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Theorem wrd2ind 13477

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

**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 13324	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → (#‘𝐴) ∈ ℕ₀)
2		eqeq2 2633	. . . . . . . . 9 ⊢ (𝑛 = 0 → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = 0))
3	2	anbi2d 740	. . . . . . . 8 ⊢ (𝑛 = 0 → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) ↔ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 0)))
4	3	imbi1d 331	. . . . . . 7 ⊢ (𝑛 = 0 → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) → 𝜑) ↔ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 0) → 𝜑)))
5	4	2ralbidv 2989	. . . . . 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 2633	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = 𝑚))
7	6	anbi2d 740	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) ↔ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚)))
8	7	imbi1d 331	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) → 𝜑) ↔ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) → 𝜑)))
9	8	2ralbidv 2989	. . . . . 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 2633	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 + 1) → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = (𝑚 + 1)))
11	10	anbi2d 740	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) ↔ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1))))
12	11	imbi1d 331	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) → 𝜑) ↔ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)) → 𝜑)))
13	12	2ralbidv 2989	. . . . . 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 2633	. . . . . . . . 9 ⊢ (𝑛 = (#‘𝐴) → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = (#‘𝐴)))
15	14	anbi2d 740	. . . . . . . 8 ⊢ (𝑛 = (#‘𝐴) → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) ↔ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴))))
16	15	imbi1d 331	. . . . . . 7 ⊢ (𝑛 = (#‘𝐴) → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) → 𝜑) ↔ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑)))
17	16	2ralbidv 2989	. . . . . 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 2626	. . . . . . . . . . . 12 ⊢ ((#‘𝑥) = 0 → ((#‘𝑥) = (#‘𝑤) ↔ 0 = (#‘𝑤)))
19	18	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ (#‘𝑥) = 0) → ((#‘𝑥) = (#‘𝑤) ↔ 0 = (#‘𝑤)))
20		eqcom 2629	. . . . . . . . . . . . . . 15 ⊢ (0 = (#‘𝑤) ↔ (#‘𝑤) = 0)
21		hasheq0 13154	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Word 𝑌 → ((#‘𝑤) = 0 ↔ 𝑤 = ∅))
22	20, 21	syl5bb 272	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Word 𝑌 → (0 = (#‘𝑤) ↔ 𝑤 = ∅))
23		hasheq0 13154	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝑋 → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
24		wrd2ind.6	. . . . . . . . . . . . . . . . . 18 ⊢ 𝜓
25		wrd2ind.1	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ∅ ∧ 𝑤 = ∅) → (𝜑 ↔ 𝜓))
26	24, 25	mpbiri 248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = ∅ ∧ 𝑤 = ∅) → 𝜑)
27	26	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝑤 = ∅ → 𝜑))
28	23, 27	syl6bi 243	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Word 𝑋 → ((#‘𝑥) = 0 → (𝑤 = ∅ → 𝜑)))
29	28	com13 88	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∅ → ((#‘𝑥) = 0 → (𝑥 ∈ Word 𝑋 → 𝜑)))
30	22, 29	syl6bi 243	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Word 𝑌 → (0 = (#‘𝑤) → ((#‘𝑥) = 0 → (𝑥 ∈ Word 𝑋 → 𝜑))))
31	30	com24 95	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Word 𝑌 → (𝑥 ∈ Word 𝑋 → ((#‘𝑥) = 0 → (0 = (#‘𝑤) → 𝜑))))
32	31	imp31 448	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ (#‘𝑥) = 0) → (0 = (#‘𝑤) → 𝜑))
33	19, 32	sylbid 230	. . . . . . . . . 10 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ (#‘𝑥) = 0) → ((#‘𝑥) = (#‘𝑤) → 𝜑))
34	33	ex 450	. . . . . . . . 9 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → ((#‘𝑥) = 0 → ((#‘𝑥) = (#‘𝑤) → 𝜑)))
35	34	com23 86	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → ((#‘𝑥) = (#‘𝑤) → ((#‘𝑥) = 0 → 𝜑)))
36	35	impd 447	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 0) → 𝜑))
37	36	rgen2 2975	. . . . . 6 ⊢ ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 0) → 𝜑)
38		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
39		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → (#‘𝑤) = (#‘𝑢))
40	38, 39	eqeqan12d 2638	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → ((#‘𝑥) = (#‘𝑤) ↔ (#‘𝑦) = (#‘𝑢)))
41	38	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((#‘𝑥) = 𝑚 ↔ (#‘𝑦) = 𝑚))
42	41	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → ((#‘𝑥) = 𝑚 ↔ (#‘𝑦) = 𝑚))
43	40, 42	anbi12d 747	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) ↔ ((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚)))
44		wrd2ind.2	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → (𝜑 ↔ 𝜒))
45	43, 44	imbi12d 334	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) → 𝜑) ↔ (((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒)))
46	45	ancoms 469	. . . . . . . . 9 ⊢ ((𝑤 = 𝑢 ∧ 𝑥 = 𝑦) → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) → 𝜑) ↔ (((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒)))
