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Theorem wrd2ind 11351

Description: Perform induction over the structure of two words of the same length. (Contributed by AV, 23-Jan-2019.) (Proof shortened by AV, 12-Oct-2022.)

**Hypotheses**
Ref	Expression
wrd2ind.1	⊢ ((𝑥 = ∅ ∧ 𝑤 = ∅) → (𝜑 ↔ 𝜓))
wrd2ind.2	⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → (𝜑 ↔ 𝜒))
wrd2ind.3	⊢ ((𝑥 = (𝑦 ++ ⟨“𝑧”⟩) ∧ 𝑤 = (𝑢 ++ ⟨“𝑠”⟩)) → (𝜑 ↔ 𝜃))
wrd2ind.4	⊢ (𝑥 = 𝐴 → (𝜌 ↔ 𝜏))
wrd2ind.5	⊢ (𝑤 = 𝐵 → (𝜑 ↔ 𝜌))
wrd2ind.6	⊢ 𝜓
wrd2ind.7	⊢ (((𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (♯‘𝑦) = (♯‘𝑢)) → (𝜒 → 𝜃))

**Assertion**
Ref	Expression
wrd2ind	⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝜏)

Distinct variable groups: 𝑥,𝑤,𝐴 𝑦,𝑤,𝑧,𝐵,𝑥 𝑢,𝑠,𝑤,𝑥,𝑦,𝑧,𝑋 𝑌,𝑠,𝑢,𝑤,𝑥,𝑦,𝑧 𝜒,𝑤,𝑥 𝜑,𝑠,𝑢,𝑦,𝑧 𝜏,𝑥 𝜃,𝑤,𝑥 𝜌,𝑤 Allowed substitution hints: 𝜑(𝑥,𝑤) 𝜓(𝑥,𝑦,𝑧,𝑤,𝑢,𝑠) 𝜒(𝑦,𝑧,𝑢,𝑠) 𝜃(𝑦,𝑧,𝑢,𝑠) 𝜏(𝑦,𝑧,𝑤,𝑢,𝑠) 𝜌(𝑥,𝑦,𝑧,𝑢,𝑠) 𝐴(𝑦,𝑧,𝑢,𝑠) 𝐵(𝑢,𝑠)

Proof of Theorem wrd2ind Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lencl 11164	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ ℕ₀)
2		eqeq2 2241	. . . . . . . . 9 ⊢ (𝑛 = 0 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 0))
3	2	anbi2d 464	. . . . . . . 8 ⊢ (𝑛 = 0 → (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) ↔ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 0)))
4	3	imbi1d 231	. . . . . . 7 ⊢ (𝑛 = 0 → ((((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) → 𝜑) ↔ (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 0) → 𝜑)))
5	4	2ralbidv 2557	. . . . . 6 ⊢ (𝑛 = 0 → (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) → 𝜑) ↔ ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 0) → 𝜑)))
6		eqeq2 2241	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 𝑚))
7	6	anbi2d 464	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) ↔ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑚)))
8	7	imbi1d 231	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) → 𝜑) ↔ (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑚) → 𝜑)))
9	8	2ralbidv 2557	. . . . . 6 ⊢ (𝑛 = 𝑚 → (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) → 𝜑) ↔ ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑚) → 𝜑)))
10		eqeq2 2241	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 + 1) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (𝑚 + 1)))
11	10	anbi2d 464	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) ↔ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1))))
12	11	imbi1d 231	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ((((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) → 𝜑) ↔ (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)) → 𝜑)))
13	12	2ralbidv 2557	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) → 𝜑) ↔ ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)) → 𝜑)))
14		eqeq2 2241	. . . . . . . . 9 ⊢ (𝑛 = (♯‘𝐴) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (♯‘𝐴)))
15	14	anbi2d 464	. . . . . . . 8 ⊢ (𝑛 = (♯‘𝐴) → (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) ↔ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴))))
16	15	imbi1d 231	. . . . . . 7 ⊢ (𝑛 = (♯‘𝐴) → ((((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) → 𝜑) ↔ (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜑)))
17	16	2ralbidv 2557	. . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑛) → 𝜑) ↔ ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜑)))
18		eqeq1 2238	. . . . . . . . . . 11 ⊢ ((♯‘𝑥) = 0 → ((♯‘𝑥) = (♯‘𝑤) ↔ 0 = (♯‘𝑤)))
19	18	adantl 277	. . . . . . . . . 10 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ (♯‘𝑥) = 0) → ((♯‘𝑥) = (♯‘𝑤) ↔ 0 = (♯‘𝑤)))
20		eqcom 2233	. . . . . . . . . . . . . 14 ⊢ (0 = (♯‘𝑤) ↔ (♯‘𝑤) = 0)
21		wrdfin 11179	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Word 𝑌 → 𝑤 ∈ Fin)
22		fihasheq0 11099	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Fin → ((♯‘𝑤) = 0 ↔ 𝑤 = ∅))
23	21, 22	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Word 𝑌 → ((♯‘𝑤) = 0 ↔ 𝑤 = ∅))
24	20, 23	bitrid 192	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Word 𝑌 → (0 = (♯‘𝑤) ↔ 𝑤 = ∅))
25		wrdfin 11179	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝑋 → 𝑥 ∈ Fin)
26		fihasheq0 11099	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Fin → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
