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Theorem wrd2ind 13329

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 13179	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → (#‘𝐴) ∈ ℕ₀)
2		eqeq2 2621	. . . . . . . . 9 ⊢ (𝑛 = 0 → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = 0))
3	2	anbi2d 736	. . . . . . . 8 ⊢ (𝑛 = 0 → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) ↔ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 0)))
4	3	imbi1d 330	. . . . . . 7 ⊢ (𝑛 = 0 → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) → 𝜑) ↔ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 0) → 𝜑)))
5	4	2ralbidv 2972	. . . . . 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 2621	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = 𝑚))
7	6	anbi2d 736	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) ↔ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚)))
8	7	imbi1d 330	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) → 𝜑) ↔ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) → 𝜑)))
9	8	2ralbidv 2972	. . . . . 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 2621	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 + 1) → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = (𝑚 + 1)))
11	10	anbi2d 736	. . . . . . . 8 ⊢ (𝑛 = (𝑚 + 1) → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) ↔ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1))))
12	11	imbi1d 330	. . . . . . 7 ⊢ (𝑛 = (𝑚 + 1) → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) → 𝜑) ↔ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)) → 𝜑)))
13	12	2ralbidv 2972	. . . . . 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 2621	. . . . . . . . 9 ⊢ (𝑛 = (#‘𝐴) → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = (#‘𝐴)))
15	14	anbi2d 736	. . . . . . . 8 ⊢ (𝑛 = (#‘𝐴) → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) ↔ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴))))
16	15	imbi1d 330	. . . . . . 7 ⊢ (𝑛 = (#‘𝐴) → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑛) → 𝜑) ↔ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑)))
17	16	2ralbidv 2972	. . . . . 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 2614	. . . . . . . . . . . 12 ⊢ ((#‘𝑥) = 0 → ((#‘𝑥) = (#‘𝑤) ↔ 0 = (#‘𝑤)))
19	18	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ (#‘𝑥) = 0) → ((#‘𝑥) = (#‘𝑤) ↔ 0 = (#‘𝑤)))
20		eqcom 2617	. . . . . . . . . . . . . . 15 ⊢ (0 = (#‘𝑤) ↔ (#‘𝑤) = 0)
21		hasheq0 13015	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Word 𝑌 → ((#‘𝑤) = 0 ↔ 𝑤 = ∅))
22	20, 21	syl5bb 271	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ Word 𝑌 → (0 = (#‘𝑤) ↔ 𝑤 = ∅))
23		hasheq0 13015	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝑋 → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
24		wrd2ind.6	. . . . . . . . . . . . . . . . . 18 ⊢ 𝜓
25		wrd2ind.1	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = ∅ ∧ 𝑤 = ∅) → (𝜑 ↔ 𝜓))
26	24, 25	mpbiri 247	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = ∅ ∧ 𝑤 = ∅) → 𝜑)
27	26	ex 449	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝑤 = ∅ → 𝜑))
28	23, 27	syl6bi 242	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ Word 𝑋 → ((#‘𝑥) = 0 → (𝑤 = ∅ → 𝜑)))
29	28	com13 86	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∅ → ((#‘𝑥) = 0 → (𝑥 ∈ Word 𝑋 → 𝜑)))
30	22, 29	syl6bi 242	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ Word 𝑌 → (0 = (#‘𝑤) → ((#‘𝑥) = 0 → (𝑥 ∈ Word 𝑋 → 𝜑))))
31	30	com24 93	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ Word 𝑌 → (𝑥 ∈ Word 𝑋 → ((#‘𝑥) = 0 → (0 = (#‘𝑤) → 𝜑))))
32	31	imp31 447	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ (#‘𝑥) = 0) → (0 = (#‘𝑤) → 𝜑))
33	19, 32	sylbid 229	. . . . . . . . . 10 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ (#‘𝑥) = 0) → ((#‘𝑥) = (#‘𝑤) → 𝜑))
34	33	ex 449	. . . . . . . . 9 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → ((#‘𝑥) = 0 → ((#‘𝑥) = (#‘𝑤) → 𝜑)))
35	34	com23 84	. . . . . . . 8 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → ((#‘𝑥) = (#‘𝑤) → ((#‘𝑥) = 0 → 𝜑)))
36	35	impd 446	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 0) → 𝜑))
37	36	rgen2 2958	. . . . . 6 ⊢ ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 0) → 𝜑)
38		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
39		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑢 → (#‘𝑤) = (#‘𝑢))
40	38, 39	eqeqan12d 2626	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → ((#‘𝑥) = (#‘𝑤) ↔ (#‘𝑦) = (#‘𝑢)))
41	38	eqeq1d 2612	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((#‘𝑥) = 𝑚 ↔ (#‘𝑦) = 𝑚))
