . . 11 ⊢ ((((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ∈ Word (𝐼 × 2_o) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o)) ∧ ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_o) ∧ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_o)) ∧ (♯‘(((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩)) = (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩))) → (((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)) ↔ ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩))))
142	112, 115, 121, 124, 140, 141	syl221anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)) ↔ ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩))))
143	106, 142	mpbid 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
144	143	simpld 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩))
145		ccatopth 13914	. . . . . . . . 9 ⊢ (((((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_o)) ∧ (((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑉(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o)) ∧ (♯‘((𝐴‘𝐾) prefix 𝑃)) = (♯‘((𝐵‘𝐿) prefix 𝑄))) → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ↔ (((𝐴‘𝐾) prefix 𝑃) = ((𝐵‘𝐿) prefix 𝑄) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ = ⟨“𝑉(𝑀‘𝑉)”⟩)))
146	109, 110, 118, 119, 130, 145	syl221anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ↔ (((𝐴‘𝐾) prefix 𝑃) = ((𝐵‘𝐿) prefix 𝑄) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ = ⟨“𝑉(𝑀‘𝑉)”⟩)))
147	144, 146	mpbid 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐴‘𝐾) prefix 𝑃) = ((𝐵‘𝐿) prefix 𝑄) ∧ ⟨“𝑈(𝑀‘𝑈)”⟩ = ⟨“𝑉(𝑀‘𝑉)”⟩))
148	147	simpld 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐴‘𝐾) prefix 𝑃) = ((𝐵‘𝐿) prefix 𝑄))
149	143	simprd 496	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) = ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩))
150	148, 149	oveq12d 7037	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → (((𝐴‘𝐾) prefix 𝑃) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
151	83, 150	mtand 812	. . . 4 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
152	151	pm2.21d 121	. . 3 ⊢ (𝜑 → (𝑃 = 𝑄 → (𝐴‘0) = (𝐵‘0)))
153		uzp1 12128	. . . 4 ⊢ (𝑃 ∈ (ℤ_≥‘(𝑄 + 1)) → (𝑃 = (𝑄 + 1) ∨ 𝑃 ∈ (ℤ_≥‘((𝑄 + 1) + 1))))
154	85	s1cld 13801	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_o))
155		ccatcl 13772	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_o)) → (((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_o))
156	108, 154, 155	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_o))
157	90	s1cld 13801	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ⟨“(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_o))
158		ccatass 13786	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_o) ∧ ⟨“(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_o) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o)) → (((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩))))
159	156, 157, 114, 158	syl3anc 1364	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩))))
160		ccatass 13786	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_o) ∧ ⟨“(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_o)) → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) = (((𝐴‘𝐾) prefix 𝑃) ++ (⟨“𝑈”⟩ ++ ⟨“(𝑀‘𝑈)”⟩)))
161	108, 154, 157, 160	syl3anc 1364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) = (((𝐴‘𝐾) prefix 𝑃) ++ (⟨“𝑈”⟩ ++ ⟨“(𝑀‘𝑈)”⟩)))
162		df-s2 14046	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨“𝑈(𝑀‘𝑈)”⟩ = (⟨“𝑈”⟩ ++ ⟨“(𝑀‘𝑈)”⟩)
163	162	oveq2i 7030	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) = (((𝐴‘𝐾) prefix 𝑃) ++ (⟨“𝑈”⟩ ++ ⟨“(𝑀‘𝑈)”⟩))
164	161, 163	syl6eqr 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) = (((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩))
165	164	oveq1d 7034	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)))
166	96	s1cld 13801	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_o))
167	100	s1cld 13801	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o))
168		ccatass 13786	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_o) ∧ ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o)) → ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ (⟨“𝑉”⟩ ++ ⟨“(𝑀‘𝑉)”⟩)))
169	117, 166, 167, 168	syl3anc 1364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ (⟨“𝑉”⟩ ++ ⟨“(𝑀‘𝑉)”⟩)))
170		df-s2 14046	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨“𝑉(𝑀‘𝑉)”⟩ = (⟨“𝑉”⟩ ++ ⟨“(𝑀‘𝑉)”⟩)
171	170	oveq2i 7030	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ (⟨“𝑉”⟩ ++ ⟨“(𝑀‘𝑉)”⟩))
172	169, 171	syl6eqr 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩))
173	172	oveq1d 7034	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
174	105, 165, 173	3eqtr4d 2840	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ ⟨“(𝑀‘𝑈)”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = (((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
175	159, 174	eqtr3d 2832	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩))) = (((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
