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Theorem wrdind 13328

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 13179	. . 3 ⊢ (𝐴 ∈ Word 𝐵 → (#‘𝐴) ∈ ℕ₀)
2		eqeq2 2621	. . . . . 6 ⊢ (𝑛 = 0 → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = 0))
3	2	imbi1d 330	. . . . 5 ⊢ (𝑛 = 0 → (((#‘𝑥) = 𝑛 → 𝜑) ↔ ((#‘𝑥) = 0 → 𝜑)))
4	3	ralbidv 2969	. . . 4 ⊢ (𝑛 = 0 → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = 0 → 𝜑)))
5		eqeq2 2621	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = 𝑚))
6	5	imbi1d 330	. . . . 5 ⊢ (𝑛 = 𝑚 → (((#‘𝑥) = 𝑛 → 𝜑) ↔ ((#‘𝑥) = 𝑚 → 𝜑)))
7	6	ralbidv 2969	. . . 4 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑚 → 𝜑)))
8		eqeq2 2621	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = (𝑚 + 1)))
9	8	imbi1d 330	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (((#‘𝑥) = 𝑛 → 𝜑) ↔ ((#‘𝑥) = (𝑚 + 1) → 𝜑)))
10	9	ralbidv 2969	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (𝑚 + 1) → 𝜑)))
11		eqeq2 2621	. . . . . 6 ⊢ (𝑛 = (#‘𝐴) → ((#‘𝑥) = 𝑛 ↔ (#‘𝑥) = (#‘𝐴)))
12	11	imbi1d 330	. . . . 5 ⊢ (𝑛 = (#‘𝐴) → (((#‘𝑥) = 𝑛 → 𝜑) ↔ ((#‘𝑥) = (#‘𝐴) → 𝜑)))
13	12	ralbidv 2969	. . . 4 ⊢ (𝑛 = (#‘𝐴) → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (#‘𝐴) → 𝜑)))
14		hasheq0 13015	. . . . . 6 ⊢ (𝑥 ∈ Word 𝐵 → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
15		wrdind.5	. . . . . . 7 ⊢ 𝜓
16		wrdind.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
17	15, 16	mpbiri 247	. . . . . 6 ⊢ (𝑥 = ∅ → 𝜑)
18	14, 17	syl6bi 242	. . . . 5 ⊢ (𝑥 ∈ Word 𝐵 → ((#‘𝑥) = 0 → 𝜑))
19	18	rgen 2906	. . . 4 ⊢ ∀𝑥 ∈ Word 𝐵((#‘𝑥) = 0 → 𝜑)
20		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
21	20	eqeq1d 2612	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((#‘𝑥) = 𝑚 ↔ (#‘𝑦) = 𝑚))
22		wrdind.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
23	21, 22	imbi12d 333	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((#‘𝑥) = 𝑚 → 𝜑) ↔ ((#‘𝑦) = 𝑚 → 𝜒)))
24	23	cbvralv 3147	. . . . 5 ⊢ (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑚 → 𝜑) ↔ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒))
25		swrdcl 13271	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝐵)
26	25	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ∈ Word 𝐵)
27		simplr 788	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒))
28		simprl 790	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Word 𝐵)
29		fzossfz 12357	. . . . . . . . . . . . . 14 ⊢ (0..^(#‘𝑥)) ⊆ (0...(#‘𝑥))
30		simprr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (#‘𝑥) = (𝑚 + 1))
31		nn0p1nn 11209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ)
32	31	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑚 + 1) ∈ ℕ)
33	30, 32	eqeltrd 2688	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (#‘𝑥) ∈ ℕ)
34		fzo0end 12426	. . . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ((#‘𝑥) − 1) ∈ (0...(#‘𝑥)))
37		swrd0len 13274	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝐵 ∧ ((#‘𝑥) − 1) ∈ (0...(#‘𝑥))) → (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = ((#‘𝑥) − 1))
38	28, 36, 37	syl2anc 691	. . . . . . . . . . . 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 6564	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ((#‘𝑥) − 1) = ((𝑚 + 1) − 1))
40		nn0cn 11179	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
41	40	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑚 ∈ ℂ)
42		ax-1cn 9873	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
43		pncan 10166	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
44	41, 42, 43	sylancl 693	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ((𝑚 + 1) − 1) = 𝑚)
45	38, 39, 44	3eqtrd 2648	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚)
46		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (#‘𝑦) = (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)))
47	46	eqeq1d 2612	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ((#‘𝑦) = 𝑚 ↔ (#‘(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩)) = 𝑚))
48		vex 3176	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
49	48, 22	sbcie 3437	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒)
50		dfsbcq 3404	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ([𝑦 / 𝑥]𝜑 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑))