42	41	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → ((#‘𝑥) = 𝑚 ↔ (#‘𝑦) = 𝑚))
43	40, 42	anbi12d 743	. . . . . . . . . . 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 333	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑤 = 𝑢) → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) → 𝜑) ↔ (((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒)))
46	45	ancoms 468	. . . . . . . . 9 ⊢ ((𝑤 = 𝑢 ∧ 𝑥 = 𝑦) → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) → 𝜑) ↔ (((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒)))
47	46	cbvraldva 3153	. . . . . . . 8 ⊢ (𝑤 = 𝑢 → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = 𝑚) → 𝜑) ↔ ∀𝑦 ∈ Word 𝑋(((#‘𝑦) = (#‘𝑢) ∧ (#‘𝑦) = 𝑚) → 𝜒)))
48	47	cbvralv 3147	. . . . . . 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 13271	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ Word 𝑌 → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌)
50	49	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌)
51	50	ad2antrl 760	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ∈ Word 𝑌)
52		simprll 798	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 𝑤 ∈ Word 𝑌)
53		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑥) = (𝑚 + 1) → ((#‘𝑥) = (#‘𝑤) ↔ (𝑚 + 1) = (#‘𝑤)))
54		nn0p1nn 11209	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ)
55		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑤) = (𝑚 + 1) → ((#‘𝑤) ∈ ℕ ↔ (𝑚 + 1) ∈ ℕ))
56	55	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 + 1) = (#‘𝑤) → ((#‘𝑤) ∈ ℕ ↔ (𝑚 + 1) ∈ ℕ))
57	54, 56	syl5ibr 235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 + 1) = (#‘𝑤) → (𝑚 ∈ ℕ₀ → (#‘𝑤) ∈ ℕ))
58	53, 57	syl6bi 242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑥) = (𝑚 + 1) → ((#‘𝑥) = (#‘𝑤) → (𝑚 ∈ ℕ₀ → (#‘𝑤) ∈ ℕ)))
59	58	impcom 445	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)) → (𝑚 ∈ ℕ₀ → (#‘𝑤) ∈ ℕ))
60	59	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑚 ∈ ℕ₀ → (#‘𝑤) ∈ ℕ))
61	60	impcom 445	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (#‘𝑤) ∈ ℕ)
62	61	nnge1d 10940	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 1 ≤ (#‘𝑤))
63		wrdlenge1n0 13195	. . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 𝑤 ≠ ∅)
66		lswcl 13208	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑤 ≠ ∅) → ( lastS ‘𝑤) ∈ 𝑌)
67	52, 65, 66	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → ( lastS ‘𝑤) ∈ 𝑌)
68	51, 67	jca 553	. . . . . . . . . . . . . 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 13271	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝑋 → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋)
70	69	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋)
71	70	ad2antrl 760	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋)
72		simprlr 799	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 𝑥 ∈ Word 𝑋)
73		eleq1 2676	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑥) = (𝑚 + 1) → ((#‘𝑥) ∈ ℕ ↔ (𝑚 + 1) ∈ ℕ))
74	54, 73	syl5ibr 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑥) = (𝑚 + 1) → (𝑚 ∈ ℕ₀ → (#‘𝑥) ∈ ℕ))
75	74	ad2antll 761	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑚 ∈ ℕ₀ → (#‘𝑥) ∈ ℕ))
76	75	impcom 445	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → (#‘𝑥) ∈ ℕ)
77	76	nnge1d 10940	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 1 ≤ (#‘𝑥))
78		wrdlenge1n0 13195	. . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → 𝑥 ≠ ∅)
81		lswcl 13208	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ Word 𝑋 ∧ 𝑥 ≠ ∅) → ( lastS ‘𝑥) ∈ 𝑋)
82	72, 80, 81	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ ((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)))) → ( lastS ‘𝑥) ∈ 𝑋)
83	68, 71, 82	jca32 556	. . . . . . . . . . . . 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 747	. . . . . . . . . . . 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 790	. . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . . . 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 6564	. . . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . . . 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 12357	. . . . . . . . . . . . . . . . . 18 ⊢ (0..^(#‘𝑥)) ⊆ (0...(#‘𝑥))
91		simprrr 801	. . . . . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . . . . 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 2688	. . . . . . . . . . . . . . . . . . 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 12426	. . . . . . . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . . . . . 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 13274	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word 𝑋 ∧ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))) → (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = ((#‘𝑥) − 1))
