Theorem 2swrd1eqwrdeq 13306

Description: Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)

Assertion

Ref	Expression
2swrd1eqwrdeq	⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Proof of Theorem 2swrd1eqwrdeq

Step	Hyp	Ref	Expression
1		lencl 13179	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
2		nn0z 11277	. . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℤ)
3		elnnz 11264	. . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℤ ∧ 0 < (#‘𝑊)))
4	3	simplbi2 653	. . . . . . 7 ⊢ ((#‘𝑊) ∈ ℤ → (0 < (#‘𝑊) → (#‘𝑊) ∈ ℕ))
5	1, 2, 4	3syl 18	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) → (#‘𝑊) ∈ ℕ))
6	5	a1d 25	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑈 ∈ Word 𝑉 → (0 < (#‘𝑊) → (#‘𝑊) ∈ ℕ)))
7	6	3imp 1249	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (#‘𝑊) ∈ ℕ)
8		fzo0end 12426	. . . 4 ⊢ ((#‘𝑊) ∈ ℕ → ((#‘𝑊) − 1) ∈ (0..^(#‘𝑊)))
9	7, 8	syl 17	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ((#‘𝑊) − 1) ∈ (0..^(#‘𝑊)))
10		2swrdeqwrdeq 13305	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ ((#‘𝑊) − 1) ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩)))))
11	9, 10	syld3an3 1363	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩)))))
12		hashneq0 13016	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
13	12	biimpd 218	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) → 𝑊 ≠ ∅))
14	13	imdistani 722	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅))
15	14	3adant2 1073	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅))
16	15	adantr 480	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅))
17		swrdlsw 13304	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑊)”⟩)
18	16, 17	syl 17	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑊)”⟩)
19		breq2 4587	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → (0 < (#‘𝑊) ↔ 0 < (#‘𝑈)))
20	19	3anbi3d 1397	. . . . . . . . 9 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ↔ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈))))
21		hashneq0 13016	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ Word 𝑉 → (0 < (#‘𝑈) ↔ 𝑈 ≠ ∅))
22	21	biimpd 218	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ Word 𝑉 → (0 < (#‘𝑈) → 𝑈 ≠ ∅))
23	22	imdistani 722	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → (𝑈 ∈ Word 𝑉 ∧ 𝑈 ≠ ∅))
24	23	3adant1 1072	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → (𝑈 ∈ Word 𝑉 ∧ 𝑈 ≠ ∅))
25		swrdlsw 13304	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑈 ≠ ∅) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩)
26	24, 25	syl 17	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩)
27	20, 26	syl6bi 242	. . . . . . . 8 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩))
28	27	impcom 445	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩)
29		oveq1 6556	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → ((#‘𝑊) − 1) = ((#‘𝑈) − 1))
30		id 22	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → (#‘𝑊) = (#‘𝑈))
31	29, 30	opeq12d 4348	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → ⟨((#‘𝑊) − 1), (#‘𝑊)⟩ = ⟨((#‘𝑈) − 1), (#‘𝑈)⟩)
32	31	oveq2d 6565	. . . . . . . . 9 ⊢ ((#‘𝑊) = (#‘𝑈) → (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩))
33	32	eqeq1d 2612	. . . . . . . 8 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑈)”⟩ ↔ (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩))
34	33	adantl 481	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑈)”⟩ ↔ (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩))
35	28, 34	mpbird 246	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑈)”⟩)
36	18, 35	eqeq12d 2625	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) ↔ ⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩))
37		hashgt0n0 13017	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → 𝑊 ≠ ∅)
38		lswcl 13208	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → ( lastS ‘𝑊) ∈ 𝑉)
39	37, 38	syldan 486	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ( lastS ‘𝑊) ∈ 𝑉)
40	39	3adant2 1073	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ( lastS ‘𝑊) ∈ 𝑉)
41	40	adantr 480	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑊) ∈ 𝑉)
42		hashgt0n0 13017	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → 𝑈 ≠ ∅)
43		lswcl 13208	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑈 ≠ ∅) → ( lastS ‘𝑈) ∈ 𝑉)
44	42, 43	syldan 486	. . . . . . . . 9 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → ( lastS ‘𝑈) ∈ 𝑉)
45	44	3adant1 1072	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → ( lastS ‘𝑈) ∈ 𝑉)
46	20, 45	syl6bi 242	. . . . . . 7 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ( lastS ‘𝑈) ∈ 𝑉))
47	46	impcom 445	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑈) ∈ 𝑉)
48		s111 13248	. . . . . 6 ⊢ ((( lastS ‘𝑊) ∈ 𝑉 ∧ ( lastS ‘𝑈) ∈ 𝑉) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
49	41, 47, 48	syl2anc 691	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
50	36, 49	bitrd 267	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
51	50	anbi2d 736	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩)) ↔ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈))))
52	51	pm5.32da 671	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩))) ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))
53	11, 52	bitrd 267	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   < clt 9953   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147  ⟨“cs1 13149   substr csubstr 13150

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-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-s1 13157  df-substr 13158

This theorem is referenced by:  wwlkextinj  26258  clwwlkf1  26324  wwlksnextinj  41105  clwwlksf1  41224