47	46	cbvraldva 3177	. . . . . . . 8 ⊢ (𝑤 = 𝑢 → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) → 𝜑) ↔ ∀𝑦 ∈ Word 𝑋(((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒)))
48	47	cbvralv 3171	. . . . . . 7 ⊢ (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) → 𝜑) ↔ ∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒))
49		swrdcl 13419	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ Word 𝑌 → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌)
50	49	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌)
51	50	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌)
52		simprll 802	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 𝑤 ∈ Word 𝑌)
53		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑥) = (𝑚 + 1) → ((#‘𝑥) = (#‘𝑤) ↔ (𝑚 + 1) = (#‘𝑤)))
54		nn0p1nn 11332	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ)
55		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑤) = (𝑚 + 1) → ((#‘𝑤) ∈ ℕ ↔ (𝑚 + 1) ∈ ℕ))
56	55	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 + 1) = (#‘𝑤) → ((#‘𝑤) ∈ ℕ ↔ (𝑚 + 1) ∈ ℕ))
57	54, 56	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 + 1) = (#‘𝑤) → (𝑚 ∈ ℕ₀ → (#‘𝑤) ∈ ℕ))
58	53, 57	syl6bi 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑥) = (𝑚 + 1) → ((#‘𝑥) = (#‘𝑤) → (𝑚 ∈ ℕ₀ → (#‘𝑤) ∈ ℕ)))
59	58	impcom 446	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)) → (𝑚 ∈ ℕ₀ → (#‘𝑤) ∈ ℕ))
60	59	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑚 ∈ ℕ₀ → (#‘𝑤) ∈ ℕ))
61	60	impcom 446	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (#‘𝑤) ∈ ℕ)
62	61	nnge1d 11063	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 1 ≤ (#‘𝑤))
63		wrdlenge1n0 13340	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ Word 𝑌 → (𝑤 ≠ ∅ ↔ 1 ≤ (#‘𝑤)))
64	52, 63	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (𝑤 ≠ ∅ ↔ 1 ≤ (#‘𝑤)))
65	62, 64	mpbird 247	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 𝑤 ≠ ∅)
66		lswcl 13355	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑤 ≠ ∅) → ( lastS ‘𝑤) ∈ 𝑌)
67	52, 65, 66	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → ( lastS ‘𝑤) ∈ 𝑌)
68	51, 67	jca 554	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌 ∧ ( lastS ‘𝑤) ∈ 𝑌))
69		swrdcl 13419	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝑋 → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋)
70	69	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋)
71	70	ad2antrl 764	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋)
72		simprlr 803	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 𝑥 ∈ Word 𝑋)
73		eleq1 2689	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑥) = (𝑚 + 1) → ((#‘𝑥) ∈ ℕ ↔ (𝑚 + 1) ∈ ℕ))
74	54, 73	syl5ibr 236	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑥) = (𝑚 + 1) → (𝑚 ∈ ℕ₀ → (#‘𝑥) ∈ ℕ))
75	74	ad2antll 765	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑚 ∈ ℕ₀ → (#‘𝑥) ∈ ℕ))
76	75	impcom 446	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (#‘𝑥) ∈ ℕ)
77	76	nnge1d 11063	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 1 ≤ (#‘𝑥))
78		wrdlenge1n0 13340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ Word 𝑋 → (𝑥 ≠ ∅ ↔ 1 ≤ (#‘𝑥)))
79	72, 78	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (𝑥 ≠ ∅ ↔ 1 ≤ (#‘𝑥)))
80	77, 79	mpbird 247	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 𝑥 ≠ ∅)
81		lswcl 13355	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Word 𝑋 ∧ 𝑥 ≠ ∅) → ( lastS ‘𝑥) ∈ 𝑋)
82	72, 80, 81	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → ( lastS ‘𝑥) ∈ 𝑋)
83	68, 71, 82	jca32 558	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌 ∧ ( lastS ‘𝑤) ∈ 𝑌) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋 ∧ ( lastS ‘𝑥) ∈ 𝑋)))
84	83	adantlr 751	. . . . . . . . . . . 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 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌 ∧ ( lastS ‘𝑤) ∈ 𝑌) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋 ∧ ( lastS ‘𝑥) ∈ 𝑋)))
85		simprl 794	. . . . . . . . . . . . . 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))
86		simplr 792	. . . . . . . . . . . . . 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))
87		simprrl 804	. . . . . . . . . . . . . . . . 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))
88	87	oveq1d 6665	. . . . . . . . . . . . . . . 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))
89		simprlr 803	. . . . . . . . . . . . . . . . 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)
90		fzossfz 12488	. . . . . . . . . . . . . . . . . 18 ⊢ (0..^(#‘𝑥)) ⊆ (0...(#‘𝑥))
91		simprrr 805	. . . . . . . . . . . . . . . . . . . 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))
92	54	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 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)
93	91, 92	eqeltrd 2701	. . . . . . . . . . . . . . . . . . 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)
94		fzo0end 12560	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑥) ∈ ℕ → ((#‘𝑥) − 1) ∈ (0..^(#‘𝑥)))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . 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)))
96	90, 95	sseldi 3601	. . . . . . . . . . . . . . . . 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)))