27	25, 26	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Word 𝑋 → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
28		wrd2ind.6	. . . . . . . . . . . . . . . . 17 ⊢ 𝜓
29		wrd2ind.1	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = ∅ ∧ 𝑤 = ∅) → (𝜑 ↔ 𝜓))
30	28, 29	mpbiri 168	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ∅ ∧ 𝑤 = ∅) → 𝜑)
31	30	ex 115	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → (𝑤 = ∅ → 𝜑))
32	27, 31	biimtrdi 163	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝑋 → ((♯‘𝑥) = 0 → (𝑤 = ∅ → 𝜑)))
33	32	com13 80	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → ((♯‘𝑥) = 0 → (𝑥 ∈ Word 𝑋 → 𝜑)))
34	24, 33	biimtrdi 163	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Word 𝑌 → (0 = (♯‘𝑤) → ((♯‘𝑥) = 0 → (𝑥 ∈ Word 𝑋 → 𝜑))))
35	34	com24 87	. . . . . . . . . . 11 ⊢ (𝑤 ∈ Word 𝑌 → (𝑥 ∈ Word 𝑋 → ((♯‘𝑥) = 0 → (0 = (♯‘𝑤) → 𝜑))))
36	35	imp31 256	. . . . . . . . . 10 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ (♯‘𝑥) = 0) → (0 = (♯‘𝑤) → 𝜑))
37	19, 36	sylbid 150	. . . . . . . . 9 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ (♯‘𝑥) = 0) → ((♯‘𝑥) = (♯‘𝑤) → 𝜑))
38	37	ex 115	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → ((♯‘𝑥) = 0 → ((♯‘𝑥) = (♯‘𝑤) → 𝜑)))
39	38	impcomd 255	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 0) → 𝜑))
40	39	rgen2 2619	. . . . . 6 ⊢ ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 0) → 𝜑)
41		fveq2 5648	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
42		fveq2 5648	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → (♯‘𝑤) = (♯‘𝑢))
43	41, 42	eqeqan12d 2247	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → ((♯‘𝑥) = (♯‘𝑤) ↔ (♯‘𝑦) = (♯‘𝑢)))
44		fveqeq2 5657	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = 𝑚 ↔ (♯‘𝑦) = 𝑚))
45	44	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → ((♯‘𝑥) = 𝑚 ↔ (♯‘𝑦) = 𝑚))
46	43, 45	anbi12d 473	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑚) ↔ ((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚)))
47		wrd2ind.2	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → (𝜑 ↔ 𝜒))
48	46, 47	imbi12d 234	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → ((((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑚) → 𝜑) ↔ (((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)))
49	48	ancoms 268	. . . . . . . . 9 ⊢ ((𝑤 = 𝑢 ∧ 𝑥 = 𝑦) → ((((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑚) → 𝜑) ↔ (((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)))
50	49	cbvraldva 2777	. . . . . . . 8 ⊢ (𝑤 = 𝑢 → (∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑚) → 𝜑) ↔ ∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)))
51	50	cbvralvw 2772	. . . . . . 7 ⊢ (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑚) → 𝜑) ↔ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒))
52		lencl 11164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ Word 𝑌 → (♯‘𝑤) ∈ ℕ₀)
53	52	nn0zd 9643	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ Word 𝑌 → (♯‘𝑤) ∈ ℤ)
54		1zzd 9549	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ Word 𝑌 → 1 ∈ ℤ)
55	53, 54	zsubcld 9650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ Word 𝑌 → ((♯‘𝑤) − 1) ∈ ℤ)
56		pfxclz 11307	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Word 𝑌 ∧ ((♯‘𝑤) − 1) ∈ ℤ) → (𝑤 prefix ((♯‘𝑤) − 1)) ∈ Word 𝑌)
57	55, 56	mpdan 421	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ Word 𝑌 → (𝑤 prefix ((♯‘𝑤) − 1)) ∈ Word 𝑌)
58	57	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (𝑤 prefix ((♯‘𝑤) − 1)) ∈ Word 𝑌)
59	58	ad2antrl 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (𝑤 prefix ((♯‘𝑤) − 1)) ∈ Word 𝑌)
60		simprll 539	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑤 ∈ Word 𝑌)
61		eqeq1 2238	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑥) = (𝑚 + 1) → ((♯‘𝑥) = (♯‘𝑤) ↔ (𝑚 + 1) = (♯‘𝑤)))
62		nn0p1nn 9484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ)
63		eleq1 2294	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑤) = (𝑚 + 1) → ((♯‘𝑤) ∈ ℕ ↔ (𝑚 + 1) ∈ ℕ))
64	63	eqcoms 2234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 + 1) = (♯‘𝑤) → ((♯‘𝑤) ∈ ℕ ↔ (𝑚 + 1) ∈ ℕ))
65	62, 64	imbitrrid 156	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 + 1) = (♯‘𝑤) → (𝑚 ∈ ℕ₀ → (♯‘𝑤) ∈ ℕ))
66	61, 65	biimtrdi 163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑥) = (𝑚 + 1) → ((♯‘𝑥) = (♯‘𝑤) → (𝑚 ∈ ℕ₀ → (♯‘𝑤) ∈ ℕ)))
