Theorem wrdind 14435

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 14236	. . 3 ⊢ (𝐴 ∈ Word 𝐵 → (♯‘𝐴) ∈ ℕ0)
2		eqeq2 2750	. . . . . 6 ⊢ (𝑛 = 0 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 0))
3	2	imbi1d 342	. . . . 5 ⊢ (𝑛 = 0 → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = 0 → 𝜑)))
4	3	ralbidv 3112	. . . 4 ⊢ (𝑛 = 0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)))
5		eqeq2 2750	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = 𝑚))
6	5	imbi1d 342	. . . . 5 ⊢ (𝑛 = 𝑚 → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = 𝑚 → 𝜑)))
7	6	ralbidv 3112	. . . 4 ⊢ (𝑛 = 𝑚 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚 → 𝜑)))
8		eqeq2 2750	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (𝑚 + 1)))
9	8	imbi1d 342	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
10	9	ralbidv 3112	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
11		eqeq2 2750	. . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → ((♯‘𝑥) = 𝑛 ↔ (♯‘𝑥) = (♯‘𝐴)))
12	11	imbi1d 342	. . . . 5 ⊢ (𝑛 = (♯‘𝐴) → (((♯‘𝑥) = 𝑛 → 𝜑) ↔ ((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
13	12	ralbidv 3112	. . . 4 ⊢ (𝑛 = (♯‘𝐴) → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑛 → 𝜑) ↔ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (♯‘𝐴) → 𝜑)))
14		hasheq0 14078	. . . . . 6 ⊢ (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
15		wrdind.5	. . . . . . 7 ⊢ 𝜓
16		wrdind.1	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
17	15, 16	mpbiri 257	. . . . . 6 ⊢ (𝑥 = ∅ → 𝜑)
18	14, 17	syl6bi 252	. . . . 5 ⊢ (𝑥 ∈ Word 𝐵 → ((♯‘𝑥) = 0 → 𝜑))
19	18	rgen 3074	. . . 4 ⊢ ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 0 → 𝜑)
20		fveqeq2 6783	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = 𝑚 ↔ (♯‘𝑦) = 𝑚))
21		wrdind.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
22	20, 21	imbi12d 345	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((♯‘𝑥) = 𝑚 → 𝜑) ↔ ((♯‘𝑦) = 𝑚 → 𝜒)))
23	22	cbvralvw 3383	. . . . 5 ⊢ (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚 → 𝜑) ↔ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒))
24		simprl 768	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Word 𝐵)
25		fzossfz 13406	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝑥)) ⊆ (0...(♯‘𝑥))
26		simprr 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) = (𝑚 + 1))
27		nn0p1nn 12272	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
28	27	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑚 + 1) ∈ ℕ)
29	26, 28	eqeltrd 2839	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘𝑥) ∈ ℕ)
30		fzo0end 13479	. . . . . . . . . . . . . . 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 3919	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) ∈ (0...(♯‘𝑥)))
33		pfxlen 14396	. . . . . . . . . . . . 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 7290	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ((♯‘𝑥) − 1) = ((𝑚 + 1) − 1))
36		nn0cn 12243	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
37	36	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑚 ∈ ℂ)
38		ax-1cn 10929	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
39		pncan 11227	. . . . . . . . . . . . 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 2782	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚)
42		fveqeq2 6783	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ((♯‘𝑦) = 𝑚 ↔ (♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚))
43		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
44	43, 21	sbcie 3759	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒)
45		dfsbcq 3718	. . . . . . . . . . . . . 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 345	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (((♯‘𝑦) = 𝑚 → 𝜒) ↔ ((♯‘(𝑥 prefix ((♯‘𝑥) − 1))) = 𝑚 → [(𝑥 prefix ((♯‘𝑥) − 1)) / 𝑥]𝜑)))
48		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒))
49		pfxcl 14390	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 prefix ((♯‘𝑥) − 1)) ∈ Word 𝐵)