98	89, 96, 97	syl2anc 691	. . . . . . . . . . . . . . . 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 798	. . . . . . . . . . . . . . . . 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 6556	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑤) = (#‘𝑥) → ((#‘𝑤) − 1) = ((#‘𝑥) − 1))
101		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑤) = (#‘𝑥) → (0...(#‘𝑤)) = (0...(#‘𝑥)))
102	100, 101	eleq12d 2682	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑤) = (#‘𝑥) → (((#‘𝑤) − 1) ∈ (0...(#‘𝑤)) ↔ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))))
103	102	eqcoms 2618	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑥) = (#‘𝑤) → (((#‘𝑤) − 1) ∈ (0...(#‘𝑤)) ↔ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))))
104	103	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)) → (((#‘𝑤) − 1) ∈ (0...(#‘𝑤)) ↔ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))))
105	104	ad2antll 761	. . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . 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 13274	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word 𝑌 ∧ ((#‘𝑤) − 1) ∈ (0...(#‘𝑤))) → (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) = ((#‘𝑤) − 1))
108	99, 106, 107	syl2anc 691	. . . . . . . . . . . . . . . 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 2654	. . . . . . . . . . . . . . 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 6564	. . . . . . . . . . . . . . . 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 11179	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
112	111	ad2antrr 758	. . . . . . . . . . . . . . . . 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 9873	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
114		pncan 10166	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
115	112, 113, 114	sylancl 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)))) → ((𝑚 + 1) − 1) = 𝑚)
116	98, 110, 115	3eqtrd 2648	. . . . . . . . . . . . . . 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 553	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) ∧ 𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝑋)
119		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (#‘𝑦) = (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)))
120		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → (#‘𝑢) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩)))
121	119, 120	eqeqan12d 2626	. . . . . . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . . . 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 2612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ((#‘𝑦) = 𝑚 ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚))
126	125	adantl 481	. . . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
129		vex 3176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑢 ∈ V
130	128, 129, 44	sbc2ie 3472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ([𝑦 / 𝑥][𝑢 / 𝑤]𝜑 ↔ 𝜒)
131	130	bicomi 213	. . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . . 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 3407	. . . . . . . . . . . . . . . . . 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 3404	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ([𝑢 / 𝑤]𝜑 ↔ [(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑))
136	135	sbcbidv 3457	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 293	. . . . . . . . . . . . . . . . 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 333	. . . . . . . . . . . . . . . 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 3285	. . . . . . . . . . . . . . 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 3283	. . . . . . . . . . . . . 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 64	. . . . . . . . . . . . 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 553	. . . . . . . . . . . 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 3404	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤][𝑦 / 𝑥]𝜑))
146		sbccom 3476	. . . . . . . . . . . . . . . 16 ⊢ ([(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑)
147	145, 146	syl6bb 275	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) / 𝑤]𝜑))
148	120	eqeq2d 2620	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → ((#‘𝑦) = (#‘𝑢) ↔ (#‘𝑦) = (#‘(𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩))))