67	66	impcom 125	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)) → (𝑚 ∈ ℕ₀ → (♯‘𝑤) ∈ ℕ))
68	67	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑚 ∈ ℕ₀ → (♯‘𝑤) ∈ ℕ))
69	68	impcom 125	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘𝑤) ∈ ℕ)
70	69	nnge1d 9229	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 1 ≤ (♯‘𝑤))
71		wrdlenge1n0 11194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ Word 𝑌 → (𝑤 ≠ ∅ ↔ 1 ≤ (♯‘𝑤)))
72	60, 71	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (𝑤 ≠ ∅ ↔ 1 ≤ (♯‘𝑤)))
73	70, 72	mpbird 167	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑤 ≠ ∅)
74		lswcl 11211	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑤 ≠ ∅) → (lastS‘𝑤) ∈ 𝑌)
75	60, 73, 74	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (lastS‘𝑤) ∈ 𝑌)
76	59, 75	jca 306	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((𝑤 prefix ((♯‘𝑤) − 1)) ∈ Word 𝑌 ∧ (lastS‘𝑤) ∈ 𝑌))
77		lencl 11164	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ Word 𝑋 → (♯‘𝑥) ∈ ℕ₀)
78	77	nn0zd 9643	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Word 𝑋 → (♯‘𝑥) ∈ ℤ)
79		1zzd 9549	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Word 𝑋 → 1 ∈ ℤ)
80	78, 79	zsubcld 9650	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ Word 𝑋 → ((♯‘𝑥) − 1) ∈ ℤ)
81		pfxclz 11307	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word 𝑋 ∧ ((♯‘𝑥) − 1) ∈ ℤ) → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝑋)
82	80, 81	mpdan 421	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝑋 → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝑋)
83	82	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝑋)
84	83	ad2antrl 490	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝑋)
85		simprlr 540	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑥 ∈ Word 𝑋)
86		eleq1 2294	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = (𝑚 + 1) → ((♯‘𝑥) ∈ ℕ ↔ (𝑚 + 1) ∈ ℕ))
87	62, 86	imbitrrid 156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑥) = (𝑚 + 1) → (𝑚 ∈ ℕ₀ → (♯‘𝑥) ∈ ℕ))
88	87	ad2antll 491	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑚 ∈ ℕ₀ → (♯‘𝑥) ∈ ℕ))
89	88	impcom 125	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘𝑥) ∈ ℕ)
90	89	nnge1d 9229	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 1 ≤ (♯‘𝑥))
91		wrdlenge1n0 11194	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ Word 𝑋 → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
92	85, 91	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
93	90, 92	mpbird 167	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑥 ≠ ∅)
94		lswcl 11211	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Word 𝑋 ∧ 𝑥 ≠ ∅) → (lastS‘𝑥) ∈ 𝑋)
95	85, 93, 94	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (lastS‘𝑥) ∈ 𝑋)
96	76, 84, 95	jca32 310	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (((𝑤 prefix ((♯‘𝑤) − 1)) ∈ Word 𝑌 ∧ (lastS‘𝑤) ∈ 𝑌) ∧ ((𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝑋 ∧ (lastS‘𝑥) ∈ 𝑋)))
97	96	adantlr 477	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (((𝑤 prefix ((♯‘𝑤) − 1)) ∈ Word 𝑌 ∧ (lastS‘𝑤) ∈ 𝑌) ∧ ((𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝑋 ∧ (lastS‘𝑥) ∈ 𝑋)))
98		simprl 531	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋))
99		simplr 529	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒))
100		simprrl 541	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘𝑥) = (♯‘𝑤))
101	100	oveq1d 6043	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((♯‘𝑥) − 1) = ((♯‘𝑤) − 1))
102		simprlr 540	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑥 ∈ Word 𝑋)
103		fzossfz 10444	. . . . . . . . . . . . . . . . . 18 ⊢ (0..^(♯‘𝑥)) ⊆ (0...(♯‘𝑥))
104		simprrr 542	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘𝑥) = (𝑚 + 1))
105	62	ad2antrr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (𝑚 + 1) ∈ ℕ)
106	104, 105	eqeltrd 2308	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘𝑥) ∈ ℕ)
107		fzo0end 10512	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑥) ∈ ℕ → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
108	106, 107	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
109	103, 108	sselid 3226	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥)))
110		pfxlen 11313	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word 𝑋 ∧ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