50	49	ad2antrl 725	. . . . . . . . . . . 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 3562	. . . . . . . . . . 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 12021	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 1 ≤ (♯‘𝑥))
54		wrdlenge1n0 14253	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
55	54	ad2antrl 725	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → (𝑥 ≠ ∅ ↔ 1 ≤ (♯‘𝑥)))
56	53, 55	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
57		lswcl 14271	. . . . . . . . . . . 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 7282	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → (𝑦 ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩))
60	59	sbceq1d 3721	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 prefix ((♯‘𝑥) − 1)) → ([(𝑦 ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑))
61	45, 60	imbi12d 345	. . . . . . . . . . . 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 14305	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (lastS‘𝑥) → ⟨“𝑧”⟩ = ⟨“(lastS‘𝑥)”⟩)
63	62	oveq2d 7291	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (lastS‘𝑥) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) = ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩))
64	63	sbceq1d 3721	. . . . . . . . . . . . 13 ⊢ (𝑧 = (lastS‘𝑥) → ([((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“𝑧”⟩) / 𝑥]𝜑 ↔ [((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) / 𝑥]𝜑))
65	64	imbi2d 341	. . . . . . . . . . . 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 7308	. . . . . . . . . . . . . 14 ⊢ (𝑦 ++ ⟨“𝑧”⟩) ∈ V
68		wrdind.3	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑 ↔ 𝜃))
69	67, 68	sbcie 3759	. . . . . . . . . . . . 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 3514	. . . . . . . . . . 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 14235	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ Word 𝐵 → 𝑥 ∈ Fin)
75	74	ad2antrl 725	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ∈ Fin)
76		hashnncl 14081	. . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝑥 ≠ ∅)
79		pfxlswccat 14426	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Word 𝐵 ∧ 𝑥 ≠ ∅) → ((𝑥 prefix ((♯‘𝑥) − 1)) ++ ⟨“(lastS‘𝑥)”⟩) = 𝑥)
80	79	eqcomd 2744	. . . . . . . . . . 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 3727	. . . . . . . . . 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 256	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ (𝑥 ∈ Word 𝐵 ∧ (♯‘𝑥) = (𝑚 + 1))) → 𝜑)
85	84	expr 457	. . . . . . 7 ⊢ (((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) ∧ 𝑥 ∈ Word 𝐵) → ((♯‘𝑥) = (𝑚 + 1) → 𝜑))
86	85	ralrimiva 3103	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ ∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒)) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑))
87	86	ex 413	. . . . 5 ⊢ (𝑚 ∈ ℕ0 → (∀𝑦 ∈ Word 𝐵((♯‘𝑦) = 𝑚 → 𝜒) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
88	23, 87	syl5bi 241	. . . 4 ⊢ (𝑚 ∈ ℕ0 → (∀𝑥 ∈ Word 𝐵((♯‘𝑥) = 𝑚 → 𝜑) → ∀𝑥 ∈ Word 𝐵((♯‘𝑥) = (𝑚 + 1) → 𝜑)))
89	4, 7, 10, 13, 19, 88	nn0ind 12415	. . 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 2739	. 2 ⊢ (𝐴 ∈ Word 𝐵 → (♯‘𝐴) = (♯‘𝐴))
92		fveqeq2 6783	. . . 4 ⊢ (𝑥 = 𝐴 → ((♯‘𝑥) = (♯‘𝐴) ↔ (♯‘𝐴) = (♯‘𝐴)))
93		wrdind.4	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
94	92, 93	imbi12d 345	. . 3 ⊢ (𝑥 = 𝐴 → (((♯‘𝑥) = (♯‘𝐴) → 𝜑) ↔ ((♯‘𝐴) = (♯‘𝐴) → 𝜏)))
95	94	rspcv 3557	. 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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  [wsbc 3716  ∅c0 4256   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  ℂcc 10869  0cc0 10871  1c1 10872   + caddc 10874   ≤ cle 11010   − cmin 11205  ℕcn 11973  ℕ₀cn0 12233  ...cfz 13239  ..^cfzo 13382  ♯chash 14044  Word cword 14217  lastSclsw 14265   ++ cconcat 14273  ⟨“cs1 14300   prefix cpfx 14383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-lsw 14266  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384

This theorem is referenced by:  frmdgsum  18501  gsumwrev  18973  gsmsymgrfix  19036  efginvrel2  19333  signstfvneq0  32551  signstfvc  32553  mrsubvrs  33484