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Theorem wrdind 13476

Description: Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

**Hypotheses**
Ref	Expression
wrdind.1	⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
wrdind.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
wrdind.3	⊢ (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑 ↔ 𝜃))
wrdind.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
wrdind.5	⊢ 𝜓
wrdind.6	⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝜒 → 𝜃))

**Assertion**
Ref	Expression
wrdind	⊢ (𝐴 ∈ Word 𝐵 → 𝜏)

Distinct variable groups: 𝑥,𝐴 𝑥,𝑦,𝑧,𝐵 𝜒,𝑥 𝜑,𝑦,𝑧 𝜏,𝑥 𝜃,𝑥 Allowed substitution hints: 𝜑(𝑥) 𝜓(𝑥,𝑦,𝑧) 𝜒(𝑦,𝑧) 𝜃(𝑦,𝑧) 𝜏(𝑦,𝑧) 𝐴(𝑦,𝑧)

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

Step	Hyp	Ref	Expression
1		lencl 13324	. . 3 ⊢ (𝐴 ∈ Word 𝐵 → (#‘𝐴) ∈ ℕ₀)
2		eqeq2 2633	. . . . . 6 ⊢ (𝑛 = 0 → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = 0))
3	2	imbi1d 331	. . . . 5 ⊢ (𝑛 = 0 → (((#‘𝑥) = 𝑛 → 𝜑) ↔ ((#‘𝑥) = 0 → 𝜑)))
4	3	ralbidv 2986	. . . 4 ⊢ (𝑛 = 0 → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = 0 → 𝜑)))
5		eqeq2 2633	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = 𝑚))
6	5	imbi1d 331	. . . . 5 ⊢ (𝑛 = 𝑚 → (((#‘𝑥) = 𝑛 → 𝜑) ↔ ((#‘𝑥) = 𝑚 → 𝜑)))
7	6	ralbidv 2986	. . . 4 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑚 → 𝜑)))
8		eqeq2 2633	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = (𝑚 + 1)))
9	8	imbi1d 331	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (((#‘𝑥) = 𝑛 → 𝜑) ↔ ((#‘𝑥) = (𝑚 + 1) → 𝜑)))
10	9	ralbidv 2986	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (𝑚 + 1) → 𝜑)))
11		eqeq2 2633	. . . . . 6 ⊢ (𝑛 = (#‘𝐴) → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = (#‘𝐴)))
12	11	imbi1d 331	. . . . 5 ⊢ (𝑛 = (#‘𝐴) → (((#‘𝑥) = 𝑛 → 𝜑) ↔ ((#‘𝑥) = (#‘𝐴) → 𝜑)))
13	12	ralbidv 2986	. . . 4 ⊢ (𝑛 = (#‘𝐴) → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (#‘𝐴) → 𝜑)))
14		hasheq0 13154	. . . . . 6 ⊢ (𝑥 ∈ Word 𝐵 → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
15		wrdind.5	. . . . . . 7 ⊢ 𝜓
16		wrdind.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
17	15, 16	mpbiri 248	. . . . . 6 ⊢ (𝑥 = ∅ → 𝜑)
18	14, 17	syl6bi 243	. . . . 5 ⊢ (𝑥 ∈ Word 𝐵 → ((#‘𝑥) = 0 → 𝜑))
19	18	rgen 2922	. . . 4 ⊢ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = 0 → 𝜑)
20		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
21	20	eqeq1d 2624	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((#‘𝑥) = 𝑚 ↔ (#‘𝑦) = 𝑚))
22		wrdind.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
23	21, 22	imbi12d 334	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((#‘𝑥) = 𝑚 → 𝜑) ↔ ((#‘𝑦) = 𝑚 → 𝜒)))
24	23	cbvralv 3171	. . . . 5 ⊢ (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑚 → 𝜑) ↔ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒))
25		swrdcl 13419	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝐵)
26	25	ad2antrl 764	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝐵)
27		simplr 792	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒))
28		simprl 794	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Word 𝐵)
29		fzossfz 12488	. . . . . . . . . . . . . 14 ⊢ (0..^(#‘𝑥)) ⊆ (0...(#‘𝑥))
30		simprr 796	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (#‘𝑥) = (𝑚 + 1))
31		nn0p1nn 11332	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ)
32	31	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑚 + 1) ∈ ℕ)
33	30, 32	eqeltrd 2701	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (#‘𝑥) ∈ ℕ)
34		fzo0end 12560	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑥) ∈ ℕ → ((#‘𝑥) − 1) ∈ (0..^(#‘𝑥)))
35	33, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ((#‘𝑥) − 1) ∈ (0..^(#‘𝑥)))
36	29, 35	sseldi 3601	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ((#‘𝑥) − 1) ∈ (0...(#‘𝑥)))
37		swrd0len 13422	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝐵 ∧ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))) → (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = ((#‘𝑥) − 1))
38	28, 36, 37	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = ((#‘𝑥) − 1))
39	30	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ((#‘𝑥) − 1) = ((𝑚 + 1) − 1))
40		nn0cn 11302	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
41	40	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑚 ∈ ℂ)
42		ax-1cn 9994	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
43		pncan 10287	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
44	41, 42, 43	sylancl 694	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ((𝑚 + 1) − 1) = 𝑚)
45	38, 39, 44	3eqtrd 2660	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚)
46		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (#‘𝑦) = (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)))
47	46	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ((#‘𝑦) = 𝑚 ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚))
48		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
49	48, 22	sbcie 3470	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒)
50		dfsbcq 3437	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ([𝑦 / 𝑥]𝜑 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑))
51	49, 50	syl5bbr 274	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (𝜒 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑))
52	47, 51	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (((#‘𝑦) = 𝑚 → 𝜒) ↔ ((#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚 → [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑)))