111	102, 109, 110	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
112		simprll 539	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑤 ∈ Word 𝑌)
113		oveq1 6035	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑤) = (♯‘𝑥) → ((♯‘𝑤) − 1) = ((♯‘𝑥) − 1))
114		oveq2 6036	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑤) = (♯‘𝑥) → (0...(♯‘𝑤)) = (0...(♯‘𝑥)))
115	113, 114	eleq12d 2302	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑤) = (♯‘𝑥) → (((♯‘𝑤) − 1) ∈ (0...(♯‘𝑤)) ↔ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))))
116	115	eqcoms 2234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = (♯‘𝑤) → (((♯‘𝑤) − 1) ∈ (0...(♯‘𝑤)) ↔ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))))
117	116	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)) → (((♯‘𝑤) − 1) ∈ (0...(♯‘𝑤)) ↔ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))))
118	117	ad2antll 491	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (((♯‘𝑤) − 1) ∈ (0...(♯‘𝑤)) ↔ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))))
119	109, 118	mpbird 167	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((♯‘𝑤) − 1) ∈ (0...(♯‘𝑤)))
120		pfxlen 11313	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word 𝑌 ∧ ((♯‘𝑤) − 1) ∈ (0...(♯‘𝑤))) → (♯‘(𝑤 prefix ((♯‘𝑤) − 1))) = ((♯‘𝑤) − 1))
121	112, 119, 120	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘(𝑤 prefix ((♯‘𝑤) − 1))) = ((♯‘𝑤) − 1))
122	101, 111, 121	3eqtr4d 2274	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))))
123	104	oveq1d 6043	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((♯‘𝑥) − 1) = ((𝑚 + 1) − 1))
124		nn0cn 9455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
125	124	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑚 ∈ ℂ)
126		ax-1cn 8168	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
127		pncan 8428	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
128	125, 126, 127	sylancl 413	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((𝑚 + 1) − 1) = 𝑚)
129	111, 123, 128	3eqtrd 2268	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚)
130	122, 129	jca 306	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))) ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚))
131	83	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝑋)
132		fveq2 5648	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (♯‘𝑦) = (♯‘(𝑥 prefix ((♯‘𝑥) − 1))))
133		fveq2 5648	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → (♯‘𝑢) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))))
134	132, 133	eqeqan12d 2247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) → ((♯‘𝑦) = (♯‘𝑢) ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))))
135	134	expcom 116	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((♯‘𝑦) = (♯‘𝑢) ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))))))
136	135	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) → (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((♯‘𝑦) = (♯‘𝑢) ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))))))
137	136	imp 124	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) ∧ 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1))) → ((♯‘𝑦) = (♯‘𝑢) ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))))
138		fveqeq2 5657	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((♯‘𝑦) = 𝑚 ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚))
139	138	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) ∧ 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1))) → ((♯‘𝑦) = 𝑚 ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚))
140	137, 139	anbi12d 473	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) ∧ 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1))) → (((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) ↔ ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))) ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚)))
141		vex 2806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
142		vex 2806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑢 ∈ V
143	141, 142, 47	sbc2ie 3104	. . . . . . . . . . . . . . . . . . . 20 ⊢ ([𝑦 / 𝑥][𝑢 / 𝑤]𝜑 ↔ 𝜒)
144	143	bicomi 132	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 ↔ [𝑦 / 𝑥][𝑢 / 𝑤]𝜑)
145	144	a1i 9	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) ∧ 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1))) → (𝜒 ↔ [𝑦 / 𝑥][𝑢 / 𝑤]𝜑))