176	175	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩))) = (((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
177	156	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_o))
178	157	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“(𝑀‘𝑈)”⟩ ∈ Word (𝐼 × 2_o))
179	114	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o))
180		ccatcl 13772	. . . . . . . . . . . . . . 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))
181	178, 179, 180	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_o))
182		ccatcl 13772	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_o)) → (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_o))
183	117, 166, 182	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_o))
184	183	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_o))
185	167	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o))
186		ccatcl 13772	. . . . . . . . . . . . . . 15 ⊢ (((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_o) ∧ ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o)) → ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_o))
187	184, 185, 186	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_o))
188	123	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_o))
189		ccatlen 13773	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_o)) → (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩)) = ((♯‘((𝐵‘𝐿) prefix 𝑄)) + (♯‘⟨“𝑉”⟩)))
190	117, 166, 189	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩)) = ((♯‘((𝐵‘𝐿) prefix 𝑄)) + (♯‘⟨“𝑉”⟩)))
191		s1len 13804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (♯‘⟨“𝑉”⟩) = 1
192	191	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘⟨“𝑉”⟩) = 1)
193	128, 192	oveq12d 7037	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((♯‘((𝐵‘𝐿) prefix 𝑄)) + (♯‘⟨“𝑉”⟩)) = (𝑄 + 1))
194	190, 193	eqtrd 2830	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩)) = (𝑄 + 1))
195	126, 194	eqeq12d 2809	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((♯‘((𝐴‘𝐾) prefix 𝑃)) = (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩)) ↔ 𝑃 = (𝑄 + 1)))
196	195	biimpar 478	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (♯‘((𝐴‘𝐾) prefix 𝑃)) = (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩)))
197		s1len 13804	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘⟨“𝑈”⟩) = 1
198		s1len 13804	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘⟨“(𝑀‘𝑉)”⟩) = 1
199	197, 198	eqtr4i 2821	. . . . . . . . . . . . . . . . 17 ⊢ (♯‘⟨“𝑈”⟩) = (♯‘⟨“(𝑀‘𝑉)”⟩)
200	199	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (♯‘⟨“𝑈”⟩) = (♯‘⟨“(𝑀‘𝑉)”⟩))
201	196, 200	oveq12d 7037	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((♯‘((𝐴‘𝐾) prefix 𝑃)) + (♯‘⟨“𝑈”⟩)) = ((♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩)) + (♯‘⟨“(𝑀‘𝑉)”⟩)))
202	108	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o))
203	154	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_o))
204		ccatlen 13773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_o)) → (♯‘(((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩)) = ((♯‘((𝐴‘𝐾) prefix 𝑃)) + (♯‘⟨“𝑈”⟩)))
205	202, 203, 204	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (♯‘(((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩)) = ((♯‘((𝐴‘𝐾) prefix 𝑃)) + (♯‘⟨“𝑈”⟩)))
206		ccatlen 13773	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_o) ∧ ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o)) → (♯‘((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩)) = ((♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩)) + (♯‘⟨“(𝑀‘𝑉)”⟩)))
207	184, 185, 206	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (♯‘((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩)) = ((♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩)) + (♯‘⟨“(𝑀‘𝑉)”⟩)))
208	201, 205, 207	3eqtr4d 2840	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (♯‘(((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩)) = (♯‘((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩)))
209		ccatopth 13914	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ∈ Word (𝐼 × 2_o) ∧ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_o)) ∧ (((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∈ Word (𝐼 × 2_o) ∧ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩) ∈ Word (𝐼 × 2_o)) ∧ (♯‘(((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩)) = (♯‘((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩))) → (((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩))) = (((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)) ↔ ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∧ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩))))
210	177, 181, 187, 188, 208, 209	syl221anc 1374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) ++ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩))) = (((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)) ↔ ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∧ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩))))
