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Theorem wrdind 14899

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 14705	. . 3 ⊢ (𝐴 ∈ Word 𝐵 → (♯‘𝐴) ∈ ℕ₀)
2		eqeq2 2775	. . . . . 6 ⊢ (𝑛 = 0 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 0))
3	2	imbi1d 344	. . . . 5 ⊢ (𝑛 = 0 → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = 0 → 𝜑)))
4	3	ralbidv 3189	. . . 4 ⊢ (𝑛 = 0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)))
5		eqeq2 2775	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 𝑚))
6	5	imbi1d 344	. . . . 5 ⊢ (𝑛 = 𝑚 → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = 𝑚 → 𝜑)))
7	6	ralbidv 3189	. . . 4 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚 → 𝜑)))
8		eqeq2 2775	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (𝑚 + 1)))
9	8	imbi1d 344	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
10	9	ralbidv 3189	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
11		eqeq2 2775	. . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (♯‘𝐴)))
12	11	imbi1d 344	. . . . 5 ⊢ (𝑛 = (♯‘𝐴) → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
13	12	ralbidv 3189	. . . 4 ⊢ (𝑛 = (♯‘𝐴) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
14		hasheq0 14534	. . . . . 6 ⊢ (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
15		wrdind.5	. . . . . . 7 ⊢ 𝜓
16		wrdind.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
17	15, 16	mpbiri 261	. . . . . 6 ⊢ (𝑥 = ∅ → 𝜑)
18	14, 17	biimtrdi 256	. . . . 5 ⊢ (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 → 𝜑))
19	18	rgen 3082	. . . 4 ⊢ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)
20		fveqeq2 6917	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = 𝑚 ↔ (♯‘𝑦) = 𝑚))
21		wrdind.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
22	20, 21	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((♯‘𝑥) = 𝑚 → 𝜑) ↔ ((♯‘𝑦) = 𝑚 → 𝜒)))
23	22	cbvralvw 3244	. . . . 5 ⊢ (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚 → 𝜑) ↔ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒))
24		simprl 783	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Word 𝐵)
25		fzossfz 13840	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝑥)) ⊆ (0...(♯‘𝑥))
26		simprr 785	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) = (𝑚 + 1))
27		nn0p1nn 12672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ)
28	27	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑚 + 1) ∈ ℕ)
29	26, 28	eqeltrd 2864	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) ∈ ℕ)
30		fzo0end 13920	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) ∈ ℕ → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
31	29, 30	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0..^(♯‘𝑥)))
32	25, 31	sselid 3911	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥)))
33		pfxlen 14861	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝐵 ∧ ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
34	24, 32, 33	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = ((♯‘𝑥) − 1))
35	26	oveq1d 7458	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) = ((𝑚 + 1) − 1))
36		nn0cn 12643	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ₀ → 𝑚 ∈ ℂ)
37	36	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑚 ∈ ℂ)
38		ax-1cn 11285	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
39		pncan 11590	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
40	37, 38, 39	sylancl 598	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((𝑚 + 1) − 1) = 𝑚)
41	34, 35, 40	3eqtrd 2802	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚)
42		fveqeq2 6917	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((♯‘𝑦) = 𝑚 ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚))
43		vex 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
44	43, 21	sbcie 3783	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒)
45		dfsbcq 3744	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([𝑦 / 𝑥]𝜑 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
46	44, 45	bitr3id 288	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝜒 ↔ [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑))
47	42, 46	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (((♯‘𝑦) = 𝑚 → 𝜒) ↔ ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚 → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑)))
48		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒))
49		pfxcl 14855	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵)
50	49	ad2antrl 741	. . . . . . . . . . . 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 3581	. . . . . . . . . . 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 16	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑)
53	29	nnge1d 12413	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 1 ≤ (♯‘𝑥))