146		simpr 110	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) ∧ 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1))) → 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)))
147	146	sbceq1d 3037	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) ∧ 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1))) → ([𝑦 / 𝑥][𝑢 / 𝑤]𝜑 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][𝑢 / 𝑤]𝜑))
148		dfsbcq 3034	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → ([𝑢 / 𝑤]𝜑 ↔ [(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑))
149	148	sbcbidv 3091	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][𝑢 / 𝑤]𝜑 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑))
150	149	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][𝑢 / 𝑤]𝜑 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑))
151	150	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) ∧ 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1))) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][𝑢 / 𝑤]𝜑 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑))
152	145, 147, 151	3bitrd 214	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) ∧ 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1))) → (𝜒 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑))
153	140, 152	imbi12d 234	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) ∧ 𝑦 = (𝑥 prefix ((♯‘𝑥) − 1))) → ((((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒) ↔ (((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))) ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚) → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑)))
154	131, 153	rspcdv 2914	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 prefix ((♯‘𝑤) − 1))) → (∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒) → (((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))) ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚) → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑)))
155	58, 154	rspcimdv 2912	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒) → (((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))) ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚) → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑)))
156	98, 99, 130, 155	syl3c 63	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑)
157	156, 122	jca 306	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))))
158		dfsbcq 3034	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤][𝑦 / 𝑥]𝜑))
159		sbccom 3108	. . . . . . . . . . . . . . . 16 ⊢ ([(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑)
160	158, 159	bitrdi 196	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑))
161	133	eqeq2d 2243	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → ((♯‘𝑦) = (♯‘𝑢) ↔ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))))
162	160, 161	anbi12d 473	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → (([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ∧ (♯‘𝑦) = (♯‘𝑢)) ↔ ([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))))))
163		oveq1 6035	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → (𝑢 ++ ⟨“𝑠”⟩) = ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“𝑠”⟩))
164	163	sbceq1d 3037	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → ([(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
165	162, 164	imbi12d 234	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑤 prefix ((♯‘𝑤) − 1)) → ((([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ∧ (♯‘𝑦) = (♯‘𝑢)) → [(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ (([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
166		s1eq 11243	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (lastS‘𝑤) → ⟨“𝑠”⟩ = ⟨“(lastS‘𝑤)”⟩)
167	166	oveq2d 6044	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (lastS‘𝑤) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“𝑠”⟩) = ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩))
168	167	sbceq1d 3037	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (lastS‘𝑤) → ([((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
169	168	imbi2d 230	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (lastS‘𝑤) → ((([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ (([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
170		sbccom 3108	. . . . . . . . . . . . . . . 16 ⊢ ([((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑)