211	176, 210	mpbid 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ∧ (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
212	211	simpld 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩))
213		ccatopth 13914	. . . . . . . . . . . 12 ⊢ (((((𝐴‘𝐾) prefix 𝑃) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑈”⟩ ∈ Word (𝐼 × 2_o)) ∧ ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ∈ Word (𝐼 × 2_o) ∧ ⟨“(𝑀‘𝑉)”⟩ ∈ Word (𝐼 × 2_o)) ∧ (♯‘((𝐴‘𝐾) prefix 𝑃)) = (♯‘(((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩))) → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ↔ (((𝐴‘𝐾) prefix 𝑃) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ∧ ⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩)))
214	202, 203, 184, 185, 196, 213	syl221anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((𝐴‘𝐾) prefix 𝑃) ++ ⟨“𝑈”⟩) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ⟨“(𝑀‘𝑉)”⟩) ↔ (((𝐴‘𝐾) prefix 𝑃) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ∧ ⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩)))
215	212, 214	mpbid 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) prefix 𝑃) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ∧ ⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩))
216	215	simpld 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐴‘𝐾) prefix 𝑃) = (((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩))
217	216	oveq1d 7034	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) prefix 𝑃) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)))
218	117	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o))
219	166	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_o))
220		ccatass 13786	. . . . . . . . 9 ⊢ ((((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o) ∧ ⟨“𝑉”⟩ ∈ Word (𝐼 × 2_o) ∧ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o)) → ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) prefix 𝑄) ++ (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩))))
221	218, 219, 179, 220	syl3anc 1364	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ((((𝐵‘𝐿) prefix 𝑄) ++ ⟨“𝑉”⟩) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) prefix 𝑄) ++ (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩))))
222	215	simprd 496	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩)
223		s111 13813	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ (𝐼 × 2_o) ∧ (𝑀‘𝑉) ∈ (𝐼 × 2_o)) → (⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩ ↔ 𝑈 = (𝑀‘𝑉)))
224	85, 100, 223	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩ ↔ 𝑈 = (𝑀‘𝑉)))
225	224	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“𝑈”⟩ = ⟨“(𝑀‘𝑉)”⟩ ↔ 𝑈 = (𝑀‘𝑉)))
226	222, 225	mpbid 233	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → 𝑈 = (𝑀‘𝑉))
227	226	fveq2d 6545	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (𝑀‘𝑈) = (𝑀‘(𝑀‘𝑉)))
228	8	efgmnvl 18567	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ (𝐼 × 2_o) → (𝑀‘(𝑀‘𝑉)) = 𝑉)
229	96, 228	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀‘(𝑀‘𝑉)) = 𝑉)
230	229	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (𝑀‘(𝑀‘𝑉)) = 𝑉)
231	227, 230	eqtrd 2830	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (𝑀‘𝑈) = 𝑉)
232	231	s1eqd 13799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → ⟨“(𝑀‘𝑈)”⟩ = ⟨“𝑉”⟩)
233	232	oveq1d 7034	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)))
234	211	simprd 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“(𝑀‘𝑈)”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩))
235	233, 234	eqtr3d 2832	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩))
236	235	oveq2d 7035	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐵‘𝐿) prefix 𝑄) ++ (⟨“𝑉”⟩ ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩))) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
237	217, 221, 236	3eqtrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 = (𝑄 + 1)) → (((𝐴‘𝐾) prefix 𝑃) ++ ((𝐴‘𝐾) substr ⟨𝑃, (♯‘(𝐴‘𝐾))⟩)) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐵‘𝐿) substr ⟨𝑄, (♯‘(𝐵‘𝐿))⟩)))
238	83, 237	mtand 812	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 = (𝑄 + 1))
239	238	pm2.21d 121	. . . . 5 ⊢ (𝜑 → (𝑃 = (𝑄 + 1) → (𝐴‘0) = (𝐵‘0)))
240		elfzelz 12758	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ (0...(♯‘(𝐵‘𝐿))) → 𝑄 ∈ ℤ)
241	71, 240	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℤ)
242	241	zcnd 11938	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℂ)
243		1cnd 10485	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
244	242, 243, 243	addassd 10512	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + 1) = (𝑄 + (1 + 1)))
245		df-2 11550	. . . . . . . . . 10 ⊢ 2 = (1 + 1)
246	245	oveq2i 7030	. . . . . . . . 9 ⊢ (𝑄 + 2) = (𝑄 + (1 + 1))
247	244, 246	syl6eqr 2848	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 1) + 1) = (𝑄 + 2))
248	247	fveq2d 6545	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘((𝑄 + 1) + 1)) = (ℤ_≥‘(𝑄 + 2)))
249	248	eleq2d 2867	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘((𝑄 + 1) + 1)) ↔ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))))
