Theorem 2swrd1eqwrdeq 13454

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 13324	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
2		nn0z 11400	. . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℤ)
3		elnnz 11387	. . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℤ ∧ 0 < (#‘𝑊)))
4	3	simplbi2 655	. . . . . . 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 1256	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (#‘𝑊) ∈ ℕ)
8		fzo0end 12560	. . . 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 13453	. . 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 1371	. 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 13155	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
13	12	biimpd 219	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) → 𝑊 ≠ ∅))
14	13	imdistani 726	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅))
15	14	3adant2 1080	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅))
16	15	adantr 481	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅))
17		swrdlsw 13452	. . . . . . 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 4657	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → (0 < (#‘𝑊) ↔ 0 < (#‘𝑈)))
20	19	3anbi3d 1405	. . . . . . . . 9 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ↔ (𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈))))
21		hashneq0 13155	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ Word 𝑉 → (0 < (#‘𝑈) ↔ 𝑈 ≠ ∅))
22	21	biimpd 219	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ Word 𝑉 → (0 < (#‘𝑈) → 𝑈 ≠ ∅))
23	22	imdistani 726	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → (𝑈 ∈ Word 𝑉 ∧ 𝑈 ≠ ∅))
24	23	3adant1 1079	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → (𝑈 ∈ Word 𝑉 ∧ 𝑈 ≠ ∅))
25		swrdlsw 13452	. . . . . . . . . 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 243	. . . . . . . 8 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩))
28	27	impcom 446	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩)
29		oveq1 6657	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → ((#‘𝑊) − 1) = ((#‘𝑈) − 1))
30		id 22	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = (#‘𝑈) → (#‘𝑊) = (#‘𝑈))
31	29, 30	opeq12d 4410	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (#‘𝑈) → ⟨((#‘𝑊) − 1), (#‘𝑊)⟩ = ⟨((#‘𝑈) − 1), (#‘𝑈)⟩)
32	31	oveq2d 6666	. . . . . . . . 9 ⊢ ((#‘𝑊) = (#‘𝑈) → (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩))
33	32	eqeq1d 2624	. . . . . . . 8 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑈)”⟩ ↔ (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩))
34	33	adantl 482	. . . . . . 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 247	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑈)”⟩)
36	18, 35	eqeq12d 2637	. . . . 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 13156	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → 𝑊 ≠ ∅)
38		lswcl 13355	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → ( lastS ‘𝑊) ∈ 𝑉)
39	37, 38	syldan 487	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ( lastS ‘𝑊) ∈ 𝑉)
40	39	3adant2 1080	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ( lastS ‘𝑊) ∈ 𝑉)
41	40	adantr 481	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑊) ∈ 𝑉)
42		hashgt0n0 13156	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → 𝑈 ≠ ∅)
43		lswcl 13355	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑈 ≠ ∅) → ( lastS ‘𝑈) ∈ 𝑉)
44	42, 43	syldan 487	. . . . . . . . 9 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → ( lastS ‘𝑈) ∈ 𝑉)
45	44	3adant1 1079	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → ( lastS ‘𝑈) ∈ 𝑉)
46	20, 45	syl6bi 243	. . . . . . 7 ⊢ ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ( lastS ‘𝑈) ∈ 𝑉))
47	46	impcom 446	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑈) ∈ 𝑉)
48		s111 13395	. . . . . 6 ⊢ ((( lastS ‘𝑊) ∈ 𝑉 ∧ ( lastS ‘𝑈) ∈ 𝑉) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
49	41, 47, 48	syl2anc 693	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
50	36, 49	bitrd 268	. . . 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 740	. . 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 673	. 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 268	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915  ⟨cop 4183   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   < clt 10074   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ..^cfzo 12465  #chash 13117  Word cword 13291   lastS clsw 13292  ⟨“cs1 13294   substr csubstr 13295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-s1 13302  df-substr 13303

This theorem is referenced by:  wwlksnextinj  26794  clwwlksf1  26917