171	170	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (lastS‘𝑤) → ([((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑))
172	171	imbi2d 230	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (lastS‘𝑤) → ((([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ (([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑)))
173	169, 172	bitrd 188	. . . . . . . . . . . . 13 ⊢ (𝑠 = (lastS‘𝑤) → ((([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ (([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑)))
174		dfsbcq 3034	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑))
175		fveqeq2 5657	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))) ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))))
176	174, 175	anbi12d 473	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) ↔ ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1))))))
177		oveq1 6035	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩))
178	177	sbceq1d 3037	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑))
179	176, 178	imbi12d 234	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((([𝑦 / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘𝑦) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑) ↔ (([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑)))
180		s1eq 11243	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (lastS‘𝑥) → ⟨“𝑧”⟩ = ⟨“(lastS‘𝑥)”⟩)
181	180	oveq2d 6044	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (lastS‘𝑥) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
182	181	sbceq1d 3037	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (lastS‘𝑥) → ([((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑))
183	182	imbi2d 230	. . . . . . . . . . . . 13 ⊢ (𝑧 = (lastS‘𝑥) → ((([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑) ↔ (([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑)))
184		simplr 529	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (♯‘𝑦) = (♯‘𝑢)) → (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋))
185		simpll 527	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (♯‘𝑦) = (♯‘𝑢)) → (𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌))
186		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (♯‘𝑦) = (♯‘𝑢)) → (♯‘𝑦) = (♯‘𝑢))
187		wrd2ind.7	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (♯‘𝑦) = (♯‘𝑢)) → (𝜒 → 𝜃))
188	184, 185, 186, 187	syl3anc 1274	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (♯‘𝑦) = (♯‘𝑢)) → (𝜒 → 𝜃))
189	47	ancoms 268	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 𝑢 ∧ 𝑥 = 𝑦) → (𝜑 ↔ 𝜒))
190	142, 141, 189	sbc2ie 3104	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ 𝜒)
191	190	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (♯‘𝑦) = (♯‘𝑢)) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ 𝜒))
192		ccatws1cl 11256	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) → (𝑢 ++ ⟨“𝑠”⟩) ∈ Word 𝑌)
193	192	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (♯‘𝑦) = (♯‘𝑢)) → (𝑢 ++ ⟨“𝑠”⟩) ∈ Word 𝑌)
194		ccatws1cl 11256	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦 ++ ⟨“𝑧”⟩) ∈ Word 𝑋)
195	184, 194	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (♯‘𝑦) = (♯‘𝑢)) → (𝑦 ++ ⟨“𝑧”⟩) ∈ Word 𝑋)
196		nfv 1577	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑤𝜃
197		nfv 1577	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝜃
198		nfv 1577	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑤(𝑦 ++ ⟨“𝑧”⟩) ∈ Word 𝑋
199		wrd2ind.3	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = (𝑦 ++ ⟨“𝑧”⟩) ∧ 𝑤 = (𝑢 ++ ⟨“𝑠”⟩)) → (𝜑 ↔ 𝜃))
200	199	ancoms 268	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = (𝑢 ++ ⟨“𝑠”⟩) ∧ 𝑥 = (𝑦 ++ ⟨“𝑧”⟩)) → (𝜑 ↔ 𝜃))
201	196, 197, 198, 200	sbc2iegf 3103	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢 ++ ⟨“𝑠”⟩) ∈ Word 𝑌 ∧ (𝑦 ++ ⟨“𝑧”⟩) ∈ Word 𝑋) → ([(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ 𝜃))
202	193, 195, 201	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (♯‘𝑦) = (♯‘𝑢)) → ([(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ 𝜃))
203	188, 191, 202	3imtr4d 203	. . . . . . . . . . . . . . 15 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (♯‘𝑦) = (♯‘𝑢)) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 → [(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