97		swrd0len 13422	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word 𝑋 ∧ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))) → (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = ((#‘𝑥) − 1))
98	89, 96, 97	syl2anc 693	. . . . . . . . . . . . . . . 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 substr ⟨0, ((#‘𝑥) − 1)⟩)) = ((#‘𝑥) − 1))
99		simprll 802	. . . . . . . . . . . . . . . . 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)
100		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑤) = (#‘𝑥) → ((#‘𝑤) − 1) = ((#‘𝑥) − 1))
101		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑤) = (#‘𝑥) → (0...(#‘𝑤)) = (0...(#‘𝑥)))
102	100, 101	eleq12d 2695	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑤) = (#‘𝑥) → (((#‘𝑤) − 1) ∈ (0...(#‘𝑤)) ↔ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))))
103	102	eqcoms 2630	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑥) = (#‘𝑤) → (((#‘𝑤) − 1) ∈ (0...(#‘𝑤)) ↔ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))))
104	103	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)) → (((#‘𝑤) − 1) ∈ (0...(#‘𝑤)) ↔ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))))
105	104	ad2antll 765	. . . . . . . . . . . . . . . . . 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))))
106	96, 105	mpbird 247	. . . . . . . . . . . . . . . . 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)))
107		swrd0len 13422	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word 𝑌 ∧ ((#‘𝑤) − 1) ∈ (0...(#‘𝑤))) → (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) = ((#‘𝑤) − 1))
108	99, 106, 107	syl2anc 693	. . . . . . . . . . . . . . . 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 substr ⟨0, ((#‘𝑤) − 1)⟩)) = ((#‘𝑤) − 1))
109	88, 98, 108	3eqtr4d 2666	. . . . . . . . . . . . . . 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 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)))
110	91	oveq1d 6665	. . . . . . . . . . . . . . . 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))
111		nn0cn 11302	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
112	111	ad2antrr 762	. . . . . . . . . . . . . . . . 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)
113		ax-1cn 9994	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
114		pncan 10287	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
115	112, 113, 114	sylancl 694	. . . . . . . . . . . . . . . 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)
116	98, 110, 115	3eqtrd 2660	. . . . . . . . . . . . . . 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 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚)
117	109, 116	jca 554	. . . . . . . . . . . . . 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 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚))
118	70	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋)
119		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (#‘𝑦) = (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)))
120		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → (#‘𝑢) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)))
121	119, 120	eqeqan12d 2638	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) → ((#‘𝑦) = (#‘𝑢) ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))))
122	121	expcom 451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ((#‘𝑦) = (#‘𝑢) ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)))))
123	122	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) → (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ((#‘𝑦) = (#‘𝑢) ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)))))
124	123	imp 445	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) → ((#‘𝑦) = (#‘𝑢) ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))))
125	119	eqeq1d 2624	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ((#‘𝑦) = 𝑚 ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚))
126	125	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) → ((#‘𝑦) = 𝑚 ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚))
127	124, 126	anbi12d 747	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) → (((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) ↔ ((#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚)))
128		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
129		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑢 ∈ V
130	128, 129, 44	sbc2ie 3505	. . . . . . . . . . . . . . . . . . . 20 ⊢ ([𝑦 / 𝑥][𝑢 / 𝑤]𝜑 ↔ 𝜒)
131	130	bicomi 214	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜒 ↔ [𝑦 / 𝑥][𝑢 / 𝑤]𝜑)
132	131	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) → (𝜒 ↔ [𝑦 / 𝑥][𝑢 / 𝑤]𝜑))
133		simpr 477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) → 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩))
134	133	sbceq1d 3440	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) → ([𝑦 / 𝑥][𝑢 / 𝑤]𝜑 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][𝑢 / 𝑤]𝜑))
135		dfsbcq 3437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ([𝑢 / 𝑤]𝜑 ↔ [(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑))