149	147, 148	anbi12d 743	. . . . . . . . . . . . . 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 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) → (𝑢 ++ ⟨“𝑠”⟩) = ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“𝑠”⟩))
151	150	sbceq1d 3407	. . . . . . . . . . . . . 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 333	. . . . . . . . . . . . 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 13233	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ( lastS ‘𝑤) → ⟨“𝑠”⟩ = ⟨“( lastS ‘𝑤)”⟩)
154	153	oveq2d 6565	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = ( lastS ‘𝑤) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“𝑠”⟩) = ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
155	154	sbceq1d 3407	. . . . . . . . . . . . . . 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 329	. . . . . . . . . . . . . 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 3476	. . . . . . . . . . . . . . . 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 329	. . . . . . . . . . . . . 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 267	. . . . . . . . . . . . 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 3404	. . . . . . . . . . . . . . 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 2612	. . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩))
165	164	sbceq1d 3407	. . . . . . . . . . . . . 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 333	. . . . . . . . . . . . 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 13233	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ( lastS ‘𝑥) → ⟨“𝑧”⟩ = ⟨“( lastS ‘𝑥)”⟩)
168	167	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ( lastS ‘𝑥) → ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
169	168	sbceq1d 3407	. . . . . . . . . . . . . 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 329	. . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (#‘𝑦) = (#‘𝑢)) → (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋))
172		simpll 786	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (#‘𝑦) = (#‘𝑢)) → (𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌))
173		simpr 476	. . . . . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑢 ∈ Word 𝑌 ∧ 𝑠 ∈ 𝑌) ∧ (𝑦 ∈ Word 𝑋 ∧ 𝑧 ∈ 𝑋)) ∧ (#‘𝑦) = (#‘𝑢)) → (𝜒 → 𝜃))
176	44	ancoms 468	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = 𝑢 ∧ 𝑥 = 𝑦) → (𝜑 ↔ 𝜒))
177	129, 128, 176	sbc2ie 3472	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑢 / 𝑤][𝑦 / 𝑥]𝜑 ↔ 𝜒)
178		ovex 6577	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ++ ⟨“𝑠”⟩) ∈ V
179		ovex 6577	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ++ ⟨“𝑧”⟩) ∈ V
180		wrd2ind.3	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = (𝑦 ++ ⟨“𝑧”⟩) ∧ 𝑤 = (𝑢 ++ ⟨“𝑠”⟩)) → (𝜑 ↔ 𝜃))
181	180	ancoms 468	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 = (𝑢 ++ ⟨“𝑠”⟩) ∧ 𝑥 = (𝑦 ++ ⟨“𝑧”⟩)) → (𝜑 ↔ 𝜃))
182	178, 179, 181	sbc2ie 3472	. . . . . . . . . . . . . . . . 17 ⊢ ([(𝑢 ++ ⟨“𝑠”⟩) / 𝑤][(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ 𝜃)
183	175, 177, 182	3imtr4g 284	. . . . . . . . . . . . . . . 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 449	. . . . . . . . . . . . . . 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 84	. . . . . . . . . . . . . 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 446	. . . . . . . . . . . . 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 3251	. . . . . . . . . . . 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 63	. . . . . . . . . . 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 2630	. . . . . . . . . . . . . . . 16 ⊢ (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (𝑚 + 1)) → (#‘𝑤) = (𝑚 + 1))
190	189	ad2antll 761	. . . . . . . . . . . . . . 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 2688	. . . . . . . . . . . . . 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 13178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ Word 𝑌 → 𝑤 ∈ Fin)
193	192	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → 𝑤 ∈ Fin)
194	193	ad2antrl 760	. . . . . . . . . . . . . . 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 13018	. . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 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 13320	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑤 ≠ ∅) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
199	198	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑤 ≠ ∅) → 𝑤 = ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
200	99, 197, 199	syl2anc 691	. . . . . . . . . . . 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 13178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ Word 𝑋 → 𝑥 ∈ Fin)