204	203	ex 115	. . . . . . . . . . . . . 14 ⊢ (((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((♯‘𝑦) = (♯‘𝑢) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 → [(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
205	204	impcomd 255	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) → (([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ∧ (♯‘𝑦) = (♯‘𝑢)) → [(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
206	165, 173, 179, 183, 205	vtocl4ga 2877	. . . . . . . . . . . 12 ⊢ ((((𝑤 prefix ((♯‘𝑤) − 1)) ∈ Word 𝑌 ∧ (lastS‘𝑤) ∈ 𝑌) ∧ ((𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝑋 ∧ (lastS‘𝑥) ∈ 𝑋)) → (([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥][(𝑤 prefix ((♯‘𝑤) − 1)) / 𝑤]𝜑 ∧ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = (♯‘(𝑤 prefix ((♯‘𝑤) − 1)))) → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑))
207	97, 157, 206	sylc 62	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑)
208		eqtr2 2250	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)) → (♯‘𝑤) = (𝑚 + 1))
209	208	ad2antll 491	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘𝑤) = (𝑚 + 1))
210	209, 105	eqeltrd 2308	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (♯‘𝑤) ∈ ℕ)
211	21	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → 𝑤 ∈ Fin)
212	211	ad2antrl 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑤 ∈ Fin)
213		hashnncl 11101	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Fin → ((♯‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
214	212, 213	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((♯‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
215	210, 214	mpbid 147	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑤 ≠ ∅)
216		pfxlswccat 11341	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑤 ≠ ∅) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
217	216	eqcomd 2237	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑤 ≠ ∅) → 𝑤 = ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩))
218	112, 215, 217	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑤 = ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩))
219	25	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → 𝑥 ∈ Fin)
220	219	ad2antrl 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑥 ∈ Fin)
221		hashnncl 11101	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Fin → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
222	220, 221	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
223	106, 222	mpbid 147	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑥 ≠ ∅)
224		pfxlswccat 11341	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word 𝑋 ∧ 𝑥 ≠ ∅) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) = 𝑥)
225	224	eqcomd 2237	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑋 ∧ 𝑥 ≠ ∅) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
226	102, 223, 225	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
227		sbceq1a 3042	. . . . . . . . . . . . 13 ⊢ (𝑤 = ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) → (𝜑 ↔ [((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑))
228		sbceq1a 3042	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) → ([((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑))
229	227, 228	sylan9bb 462	. . . . . . . . . . . 12 ⊢ ((𝑤 = ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) ∧ 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩)) → (𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑))
230	218, 226, 229	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → (𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥][((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) / 𝑤]𝜑))
231	207, 230	mpbird 167	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)))) → 𝜑)
232	231	expr 375	. . . . . . . . 9 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) ∧ (𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋)) → (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)) → 𝜑))
233	232	ralrimivva 2615	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ₀ ∧ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒)) → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)) → 𝜑))
234	233	ex 115	. . . . . . 7 ⊢ (𝑚 ∈ ℕ₀ → (∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((♯‘𝑦) = (♯‘𝑢) ∧ (♯‘𝑦) = 𝑚) → 𝜒) → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)) → 𝜑)))
