Theorem wrdind 14631

Description: Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 12-Oct-2022.)

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 14442	. . 3 ⊢ (𝐴 ∈ Word 𝐵 → (♯‘𝐴) ∈ ℕ0)
2		eqeq2 2745	. . . . . 6 ⊢ (𝑛 = 0 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 0))
3	2	imbi1d 341	. . . . 5 ⊢ (𝑛 = 0 → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = 0 → 𝜑)))
4	3	ralbidv 3156	. . . 4 ⊢ (𝑛 = 0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)))
5		eqeq2 2745	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 𝑚))
6	5	imbi1d 341	. . . . 5 ⊢ (𝑛 = 𝑚 → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = 𝑚 → 𝜑)))
7	6	ralbidv 3156	. . . 4 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚 → 𝜑)))
8		eqeq2 2745	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (𝑚 + 1)))
9	8	imbi1d 341	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
10	9	ralbidv 3156	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
11		eqeq2 2745	. . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (♯‘𝐴)))
12	11	imbi1d 341	. . . . 5 ⊢ (𝑛 = (♯‘𝐴) → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
13	12	ralbidv 3156	. . . 4 ⊢ (𝑛 = (♯‘𝐴) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
14		hasheq0 14272	. . . . . 6 ⊢ (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
15		wrdind.5	. . . . . . 7 ⊢ 𝜓
16		wrdind.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
17	15, 16	mpbiri 258	. . . . . 6 ⊢ (𝑥 = ∅ → 𝜑)
18	14, 17	biimtrdi 253	. . . . 5 ⊢ (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 → 𝜑))
19	18	rgen 3050	. . . 4 ⊢ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)
20		fveqeq2 6837	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = 𝑚 ↔ (♯‘𝑦) = 𝑚))
21		wrdind.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
22	20, 21	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((♯‘𝑥) = 𝑚 → 𝜑) ↔ ((♯‘𝑦) = 𝑚 → 𝜒)))
23	22	cbvralvw 3211	. . . . 5 ⊢ (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚 → 𝜑) ↔ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒))
24		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Word 𝐵)
25		fzossfz 13580	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝑥)) ⊆ (0...(♯‘𝑥))
26		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) = (𝑚 + 1))
27		nn0p1nn 12427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
28	27	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑚 + 1) ∈ ℕ)
29	26, 28	eqeltrd 2833	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) ∈ ℕ)
30		fzo0end 13660	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) ∈ ℕ → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
31	29, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
32	25, 31	sselid 3928	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥)))
33		pfxlen 14593	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝐵 ∧ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
34	24, 32, 33	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
35	26	oveq1d 7367	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) = ((𝑚 + 1) − 1))
36		nn0cn 12398	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
37	36	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑚 ∈ ℂ)
38		ax-1cn 11071	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
39		pncan 11373	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
40	37, 38, 39	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((𝑚 + 1) − 1) = 𝑚)
41	34, 35, 40	3eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚)
42		fveqeq2 6837	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((♯‘𝑦) = 𝑚 ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚))
43		vex 3441	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
44	43, 21	sbcie 3779	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒)
45		dfsbcq 3739	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([𝑦 / 𝑥]𝜑 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
46	44, 45	bitr3id 285	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝜒 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
47	42, 46	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (((♯‘𝑦) = 𝑚 → 𝜒) ↔ ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚 → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑)))
48		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒))
49		pfxcl 14587	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵)
50	49	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵)
51	47, 48, 50	rspcdva 3574	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚 → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
52	41, 51	mpd 15	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑)
53	29	nnge1d 12180	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 1 ≤ (♯‘𝑥))
54		wrdlenge1n0 14459	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
55	54	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
56	53, 55	mpbird 257	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
57		lswcl 14477	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → (lastS‘𝑥) ∈ 𝐵)