54		wrdlenge1n0 14722	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
55	54	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
56	53, 55	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
57		lswcl 14740	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → (lastS‘𝑥) ∈ 𝐵)
58	24, 56, 57	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (lastS‘𝑥) ∈ 𝐵)
59		oveq1 7450	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩))
60	59	sbceq1d 3747	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
61	45, 60	imbi12d 347	. . . . . . . . . . . 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 14774	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (lastS‘𝑥) → ⟨“𝑧”⟩ = ⟨“(lastS‘𝑥)”⟩)
63	62	oveq2d 7459	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (lastS‘𝑥) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
64	63	sbceq1d 3747	. . . . . . . . . . . . 13 ⊢ (𝑧 = (lastS‘𝑥) → ([((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
65	64	imbi2d 343	. . . . . . . . . . . 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 7476	. . . . . . . . . . . . . 14 ⊢ (𝑦 ++ ⟨“𝑧”⟩) ∈ V
68		wrdind.3	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑 ↔ 𝜃))
69	67, 68	sbcie 3783	. . . . . . . . . . . . 13 ⊢ ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ 𝜃)
70	66, 44, 69	3imtr4g 299	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑧 ∈ 𝐵) → ([𝑦 / 𝑥]𝜑 → [(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
71	61, 65, 70	vtocl2ga 3541	. . . . . . . . . . 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 596	. . . . . . . . . 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 16	. . . . . . . . 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 14704	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → 𝑥 ∈ Fin)
75	74	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Fin)
76		hashnncl 14537	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
77	75, 76	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅))
78	29, 77	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
79		pfxlswccat 14890	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) = 𝑥)
80	79	eqcomd 2769	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
81	24, 78, 80	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
82		sbceq1a 3753	. . . . . . . . . 10 ⊢ (𝑥 = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) → (𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
83	81, 82	syl 18	. . . . . . . . 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 260	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝜑)
85	84	expr 462	. . . . . . 7 ⊢ (((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ 𝑥 ∈ Word 𝐵) → ((♯‘𝑥) = (𝑚 + 1) → 𝜑))
86	85	ralrimiva 3158	. . . . . 6 ⊢ ((𝑚 ∈ ℕ₀ ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑))
87	86	ex 418	. . . . 5 ⊢ (𝑚 ∈ ℕ₀ → (∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
88	23, 87	biimtrid 245	. . . 4 ⊢ (𝑚 ∈ ℕ₀ → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚 → 𝜑) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
89	4, 7, 10, 13, 19, 88	nn0ind 12821	. . 3 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
90	1, 89	syl 18	. 2 ⊢ (𝐴 ∈ Word 𝐵 → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑))
91		eqidd 2764	. 2 ⊢ (𝐴 ∈ Word 𝐵 → (♯‘𝐴) = (♯‘𝐴))
92		fveqeq2 6917	. . . 4 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) = (♯‘𝐴) ↔ (♯‘𝐴) = (♯‘𝐴)))
93		wrdind.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
94	92, 93	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → (((♯‘𝑥) = (♯‘𝐴) → 𝜑) ↔ ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
95	94	rspcv 3576	. 2 ⊢ (𝐴 ∈ Word 𝐵 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑) → ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
96	90, 91, 95	mp2d 50	1 ⊢ (𝐴 ∈ Word 𝐵 → 𝜏)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 [wsbc 3742 ∅c0 4281 class class class wbr 5106 ‘cfv 6558 (class class class)co 7443 Fincfn 9001 ℂcc 11225 0cc0 11227 1c1 11228 + caddc 11230 ≤ cle 11371 − cmin 11568 ℕcn 12362 ℕ₀cn0 12633 ...cfz 13666 ..^cfzo 13815 ♯chash 14501 Word cword 14685 lastSclsw 14734 ++ cconcat 14742 ⟨“cs1 14769 prefix cpfx 14848

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-n0 12634 df-xnn0 12707 df-z 12721 df-uz 12993 df-fz 13667 df-fzo 13816 df-hash 14502 df-word 14686 df-lsw 14735 df-concat 14743 df-s1 14770 df-substr 14817 df-pfx 14849

This theorem is used by: chnind 18824 frmdgsum 19087 gsumwrev 19610 gsmsymgrfix 19672 efginvrel2 19971 gsumwun 33707 domnprodn0 33909 unitprodclb 34014 1arithidom 34139 1arithufdlem3 34148 dfufd2lem 34151 signstfvneq0 35271 signstfvc 35273 mrsubvrs 36454

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