136	135	sbcbidv 3490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][𝑢 / 𝑤]𝜑 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑))
137	136	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) → ([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][𝑢 / 𝑤]𝜑 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑))
138	137	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) → ([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][𝑢 / 𝑤]𝜑 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑))
139	132, 134, 138	3bitrd 294	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) → (𝜒 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑))
140	127, 139	imbi12d 334	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ 𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) → ((((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒) ↔ (((#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚) → [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑)))
141	118, 140	rspcdv 3312	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) → (∀𝑦 ∈ Word 𝑋(((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒) → (((#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚) → [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑)))
142	50, 141	rspcimdv 3310	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (∀𝑢 ∈ Word 𝑌∀𝑦 ∈ Word 𝑋(((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒) → (((#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚) → [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑)))
143	85, 86, 117, 142	syl3c 66	. . . . . . . . . . . . 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 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑)
144	143, 109	jca 554	. . . . . . . . . . . 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 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))))
145		dfsbcq 3437	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤][𝑦 / 𝑥]𝜑))
146		sbccom 3509	. . . . . . . . . . . . . . . 16 ⊢ ([(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑)
147	145, 146	syl6bb 276	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑))
148	120	eqeq2d 2632	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ((#‘𝑦) = (#‘𝑢) ↔ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))))
149	147, 148	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → (([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ∧ (#‘𝑦) = (#‘𝑢)) ↔ ([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)))))
150		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → (𝑢 ++ ⟨“𝑠”⟩) = ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“𝑠”⟩))
151	150	sbceq1d 3440	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ([(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
152	149, 151	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ((([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ∧ (#‘𝑦) = (#‘𝑢)) → [(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ (([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
153		s1eq 13380	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ( lastS ‘𝑤) → ⟨“𝑠”⟩ = ⟨“( lastS ‘𝑤)”⟩)
154	153	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = ( lastS ‘𝑤) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“𝑠”⟩) = ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
155	154	sbceq1d 3440	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = ( lastS ‘𝑤) → ([((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
156	155	imbi2d 330	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ( lastS ‘𝑤) → ((([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ (([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
157		sbccom 3509	. . . . . . . . . . . . . . . 16 ⊢ ([((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑)
158	157	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = ( lastS ‘𝑤) → ([((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑))
159	158	imbi2d 330	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ( lastS ‘𝑤) → ((([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ (([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑)))
160	156, 159	bitrd 268	. . . . . . . . . . . . 13 ⊢ (𝑠 = ( lastS ‘𝑤) → ((([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ (([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑)))
161		dfsbcq 3437	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑))
162	119	eqeq1d 2624	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ((#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))))
163	161, 162	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) ↔ ([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)))))
164		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩))
165	164	sbceq1d 3440	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑))
166	163, 165	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ((([𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑) ↔ (([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑)))
167		s1eq 13380	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ( lastS ‘𝑥) → ⟨“𝑧”⟩ = ⟨“( lastS ‘𝑥)”⟩)
168	167	oveq2d 6666	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ( lastS ‘𝑥) → ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