58	24, 56, 57	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (lastS‘𝑥) ∈ 𝐵)
59		oveq1 7359	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩))
60	59	sbceq1d 3742	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
61	45, 60	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (([𝑦 / 𝑥]𝜑 → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑 → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑)))
62		s1eq 14510	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (lastS‘𝑥) → ⟨“𝑧”⟩ = ⟨“(lastS‘𝑥)”⟩)
63	62	oveq2d 7368	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (lastS‘𝑥) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
64	63	sbceq1d 3742	. . . . . . . . . . . . 13 ⊢ (𝑧 = (lastS‘𝑥) → ([((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
65	64	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑧 = (lastS‘𝑥) → (([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑 → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑) ↔ ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑 → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑)))
66		wrdind.6	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝜒 → 𝜃))
67		ovex 7385	. . . . . . . . . . . . . 14 ⊢ (𝑦 ++ ⟨“𝑧”⟩) ∈ V
68		wrdind.3	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑 ↔ 𝜃))
69	67, 68	sbcie 3779	. . . . . . . . . . . . 13 ⊢ ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ 𝜃)
70	66, 44, 69	3imtr4g 296	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑧 ∈ 𝐵) → ([𝑦 / 𝑥]𝜑 → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
71	61, 65, 70	vtocl2ga 3530	. . . . . . . . . . 11 ⊢ (((𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵 ∧ (lastS‘𝑥) ∈ 𝐵) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑 → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
72	50, 58, 71	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ([(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑 → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
73	52, 72	mpd 15	. . . . . . . . 9 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑)
74		wrdfin 14441	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → 𝑥 ∈ Fin)
75	74	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Fin)
76		hashnncl 14275	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
77	75, 76	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
78	29, 77	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
79		pfxlswccat 14622	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) = 𝑥)
80	79	eqcomd 2739	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
81	24, 78, 80	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
82		sbceq1a 3748	. . . . . . . . . 10 ⊢ (𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) → (𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
83	81, 82	syl 17	. . . . . . . . 9 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
84	73, 83	mpbird 257	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝜑)
85	84	expr 456	. . . . . . 7 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ 𝑥 ∈ Word 𝐵) → ((♯‘𝑥) = (𝑚 + 1) → 𝜑))
86	85	ralrimiva 3125	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑))
87	86	ex 412	. . . . 5 ⊢ (𝑚 ∈ ℕ0 → (∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
88	23, 87	biimtrid 242	. . . 4 ⊢ (𝑚 ∈ ℕ0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚 → 𝜑) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
89	4, 7, 10, 13, 19, 88	nn0ind 12574	. . 3 ⊢ ((♯‘𝐴) ∈ ℕ0 → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
90	1, 89	syl 17	. 2 ⊢ (𝐴 ∈ Word 𝐵 → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
91		eqidd 2734	. 2 ⊢ (𝐴 ∈ Word 𝐵 → (♯‘𝐴) = (♯‘𝐴))
92		fveqeq2 6837	. . . 4 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) = (♯‘𝐴) ↔ (♯‘𝐴) = (♯‘𝐴)))
93		wrdind.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
94	92, 93	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → (((♯‘𝑥) = (♯‘𝐴) → 𝜑) ↔ ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
95	94	rspcv 3569	. 2 ⊢ (𝐴 ∈ Word 𝐵 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑) → ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
96	90, 91, 95	mp2d 49	1 ⊢ (𝐴 ∈ Word 𝐵 → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  [wsbc 3737  ∅c0 4282   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  Fincfn 8875  ℂcc 11011  0cc0 11013  1c1 11014   + caddc 11016   ≤ cle 11154   − cmin 11351  ℕcn 12132  ℕ₀cn0 12388  ...cfz 13409  ..^cfzo 13556  ♯chash 14239  Word cword 14422  lastSclsw 14471   ++ cconcat 14479  ⟨“cs1 14505   prefix cpfx 14580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-n0 12389  df-xnn0 12462  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-hash 14240  df-word 14423  df-lsw 14472  df-concat 14480  df-s1 14506  df-substr 14551  df-pfx 14581

This theorem is referenced by:  chnind  18529  frmdgsum  18772  gsumwrev  19280  gsmsymgrfix  19342  efginvrel2  19641  gsumwun  33052  domnprodn0  33249  unitprodclb  33361  1arithidom  33509  1arithufdlem3  33518  dfufd2lem  33521  signstfvneq0  34606  signstfvc  34608  mrsubvrs  35587