202	201	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑌 ∧ 𝑥 ∈ Word 𝑋) → 𝑥 ∈ Fin)
203	202	ad2antrl 760	. . . . . . . . . . . . . . 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 13018	. . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 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 13320	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word 𝑋 ∧ 𝑥 ≠ ∅) → ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) = 𝑥)
208	207	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑋 ∧ 𝑥 ≠ ∅) → 𝑥 = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
209	89, 206, 208	syl2anc 691	. . . . . . . . . . . 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 3413	. . . . . . . . . . . . 13 ⊢ (𝑤 = ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) → (𝜑 ↔ [((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) / 𝑤]𝜑))
211		sbceq1a 3413	. . . . . . . . . . . . 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 732	. . . . . . . . . . . 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 691	. . . . . . . . . . 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 246	. . . . . . . . . 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 641	. . . . . . . . 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 2954	. . . . . . . 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 449	. . . . . . 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 231	. . . . . 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 11348	. . . . 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 1075	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (#‘𝐴) = (#‘𝐵)) → ∀𝑤 ∈ Word 𝑌∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑))
222		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = 𝐵 → (#‘𝑤) = (#‘𝐵))
223	222	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → ((#‘𝑥) = (#‘𝑤) ↔ (#‘𝑥) = (#‘𝐵)))
224	223	anbi1d 737	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) ↔ ((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴))))
225		wrd2ind.5	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝜑 ↔ 𝜌))
226	224, 225	imbi12d 333	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑) ↔ (((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌)))
227	226	ralbidv 2969	. . . . 5 ⊢ (𝑤 = 𝐵 → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝑤) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜑) ↔ ∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌)))
228	227	rspcv 3278	. . . 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 1076	. . 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 2611	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (#‘𝐴) = (#‘𝐵)) → (#‘𝐴) = (#‘𝐴))
232		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (#‘𝑥) = (#‘𝐴))
233	232	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((#‘𝑥) = (#‘𝐵) ↔ (#‘𝐴) = (#‘𝐵)))
234	232	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((#‘𝑥) = (#‘𝐴) ↔ (#‘𝐴) = (#‘𝐴)))
235	233, 234	anbi12d 743	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) ↔ ((#‘𝐴) = (#‘𝐵) ∧ (#‘𝐴) = (#‘𝐴))))
236		wrd2ind.4	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝜌 ↔ 𝜏))
237	235, 236	imbi12d 333	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) ↔ (((#‘𝐴) = (#‘𝐵) ∧ (#‘𝐴) = (#‘𝐴)) → 𝜏)))
238	237	rspcv 3278	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑋 → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → (((#‘𝐴) = (#‘𝐵) ∧ (#‘𝐴) = (#‘𝐴)) → 𝜏)))
239	238	com23 84	. . . . . 6 ⊢ (𝐴 ∈ Word 𝑋 → (((#‘𝐴) = (#‘𝐵) ∧ (#‘𝐴) = (#‘𝐴)) → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → 𝜏)))
240	239	expd 451	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → ((#‘𝐴) = (#‘𝐵) → ((#‘𝐴) = (#‘𝐴) → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → 𝜏))))
241	240	com34 89	. . . 4 ⊢ (𝐴 ∈ Word 𝑋 → ((#‘𝐴) = (#‘𝐵) → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → ((#‘𝐴) = (#‘𝐴) → 𝜏))))
242	241	imp 444	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ (#‘𝐴) = (#‘𝐵)) → (∀𝑥 ∈ Word 𝑋(((#‘𝑥) = (#‘𝐵) ∧ (#‘𝑥) = (#‘𝐴)) → 𝜌) → ((#‘𝐴) = (#‘𝐴) → 𝜏)))
243	242	3adant2 1073	. 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 47	1 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (#‘𝐴) = (#‘𝐵)) → 𝜏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 [wsbc 3402 ∅c0 3874 ⟨cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 lastS clsw 13147 ++ cconcat 13148 ⟨“cs1 13149 substr csubstr 13150

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158

This theorem is referenced by: gsmsymgreq 17675

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