53	52	rspcv 3305	. . . . . . . . . . 11 ⊢ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝐵 → (∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒) → ((#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚 → [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑)))
54	26, 27, 45, 53	syl3c 66	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑)
55	33	nnge1d 11063	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 1 ≤ (#‘𝑥))
56		wrdlenge1n0 13340	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 ≠ ∅ ↔ 1 ≤ (#‘𝑥)))
57	56	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑥 ≠ ∅ ↔ 1 ≤ (#‘𝑥)))
58	55, 57	mpbird 247	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
59		lswcl 13355	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → ( lastS ‘𝑥) ∈ 𝐵)
60	28, 58, 59	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ( lastS ‘𝑥) ∈ 𝐵)
61		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩))
62	61	sbceq1d 3440	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
63	50, 62	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (([𝑦 / 𝑥]𝜑 → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑 → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
64		s1eq 13380	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ( lastS ‘𝑥) → ⟨“𝑧”⟩ = ⟨“( lastS ‘𝑥)”⟩)
65	64	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ( lastS ‘𝑥) → ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
66	65	sbceq1d 3440	. . . . . . . . . . . . 13 ⊢ (𝑧 = ( lastS ‘𝑥) → ([((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥]𝜑))
67	66	imbi2d 330	. . . . . . . . . . . 12 ⊢ (𝑧 = ( lastS ‘𝑥) → (([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑 → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑 → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥]𝜑)))
68		wrdind.6	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝜒 → 𝜃))
69		ovex 6678	. . . . . . . . . . . . . 14 ⊢ (𝑦 ++ ⟨“𝑧”⟩) ∈ V
70		wrdind.3	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑 ↔ 𝜃))
71	69, 70	sbcie 3470	. . . . . . . . . . . . 13 ⊢ ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ 𝜃)
72	68, 49, 71	3imtr4g 285	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑧 ∈ 𝐵) → ([𝑦 / 𝑥]𝜑 → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
73	63, 67, 72	vtocl2ga 3274	. . . . . . . . . . 11 ⊢ (((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝐵 ∧ ( lastS ‘𝑥) ∈ 𝐵) → ([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑 → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥]𝜑))
74	26, 60, 73	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ([(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑 → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥]𝜑))
75	54, 74	mpd 15	. . . . . . . . 9 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥]𝜑)
76		wrdfin 13323	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → 𝑥 ∈ Fin)
77	76	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Fin)
78		hashnncl 13157	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → ((#‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
79	77, 78	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ((#‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
80	33, 79	mpbid 222	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
81		swrdccatwrd 13468	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) = 𝑥)
82	81	eqcomd 2628	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → 𝑥 = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
83	28, 80, 82	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑥 = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
84		sbceq1a 3446	. . . . . . . . . 10 ⊢ (𝑥 = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) → (𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥]𝜑))
85	83, 84	syl 17	. . . . . . . . 9 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥]𝜑))
86	75, 85	mpbird 247	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝜑)
87	86	expr 643	. . . . . . 7 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ 𝑥 ∈ Word 𝐵) → ((#‘𝑥) = (𝑚 + 1) → 𝜑))
88	87	ralrimiva 2966	. . . . . 6 ⊢ ((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) → ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (𝑚 + 1) → 𝜑))
89	88	ex 450	. . . . 5 ⊢ (𝑚 ∈ ℕ₀ → (∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒) → ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (𝑚 + 1) → 𝜑)))
90	24, 89	syl5bi 232	. . . 4 ⊢ (𝑚 ∈ ℕ₀ → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑚 → 𝜑) → ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (𝑚 + 1) → 𝜑)))
91	4, 7, 10, 13, 19, 90	nn0ind 11472	. . 3 ⊢ ((#‘𝐴) ∈ ℕ₀ → ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (#‘𝐴) → 𝜑))
92	1, 91	syl 17	. 2 ⊢ (𝐴 ∈ Word 𝐵 → ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (#‘𝐴) → 𝜑))
93		eqidd 2623	. 2 ⊢ (𝐴 ∈ Word 𝐵 → (#‘𝐴) = (#‘𝐴))
94		fveq2 6191	. . . . 5 ⊢ (𝑥 = 𝐴 → (#‘𝑥) = (#‘𝐴))
95	94	eqeq1d 2624	. . . 4 ⊢ (𝑥 = 𝐴 → ((#‘𝑥) = (#‘𝐴) ↔ (#‘𝐴) = (#‘𝐴)))
96		wrdind.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
97	95, 96	imbi12d 334	. . 3 ⊢ (𝑥 = 𝐴 → (((#‘𝑥) = (#‘𝐴) → 𝜑) ↔ ((#‘𝐴) = (#‘𝐴) → 𝜏)))
98	97	rspcv 3305	. 2 ⊢ (𝐴 ∈ Word 𝐵 → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = (#‘𝐴) → 𝜑) → ((#‘𝐴) = (#‘𝐴) → 𝜏)))
99	92, 93, 98	mp2d 49	1 ⊢ (𝐴 ∈ Word 𝐵 → 𝜏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = 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: frmdgsum 17399 gsumwrev 17796 gsmsymgrfix 17848 efginvrel2 18140 signstfvneq0 30649 signstfvc 30651 mrsubvrs 31419

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