250	3, 7, 8, 9, 10, 11	efgsfo 18592	. . . . . . . . . 10 ⊢ 𝑆:dom 𝑆–onto→𝑊
251		swrdcl 13843	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2_o) → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o))
252	37, 251	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o))
253		ccatcl 13772	. . . . . . . . . . . 12 ⊢ ((((𝐵‘𝐿) prefix 𝑄) ∈ Word (𝐼 × 2_o) ∧ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩) ∈ Word (𝐼 × 2_o)) → (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_o))
254	117, 252, 253	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) ∈ Word (𝐼 × 2_o))
255	3	efgrcl 18568	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝐾) ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_o)))
256	36, 255	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_o)))
257	256	simprd 496	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2_o))
258	254, 257	eleqtrrd 2885	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) ∈ 𝑊)
259		foelrn 6738	. . . . . . . . . 10 ⊢ ((𝑆:dom 𝑆–onto→𝑊 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) ∈ 𝑊) → ∃𝑐 ∈ dom 𝑆(((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))
260	250, 258, 259	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑐 ∈ dom 𝑆(((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))
261	260	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) → ∃𝑐 ∈ dom 𝑆(((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))
262	20	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 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))))
263	6	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝐴 ∈ dom 𝑆)
264	21	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝐵 ∈ dom 𝑆)
265	22	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝑆‘𝐴) = (𝑆‘𝐵))
266	23	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → ¬ (𝐴‘0) = (𝐵‘0))
267	38	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))
268	71	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))
269	85	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑈 ∈ (𝐼 × 2_o))
270	96	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑉 ∈ (𝐼 × 2_o))
271	84	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))
272	95	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))
273	2	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))
274		simplr 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑃 ∈ (ℤ_≥‘(𝑄 + 2)))
275		simprl 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → 𝑐 ∈ dom 𝑆)
276		simprr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))
277	276	eqcomd 2800	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝑆‘𝑐) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)))
278	3, 7, 8, 9, 10, 11, 262, 263, 264, 265, 266, 19, 57, 267, 268, 269, 270, 271, 272, 273, 274, 275, 277	efgredlemd 18597	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) ∧ (𝑐 ∈ dom 𝑆 ∧ (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)) = (𝑆‘𝑐))) → (𝐴‘0) = (𝐵‘0))
279	261, 278	rexlimddv 3253	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘(𝑄 + 2))) → (𝐴‘0) = (𝐵‘0))
280	279	ex 413	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘(𝑄 + 2)) → (𝐴‘0) = (𝐵‘0)))
281	249, 280	sylbid 241	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘((𝑄 + 1) + 1)) → (𝐴‘0) = (𝐵‘0)))
282	239, 281	jaod 854	. . . 4 ⊢ (𝜑 → ((𝑃 = (𝑄 + 1) ∨ 𝑃 ∈ (ℤ_≥‘((𝑄 + 1) + 1))) → (𝐴‘0) = (𝐵‘0)))
283	153, 282	syl5 34	. . 3 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘(𝑄 + 1)) → (𝐴‘0) = (𝐵‘0)))
284	152, 283	jaod 854	. 2 ⊢ (𝜑 → ((𝑃 = 𝑄 ∨ 𝑃 ∈ (ℤ_≥‘(𝑄 + 1))) → (𝐴‘0) = (𝐵‘0)))
285	1, 284	syl5 34	1 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘𝑄) → (𝐴‘0) = (𝐵‘0)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 842 = wceq 1522 ∈ wcel 2080 ∀wral 3104 ∃wrex 3105 {crab 3108 Vcvv 3436 ∖ cdif 3858 ∅c0 4213 {csn 4474 ⟨cop 4480 ⟨cotp 4482 ∪ ciun 4827 class class class wbr 4964 ↦ cmpt 5043 I cid 5350 × cxp 5444 dom cdm 5446 ran crn 5447 ⟶wf 6224 –onto→wfo 6226 ‘cfv 6228 (class class class)co 7019 ∈ cmpo 7021 1_oc1o 7949 2_oc2o 7950 0cc0 10386 1c1 10387 + caddc 10389 < clt 10524 − cmin 10719 ℕcn 11488 2c2 11542 ℕ₀cn0 11747 ℤcz 11831 ℤ_≥cuz 12093 ...cfz 12742 ..^cfzo 12883 ♯chash 13540 Word cword 13707 ++ cconcat 13768 ⟨“cs1 13793 substr csubstr 13838 prefix cpfx 13868 splice csplice 13947 ⟨“cs2 14039 ~_FG cefg 18559

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-ot 4483 df-uni 4748 df-int 4785 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-2o 7957 df-oadd 7960 df-er 8142 df-map 8261 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-card 9217 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-nn 11489 df-2 11550 df-n0 11748 df-z 11832 df-uz 12094 df-rp 12240 df-fz 12743 df-fzo 12884 df-hash 13541 df-word 13708 df-concat 13769 df-s1 13794 df-substr 13839 df-pfx 13869 df-splice 13948 df-s2 14046

This theorem is referenced by: efgredlemb 18599

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