Theorem pfxsuffeqwrdeq 40269

Description: Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 13305. (Contributed by AV, 5-May-2020.)

Assertion

Ref	Expression
pfxsuffeqwrdeq	⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))

Proof of Theorem pfxsuffeqwrdeq

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqwrd 13201	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖))))
2	1	3adant3 1074	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖))))
3		elfzofz 12354	. . . . . . . . 9 ⊢ (𝐼 ∈ (0..^(#‘𝑊)) → 𝐼 ∈ (0...(#‘𝑊)))
4		fzosplit 12370	. . . . . . . . 9 ⊢ (𝐼 ∈ (0...(#‘𝑊)) → (0..^(#‘𝑊)) = ((0..^𝐼) ∪ (𝐼..^(#‘𝑊))))
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝐼 ∈ (0..^(#‘𝑊)) → (0..^(#‘𝑊)) = ((0..^𝐼) ∪ (𝐼..^(#‘𝑊))))
6	5	3ad2ant3 1077	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (0..^(#‘𝑊)) = ((0..^𝐼) ∪ (𝐼..^(#‘𝑊))))
7	6	adantr 480	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (0..^(#‘𝑊)) = ((0..^𝐼) ∪ (𝐼..^(#‘𝑊))))
8	7	raleqdv 3121	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖) ↔ ∀𝑖 ∈ ((0..^𝐼) ∪ (𝐼..^(#‘𝑊)))(𝑊‘𝑖) = (𝑆‘𝑖)))
9		ralunb 3756	. . . . 5 ⊢ (∀𝑖 ∈ ((0..^𝐼) ∪ (𝐼..^(#‘𝑊)))(𝑊‘𝑖) = (𝑆‘𝑖) ↔ (∀𝑖 ∈ (0..^𝐼)(𝑊‘𝑖) = (𝑆‘𝑖) ∧ ∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖)))
10	8, 9	syl6bb 275	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖) ↔ (∀𝑖 ∈ (0..^𝐼)(𝑊‘𝑖) = (𝑆‘𝑖) ∧ ∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖))))
11		eqidd 2611	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → 𝐼 = 𝐼)
12	11	biantrurd 528	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^𝐼)(𝑊‘𝑖) = (𝑆‘𝑖) ↔ (𝐼 = 𝐼 ∧ ∀𝑖 ∈ (0..^𝐼)(𝑊‘𝑖) = (𝑆‘𝑖))))
13		3simpa 1051	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉))
14	13	adantr 480	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉))
15		elfzonn0 12380	. . . . . . . . . 10 ⊢ (𝐼 ∈ (0..^(#‘𝑊)) → 𝐼 ∈ ℕ0)
16	15, 15	jca 553	. . . . . . . . 9 ⊢ (𝐼 ∈ (0..^(#‘𝑊)) → (𝐼 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0))
17	16	3ad2ant3 1077	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (𝐼 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0))
18	17	adantr 480	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (𝐼 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0))
19		elfzo0le 12379	. . . . . . . . 9 ⊢ (𝐼 ∈ (0..^(#‘𝑊)) → 𝐼 ≤ (#‘𝑊))
20	19	3ad2ant3 1077	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → 𝐼 ≤ (#‘𝑊))
21	20	adantr 480	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → 𝐼 ≤ (#‘𝑊))
22		breq2 4587	. . . . . . . . . 10 ⊢ ((#‘𝑆) = (#‘𝑊) → (𝐼 ≤ (#‘𝑆) ↔ 𝐼 ≤ (#‘𝑊)))
23	22	eqcoms 2618	. . . . . . . . 9 ⊢ ((#‘𝑊) = (#‘𝑆) → (𝐼 ≤ (#‘𝑆) ↔ 𝐼 ≤ (#‘𝑊)))
24	23	adantl 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (𝐼 ≤ (#‘𝑆) ↔ 𝐼 ≤ (#‘𝑊)))
25	21, 24	mpbird 246	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → 𝐼 ≤ (#‘𝑆))
26		pfxeq 40267	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉) ∧ (𝐼 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) ∧ (𝐼 ≤ (#‘𝑊) ∧ 𝐼 ≤ (#‘𝑆))) → ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ↔ (𝐼 = 𝐼 ∧ ∀𝑖 ∈ (0..^𝐼)(𝑊‘𝑖) = (𝑆‘𝑖))))
27	14, 18, 21, 25, 26	syl112anc 1322	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ↔ (𝐼 = 𝐼 ∧ ∀𝑖 ∈ (0..^𝐼)(𝑊‘𝑖) = (𝑆‘𝑖))))
28	12, 27	bitr4d 270	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^𝐼)(𝑊‘𝑖) = (𝑆‘𝑖) ↔ (𝑊 prefix 𝐼) = (𝑆 prefix 𝐼)))
29		lencl 13179	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
30	29, 15	anim12i 588	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → ((#‘𝑊) ∈ ℕ0 ∧ 𝐼 ∈ ℕ0))
31	30	3adant2 1073	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → ((#‘𝑊) ∈ ℕ0 ∧ 𝐼 ∈ ℕ0))
32	31	ancomd 466	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (𝐼 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0))
33	32	adantr 480	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (𝐼 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0))
34	29	nn0red 11229	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℝ)
35	34	leidd 10473	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ≤ (#‘𝑊))
36	35	3ad2ant1 1075	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (#‘𝑊) ≤ (#‘𝑊))
37	36	adantr 480	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (#‘𝑊) ≤ (#‘𝑊))
38	34	3ad2ant1 1075	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (#‘𝑊) ∈ ℝ)
39		eqle 10018	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℝ ∧ (#‘𝑊) = (#‘𝑆)) → (#‘𝑊) ≤ (#‘𝑆))
40	38, 39	sylan 487	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (#‘𝑊) ≤ (#‘𝑆))
41		swrdspsleq 13301	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉) ∧ (𝐼 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) ∧ ((#‘𝑊) ≤ (#‘𝑊) ∧ (#‘𝑊) ≤ (#‘𝑆))) → ((𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩) ↔ ∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖)))
42	41	bicomd 212	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉) ∧ (𝐼 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) ∧ ((#‘𝑊) ≤ (#‘𝑊) ∧ (#‘𝑊) ≤ (#‘𝑆))) → (∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖) ↔ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))
43	14, 33, 37, 40, 42	syl112anc 1322	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖) ↔ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))
44	28, 43	anbi12d 743	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → ((∀𝑖 ∈ (0..^𝐼)(𝑊‘𝑖) = (𝑆‘𝑖) ∧ ∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖)) ↔ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩))))
45	10, 44	bitrd 267	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖) ↔ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩))))
46	45	pm5.32da 671	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (((#‘𝑊) = (#‘𝑆) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑆‘𝑖)) ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))
47	2, 46	bitrd 267	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∪ cun 3538  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815   ≤ cle 9954  ℕ₀cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   substr csubstr 13150   prefix cpfx 40244

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-fal 1481  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-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-substr 13158  df-pfx 40245

This theorem is referenced by:  pfxsuff1eqwrdeq  40270