169	168	sbceq1d 3440	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ( lastS ‘𝑥) → ([((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑))
170	169	imbi2d 330	. . . . . . . . . . . . 13 ⊢ (𝑧 = ( lastS ‘𝑥) → ((([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑) ↔ (([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑)))
171		simplr 792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (#‘𝑦) = (#‘𝑢)) → (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋))
172		simpll 790	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (#‘𝑦) = (#‘𝑢)) → (𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌))
173		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (#‘𝑦) = (#‘𝑢)) → (#‘𝑦) = (#‘𝑢))
174		wrd2ind.7	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ (𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (#‘𝑦) = (#‘𝑢)) → (𝜒 → 𝜃))
175	171, 172, 173, 174	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (#‘𝑦) = (#‘𝑢)) → (𝜒 → 𝜃))
176	44	ancoms 469	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 𝑢 ∧ 𝑥 = 𝑦) → (𝜑 ↔ 𝜒))
177	129, 128, 176	sbc2ie 3505	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ 𝜒)
178		ovex 6678	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ++ ⟨“𝑠”⟩) ∈ V
179		ovex 6678	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ++ ⟨“𝑧”⟩) ∈ V
180		wrd2ind.3	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = (𝑦 ++ ⟨“𝑧”⟩) ∧ 𝑤 = (𝑢 ++ ⟨“𝑠”⟩)) → (𝜑 ↔ 𝜃))
181	180	ancoms 469	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = (𝑢 ++ ⟨“𝑠”⟩) ∧ 𝑥 = (𝑦 ++ ⟨“𝑧”⟩)) → (𝜑 ↔ 𝜃))
182	178, 179, 181	sbc2ie 3505	. . . . . . . . . . . . . . . . 17 ⊢ ([(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ 𝜃)
183	175, 177, 182	3imtr4g 285	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (#‘𝑦) = (#‘𝑢)) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 → [(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
184	183	ex 450	. . . . . . . . . . . . . . 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)))
185	184	com23 86	. . . . . . . . . . . . . 14 ⊢ (((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 → ((#‘𝑦) = (#‘𝑢) → [(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
186	185	impd 447	. . . . . . . . . . . . 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))
187	152, 160, 166, 170, 186	vtocl4ga 3278	. . . . . . . . . . . 12 ⊢ ((((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌 ∧ ( lastS ‘𝑤) ∈ 𝑌) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋 ∧ ( lastS ‘𝑥) ∈ 𝑋)) → (([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑 ∧ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))) → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑))
188	84, 144, 187	sylc 65	. . . . . . . . . . 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 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑)
189		eqtr2 2642	. . . . . . . . . . . . . . . 16 ⊢ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)) → (#‘𝑤) = (𝑚 + 1))
190	189	ad2antll 765	. . . . . . . . . . . . . . 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))
191	190, 92	eqeltrd 2701	. . . . . . . . . . . . . 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)
192		wrdfin 13323	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ Word 𝑌 → 𝑤 ∈ Fin)
193	192	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → 𝑤 ∈ Fin)
194	193	ad2antrl 764	. . . . . . . . . . . . . . 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)
195		hashnncl 13157	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Fin → ((#‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
196	194, 195	syl 17	. . . . . . . . . . . . . 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 ≠ ∅))
197	191, 196	mpbid 222	. . . . . . . . . . . . 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 ≠ ∅)
198		swrdccatwrd 13468	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑤 ≠ ∅) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
199	198	eqcomd 2628	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑤 ≠ ∅) → 𝑤 = ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
200	99, 197, 199	syl2anc 693	. . . . . . . . . . . 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 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
201		wrdfin 13323	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ Word 𝑋 → 𝑥 ∈ Fin)
202	201	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → 𝑥 ∈ Fin)
203	202	ad2antrl 764	. . . . . . . . . . . . . . 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)
204		hashnncl 13157	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Fin → ((#‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
205	203, 204	syl 17	. . . . . . . . . . . . . 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 ≠ ∅))
206	93, 205	mpbid 222	. . . . . . . . . . . . 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 ≠ ∅)
207		swrdccatwrd 13468	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word 𝑋 ∧ 𝑥 ≠ ∅) → ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) = 𝑥)
208	207	eqcomd 2628	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑋 ∧ 𝑥 ≠ ∅) → 𝑥 = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