51	49, 50	syl5bbr 273	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (𝜒 ↔ [(𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) / 𝑥]𝜑))
52	47, 51	imbi12d 333	. . . . . . . . . . . 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 3278	. . . . . . . . . . 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 64	. . . . . . . . . 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 10940	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 1 ≤ (#‘𝑥))
56		wrdlenge1n0 13195	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 ≠ ∅ ↔ 1 ≤ (#‘𝑥)))
57	56	ad2antrl 760	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → (𝑥 ≠ ∅ ↔ 1 ≤ (#‘𝑥)))
58	55, 57	mpbird 246	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
59		lswcl 13208	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → ( lastS ‘𝑥) ∈ 𝐵)
60	28, 58, 59	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → ( lastS ‘𝑥) ∈ 𝐵)
61		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩))
62	61	sbceq1d 3407	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
63	50, 62	imbi12d 333	. . . . . . . . . . . 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 13233	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ( lastS ‘𝑥) → ⟨“𝑧”⟩ = ⟨“( lastS ‘𝑥)”⟩)
65	64	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ( lastS ‘𝑥) → ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
66	65	sbceq1d 3407	. . . . . . . . . . . . 13 ⊢ (𝑧 = ( lastS ‘𝑥) → ([((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) / 𝑥]𝜑))
67	66	imbi2d 329	. . . . . . . . . . . 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 6577	. . . . . . . . . . . . . 14 ⊢ (𝑦 ++ ⟨“𝑧”⟩) ∈ V
70		wrdind.3	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑 ↔ 𝜃))
71	69, 70	sbcie 3437	. . . . . . . . . . . . 13 ⊢ ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ 𝜃)
72	68, 49, 71	3imtr4g 284	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑧 ∈ 𝐵) → ([𝑦 / 𝑥]𝜑 → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
73	63, 67, 72	vtocl2ga 3247	. . . . . . . . . . 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 691	. . . . . . . . . 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 13178	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → 𝑥 ∈ Fin)
77	76	ad2antrl 760	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Fin)
78		hashnncl 13018	. . . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
81		swrdccatwrd 13320	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩) = 𝑥)
82	81	eqcomd 2616	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → 𝑥 = ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) ++ ⟨“( lastS ‘𝑥)”⟩))
83	28, 80, 82	syl2anc 691	. . . . . . . . . 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 3413	. . . . . . . . . 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 246	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (#‘𝑥) = (𝑚 + 1))) → 𝜑)
87	86	expr 641	. . . . . . 7 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) ∧ 𝑥 ∈ Word 𝐵) → ((#‘𝑥) = (𝑚 + 1) → 𝜑))
88	87	ralrimiva 2949	. . . . . 6 ⊢ ((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒)) → ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (𝑚 + 1) → 𝜑))
89	88	ex 449	. . . . 5 ⊢ (𝑚 ∈ ℕ₀ → (∀𝑦 ∈ Word 𝐵((#‘𝑦) = 𝑚 → 𝜒) → ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (𝑚 + 1) → 𝜑)))
90	24, 89	syl5bi 231	. . . 4 ⊢ (𝑚 ∈ ℕ₀ → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = 𝑚 → 𝜑) → ∀𝑥 ∈ Word 𝐵((#‘𝑥) = (𝑚 + 1) → 𝜑)))
91	4, 7, 10, 13, 19, 90	nn0ind 11348	. . 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 2611	. 2 ⊢ (𝐴 ∈ Word 𝐵 → (#‘𝐴) = (#‘𝐴))
94		fveq2 6103	. . . . 5 ⊢ (𝑥 = 𝐴 → (#‘𝑥) = (#‘𝐴))
95	94	eqeq1d 2612	. . . 4 ⊢ (𝑥 = 𝐴 → ((#‘𝑥) = (#‘𝐴) ↔ (#‘𝐴) = (#‘𝐴)))
96		wrdind.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
97	95, 96	imbi12d 333	. . 3 ⊢ (𝑥 = 𝐴 → (((#‘𝑥) = (#‘𝐴) → 𝜑) ↔ ((#‘𝐴) = (#‘𝐴) → 𝜏)))
98	97	rspcv 3278	. 2 ⊢ (𝐴 ∈ Word 𝐵 → (∀𝑥 ∈ Word 𝐵((#‘𝑥) = (#‘𝐴) → 𝜑) → ((#‘𝐴) = (#‘𝐴) → 𝜏)))
99	92, 93, 98	mp2d 47	1 ⊢ (𝐴 ∈ Word 𝐵 → 𝜏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = 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: frmdgsum 17222 gsumwrev 17619 gsmsymgrfix 17671 efginvrel2 17963 signstfvneq0 29975 signstfvc 29977 mrsubvrs 30673

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