Theorem pfxeq 11187

Description: The prefixes of two words are equal iff they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 4-May-2020.)

Assertion

Ref	Expression
pfxeq	⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 prefix 𝑀) = (𝑈 prefix 𝑁) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖))))

Distinct variable groups:   𝑖,𝑀   𝑖,𝑁   𝑈,𝑖   𝑖,𝑉   𝑖,𝑊

Proof of Theorem pfxeq

Step	Hyp	Ref	Expression
1		simp2l 1026	. . . . 5 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑊 ∈ Word 𝑉)
2		simp1l 1024	. . . . 5 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑀 ∈ ℕ0)
3		pfxclg 11169	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℕ0) → (𝑊 prefix 𝑀) ∈ Word 𝑉)
4	1, 2, 3	syl2anc 411	. . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑊 prefix 𝑀) ∈ Word 𝑉)
5		simp2r 1027	. . . . 5 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑈 ∈ Word 𝑉)
6		simp1r 1025	. . . . 5 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑁 ∈ ℕ0)
7		pfxclg 11169	. . . . 5 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑈 prefix 𝑁) ∈ Word 𝑉)
8	5, 6, 7	syl2anc 411	. . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑈 prefix 𝑁) ∈ Word 𝑉)
9		eqwrd 11071	. . . 4 ⊢ (((𝑊 prefix 𝑀) ∈ Word 𝑉 ∧ (𝑈 prefix 𝑁) ∈ Word 𝑉) → ((𝑊 prefix 𝑀) = (𝑈 prefix 𝑁) ↔ ((♯‘(𝑊 prefix 𝑀)) = (♯‘(𝑈 prefix 𝑁)) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 𝑀)))((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖))))
10	4, 8, 9	syl2anc 411	. . 3 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 prefix 𝑀) = (𝑈 prefix 𝑁) ↔ ((♯‘(𝑊 prefix 𝑀)) = (♯‘(𝑈 prefix 𝑁)) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 𝑀)))((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖))))
11		simpl 109	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑀 ∈ ℕ0)
12		lencl 11035	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0)
13	12	adantr 276	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → (♯‘𝑊) ∈ ℕ0)
14		simpl 109	. . . . . . . 8 ⊢ ((𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)) → 𝑀 ≤ (♯‘𝑊))
15	11, 13, 14	3anim123i 1187	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑀 ∈ ℕ0 ∧ (♯‘𝑊) ∈ ℕ0 ∧ 𝑀 ≤ (♯‘𝑊)))
16		elfz2nn0 10269	. . . . . . 7 ⊢ (𝑀 ∈ (0...(♯‘𝑊)) ↔ (𝑀 ∈ ℕ0 ∧ (♯‘𝑊) ∈ ℕ0 ∧ 𝑀 ≤ (♯‘𝑊)))
17	15, 16	sylibr 134	. . . . . 6 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑀 ∈ (0...(♯‘𝑊)))
18		pfxlen 11176	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝑀)) = 𝑀)
19	1, 17, 18	syl2anc 411	. . . . 5 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (♯‘(𝑊 prefix 𝑀)) = 𝑀)
20		simpr 110	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
21		lencl 11035	. . . . . . . . 9 ⊢ (𝑈 ∈ Word 𝑉 → (♯‘𝑈) ∈ ℕ0)
22	21	adantl 277	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → (♯‘𝑈) ∈ ℕ0)
23		simpr 110	. . . . . . . 8 ⊢ ((𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)) → 𝑁 ≤ (♯‘𝑈))
24	20, 22, 23	3anim123i 1187	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑁 ∈ ℕ0 ∧ (♯‘𝑈) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝑈)))
25		elfz2nn0 10269	. . . . . . 7 ⊢ (𝑁 ∈ (0...(♯‘𝑈)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝑈) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝑈)))
26	24, 25	sylibr 134	. . . . . 6 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑁 ∈ (0...(♯‘𝑈)))
27		pfxlen 11176	. . . . . 6 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑈))) → (♯‘(𝑈 prefix 𝑁)) = 𝑁)
28	5, 26, 27	syl2anc 411	. . . . 5 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (♯‘(𝑈 prefix 𝑁)) = 𝑁)
29	19, 28	eqeq12d 2222	. . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((♯‘(𝑊 prefix 𝑀)) = (♯‘(𝑈 prefix 𝑁)) ↔ 𝑀 = 𝑁))
30	29	anbi1d 465	. . 3 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (((♯‘(𝑊 prefix 𝑀)) = (♯‘(𝑈 prefix 𝑁)) ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 𝑀)))((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 𝑀)))((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖))))
31	19	adantr 276	. . . . . . 7 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) → (♯‘(𝑊 prefix 𝑀)) = 𝑀)
32	31	oveq2d 5983	. . . . . 6 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) → (0..^(♯‘(𝑊 prefix 𝑀))) = (0..^𝑀))
33	32	raleqdv 2711	. . . . 5 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) → (∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 𝑀)))((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖) ↔ ∀𝑖 ∈ (0..^𝑀)((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖)))
34	1	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ Word 𝑉)
35	17	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ (0...(♯‘𝑊)))
36		simpr 110	. . . . . . . 8 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
37		pfxfv 11175	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(♯‘𝑊)) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊 prefix 𝑀)‘𝑖) = (𝑊‘𝑖))
38	34, 35, 36, 37	syl3anc 1250	. . . . . . 7 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊 prefix 𝑀)‘𝑖) = (𝑊‘𝑖))
39	5	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ Word 𝑉)
40	26	ad2antrr 488	. . . . . . . 8 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑁 ∈ (0...(♯‘𝑈)))
41		oveq2 5975	. . . . . . . . . . 11 ⊢ (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
42	41	eleq2d 2277	. . . . . . . . . 10 ⊢ (𝑀 = 𝑁 → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ (0..^𝑁)))
43	42	adantl 277	. . . . . . . . 9 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ (0..^𝑁)))
44	43	biimpa 296	. . . . . . . 8 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑁))
45		pfxfv 11175	. . . . . . . 8 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑈 prefix 𝑁)‘𝑖) = (𝑈‘𝑖))
46	39, 40, 44, 45	syl3anc 1250	. . . . . . 7 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑈 prefix 𝑁)‘𝑖) = (𝑈‘𝑖))
47	38, 46	eqeq12d 2222	. . . . . 6 ⊢ (((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖) ↔ (𝑊‘𝑖) = (𝑈‘𝑖)))
48	47	ralbidva 2504	. . . . 5 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) → (∀𝑖 ∈ (0..^𝑀)((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖)))
49	33, 48	bitrd 188	. . . 4 ⊢ ((((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) → (∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 𝑀)))((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖)))
50	49	pm5.32da 452	. . 3 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^(♯‘(𝑊 prefix 𝑀)))((𝑊 prefix 𝑀)‘𝑖) = ((𝑈 prefix 𝑁)‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖))))
51	10, 30, 50	3bitrd 214	. 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 prefix 𝑀) = (𝑈 prefix 𝑁) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖))))
52	51	3com12 1210	1 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 prefix 𝑀) = (𝑈 prefix 𝑁) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2178  ∀wral 2486   class class class wbr 4059  ‘cfv 5290  (class class class)co 5967  0cc0 7960   ≤ cle 8143  ℕ₀cn0 9330  ...cfz 10165  ..^cfzo 10299  ♯chash 10957  Word cword 11031   prefix cpfx 11163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300  df-ihash 10958  df-word 11032  df-substr 11137  df-pfx 11164

This theorem is referenced by:  pfxsuffeqwrdeq  11189