209	89, 206, 208	syl2anc 693	. . . . . . . . . . . 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 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
210		sbceq1a 3446	. . . . . . . . . . . . 13 ⊢ (𝑤 = ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) → (𝜑 ↔ [((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑))
211		sbceq1a 3446	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) → ([((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑))
212	210, 211	sylan9bb 736	. . . . . . . . . . . 12 ⊢ ((𝑤 = ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) ∧ 𝑥 = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩)) → (𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑))
213	200, 209, 212	syl2anc 693	. . . . . . . . . . 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 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥][((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑))
214	188, 213	mpbird 247	. . . . . . . . . 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)
215	214	expr 643	. . . . . . . . 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))
216	215	ralrimivva 2971	. . . . . . . 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))
217	216	ex 450	. . . . . . 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)))
218	48, 217	syl5bi 232	. . . . . 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)))
219	5, 9, 13, 17, 37, 218	nn0ind 11472	. . . . 5 ⊢ ((#‘𝐴) ∈ ℕ₀ → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑))
220	1, 219	syl 17	. . . 4 ⊢ (𝐴 ∈ Word 𝑋 → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑))
221	220	3ad2ant1 1082	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (#‘𝐴) = (#‘𝐵)) → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑))
222		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → (#‘𝑤) = (#‘𝐵))
223	222	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → ((#‘𝑥) = (#‘𝑤) ↔ (#‘𝑥) = (#‘𝐵)))
224	223	anbi1d 741	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) ↔ ((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴))))
225		wrd2ind.5	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝜑 ↔ 𝜌))
226	224, 225	imbi12d 334	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑) ↔ (((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌)))
227	226	ralbidv 2986	. . . . 5 ⊢ (𝑤 = 𝐵 → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑) ↔ ∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌)))
228	227	rspcv 3305	. . . 4 ⊢ (𝐵 ∈ Word 𝑌 → (∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑) → ∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌)))
229	228	3ad2ant2 1083	. . 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)))
230	221, 229	mpd 15	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (#‘𝐴) = (#‘𝐵)) → ∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌))
231		eqidd 2623	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (#‘𝐴) = (#‘𝐵)) → (#‘𝐴) = (#‘𝐴))
232		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (#‘𝑥) = (#‘𝐴))
233	232	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((#‘𝑥) = (#‘𝐵) ↔ (#‘𝐴) = (#‘𝐵)))
234	232	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((#‘𝑥) = (#‘𝐴) ↔ (#‘𝐴) = (#‘𝐴)))
235	233, 234	anbi12d 747	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) ↔ ((#‘𝐴) = (#‘𝐵) ∧ (#‘𝐴) = (#‘𝐴))))
236		wrd2ind.4	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝜌 ↔ 𝜏))
237	235, 236	imbi12d 334	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) ↔ (((#‘𝐴) = (#‘𝐵) ∧ (#‘𝐴) = (#‘𝐴)) → 𝜏)))
238	237	rspcv 3305	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑋 → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → (((#‘𝐴) = (#‘𝐵) ∧ (#‘𝐴) = (#‘𝐴)) → 𝜏)))
239	238	com23 86	. . . . . 6 ⊢ (𝐴 ∈ Word 𝑋 → (((#‘𝐴) = (#‘𝐵) ∧ (#‘𝐴) = (#‘𝐴)) → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → 𝜏)))
240	239	expd 452	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → ((#‘𝐴) = (#‘𝐵) → ((#‘𝐴) = (#‘𝐴) → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → 𝜏))))
241	240	com34 91	. . . 4 ⊢ (𝐴 ∈ Word 𝑋 → ((#‘𝐴) = (#‘𝐵) → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → ((#‘𝐴) = (#‘𝐴) → 𝜏))))
242	241	imp 445	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ (#‘𝐴) = (#‘𝐵)) → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → ((#‘𝐴) = (#‘𝐴) → 𝜏)))
243	242	3adant2 1080	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (#‘𝐴) = (#‘𝐵)) → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → ((#‘𝐴) = (#‘𝐴) → 𝜏)))
244	230, 231, 243	mp2d 49	1 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (#‘𝐴) = (#‘𝐵)) → 𝜏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 [wsbc 3435 ∅c0 3915 ⟨cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 lastS clsw 13292 ++ cconcat 13293 ⟨“cs1 13294 substr csubstr 13295

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-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-oadd 7564 df-er 7742 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-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303

This theorem is referenced by: gsmsymgreq 17852

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