235	51, 234	biimtrid 152	. . . . . 6 ⊢ (𝑚 ∈ ℕ₀ → (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = 𝑚) → 𝜑) → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (𝑚 + 1)) → 𝜑)))
236	5, 9, 13, 17, 40, 235	nn0ind 9637	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜑))
237	1, 236	syl 14	. . . 4 ⊢ (𝐴 ∈ Word 𝑋 → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜑))
238	237	3ad2ant1 1045	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (♯‘𝐴) = (♯‘𝐵)) → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜑))
239		fveq2 5648	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → (♯‘𝑤) = (♯‘𝐵))
240	239	eqeq2d 2243	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → ((♯‘𝑥) = (♯‘𝑤) ↔ (♯‘𝑥) = (♯‘𝐵)))
241	240	anbi1d 465	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) ↔ ((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴))))
242		wrd2ind.5	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝜑 ↔ 𝜌))
243	241, 242	imbi12d 234	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜑) ↔ (((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌)))
244	243	ralbidv 2533	. . . . 5 ⊢ (𝑤 = 𝐵 → (∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜑) ↔ ∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌)))
245	244	rspcv 2907	. . . 4 ⊢ (𝐵 ∈ Word 𝑌 → (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜑) → ∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌)))
246	245	3ad2ant2 1046	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (♯‘𝐴) = (♯‘𝐵)) → (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝑤) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜑) → ∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌)))
247	238, 246	mpd 13	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (♯‘𝐴) = (♯‘𝐵)) → ∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌))
248		eqidd 2232	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (♯‘𝐴) = (♯‘𝐵)) → (♯‘𝐴) = (♯‘𝐴))
249		fveqeq2 5657	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) = (♯‘𝐵) ↔ (♯‘𝐴) = (♯‘𝐵)))
250		fveqeq2 5657	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) = (♯‘𝐴) ↔ (♯‘𝐴) = (♯‘𝐴)))
251	249, 250	anbi12d 473	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) ↔ ((♯‘𝐴) = (♯‘𝐵) ∧ (♯‘𝐴) = (♯‘𝐴))))
252		wrd2ind.4	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝜌 ↔ 𝜏))
253	251, 252	imbi12d 234	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌) ↔ (((♯‘𝐴) = (♯‘𝐵) ∧ (♯‘𝐴) = (♯‘𝐴)) → 𝜏)))
254	253	rspcv 2907	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑋 → (∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌) → (((♯‘𝐴) = (♯‘𝐵) ∧ (♯‘𝐴) = (♯‘𝐴)) → 𝜏)))
255	254	com23 78	. . . . . 6 ⊢ (𝐴 ∈ Word 𝑋 → (((♯‘𝐴) = (♯‘𝐵) ∧ (♯‘𝐴) = (♯‘𝐴)) → (∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌) → 𝜏)))
256	255	expd 258	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → ((♯‘𝐴) = (♯‘𝐵) → ((♯‘𝐴) = (♯‘𝐴) → (∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌) → 𝜏))))
257	256	com34 83	. . . 4 ⊢ (𝐴 ∈ Word 𝑋 → ((♯‘𝐴) = (♯‘𝐵) → (∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌) → ((♯‘𝐴) = (♯‘𝐴) → 𝜏))))
258	257	imp 124	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ (♯‘𝐴) = (♯‘𝐵)) → (∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌) → ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
259	258	3adant2 1043	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (♯‘𝐴) = (♯‘𝐵)) → (∀𝑥 ∈ Word 𝑋(((♯‘𝑥) = (♯‘𝐵) ∧ (♯‘𝑥) = (♯‘𝐴)) → 𝜌) → ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
260	247, 248, 259	mp2d 47	1 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝜏)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∈ wcel 2202 ≠ wne 2403 ∀wral 2511 [wsbc 3032 ∅c0 3496 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 Fincfn 6952 ℂcc 8073 0cc0 8075 1c1 8076 + caddc 8078 ≤ cle 8258 − cmin 8393 ℕcn 9186 ℕ₀cn0 9445 ℤcz 9522 ...cfz 10286 ..^cfzo 10420 ♯chash 11081 Word cword 11160 lastSclsw 11205 ++ cconcat 11214 ⟨“cs1 11239 prefix cpfx 11300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-1o 6625 df-er 6745 df-en 6953 df-dom 6954 df-fin 6955 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-inn 9187 df-n0 9446 df-z 9523 df-uz 9799 df-fz 10287 df-fzo 10421 df-ihash 11082 df-word 11161 df-lsw 11206 df-concat 11215 df-s1 11240 df-substr 11274 df-pfx 11301

This theorem is referenced by: (None)

W3C validator