cshwlen - Metamath Proof Explorer

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Theorem cshwlen 13545

Description: The length of a cyclically shifted word is the same as the length of the original word. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 27-Oct-2018.)

**Assertion**
Ref	Expression
cshwlen	⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))

**Proof of Theorem cshwlen**
Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . 5 ⊢ (𝑊 = ∅ → (𝑊 cyclShift 𝑁) = (∅ cyclShift 𝑁))
2		0csh0 13539	. . . . . 6 ⊢ (∅ cyclShift 𝑁) = ∅
3	2	a1i 11	. . . . 5 ⊢ (𝑊 = ∅ → (∅ cyclShift 𝑁) = ∅)
4		eqcom 2629	. . . . . 6 ⊢ (𝑊 = ∅ ↔ ∅ = 𝑊)
5	4	biimpi 206	. . . . 5 ⊢ (𝑊 = ∅ → ∅ = 𝑊)
6	1, 3, 5	3eqtrd 2660	. . . 4 ⊢ (𝑊 = ∅ → (𝑊 cyclShift 𝑁) = 𝑊)
7	6	fveq2d 6195	. . 3 ⊢ (𝑊 = ∅ → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))
8	7	a1d 25	. 2 ⊢ (𝑊 = ∅ → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊)))
9		cshword 13537	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
10	9	fveq2d 6195	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
11	10	adantr 481	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (#‘(𝑊 cyclShift 𝑁)) = (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
12		swrdcl 13419	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ∈ Word 𝑉)
13		swrdcl 13419	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩) ∈ Word 𝑉)
14		ccatlen 13360	. . . . . . 7 ⊢ (((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ∈ Word 𝑉 ∧ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩) ∈ Word 𝑉) → (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
15	12, 13, 14	syl2anc 693	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
16	15	adantr 481	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
17	16	adantr 481	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
18		lennncl 13325	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ)
19		pm3.21 464	. . . . . . . . . . 11 ⊢ (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ))))
20	19	ex 450	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ → (𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))))
21	18, 20	syl 17	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))))
22	21	ex 450	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ → (𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ))))))
23	22	com24 95	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 → (𝑁 ∈ ℤ → (𝑊 ≠ ∅ → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ))))))
24	23	pm2.43i 52	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑁 ∈ ℤ → (𝑊 ≠ ∅ → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))))
25	24	imp31 448	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))
26		simpl 473	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → 𝑊 ∈ Word 𝑉)
27		pm3.22 465	. . . . . . . . . 10 ⊢ (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
28	27	adantl 482	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
29		zmodfzp1 12694	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → (𝑁 mod (#‘𝑊)) ∈ (0...(#‘𝑊)))
30	28, 29	syl 17	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ (0...(#‘𝑊)))
31		lencl 13324	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ₀)
32		nn0fz0 12437	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ₀ ↔ (#‘𝑊) ∈ (0...(#‘𝑊)))
33	31, 32	sylib 208	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ (0...(#‘𝑊)))
34	33	adantr 481	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (#‘𝑊) ∈ (0...(#‘𝑊)))
35		swrdlen 13423	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑁 mod (#‘𝑊)) ∈ (0...(#‘𝑊)) ∧ (#‘𝑊) ∈ (0...(#‘𝑊))) → (#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) = ((#‘𝑊) − (𝑁 mod (#‘𝑊))))
36	26, 30, 34, 35	syl3anc 1326	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) = ((#‘𝑊) − (𝑁 mod (#‘𝑊))))
37		zmodcl 12690	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → (𝑁 mod (#‘𝑊)) ∈ ℕ₀)
38	37	ancoms 469	. . . . . . . . . 10 ⊢ (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 mod (#‘𝑊)) ∈ ℕ₀)
39	38	adantl 482	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℕ₀)
40		0elfz 12436	. . . . . . . . 9 ⊢ ((𝑁 mod (#‘𝑊)) ∈ ℕ₀ → 0 ∈ (0...(𝑁 mod (#‘𝑊))))
41	39, 40	syl 17	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → 0 ∈ (0...(𝑁 mod (#‘𝑊))))
42		swrdlen 13423	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0...(𝑁 mod (#‘𝑊))) ∧ (𝑁 mod (#‘𝑊)) ∈ (0...(#‘𝑊))) → (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)) = ((𝑁 mod (#‘𝑊)) − 0))
43	26, 41, 30, 42	syl3anc 1326	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)) = ((𝑁 mod (#‘𝑊)) − 0))
44	36, 43	oveq12d 6668	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + ((𝑁 mod (#‘𝑊)) − 0)))
45	37	nn0cnd 11353	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → (𝑁 mod (#‘𝑊)) ∈ ℂ)
46	45	ancoms 469	. . . . . . . . 9 ⊢ (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 mod (#‘𝑊)) ∈ ℂ)
47	46	adantl 482	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℂ)
48	47	subid1d 10381	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → ((𝑁 mod (#‘𝑊)) − 0) = (𝑁 mod (#‘𝑊)))
49	48	oveq2d 6666	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + ((𝑁 mod (#‘𝑊)) − 0)) = (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + (𝑁 mod (#‘𝑊))))
50	31	nn0cnd 11353	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℂ)
51		npcan 10290	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℂ ∧ (𝑁 mod (#‘𝑊)) ∈ ℂ) → (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + (𝑁 mod (#‘𝑊))) = (#‘𝑊))
52	50, 46, 51	syl2an 494	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + (𝑁 mod (#‘𝑊))) = (#‘𝑊))
53	44, 49, 52	3eqtrd 2660	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = (#‘𝑊))
54	25, 53	syl 17	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → ((#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))) = (#‘𝑊))
55	11, 17, 54	3eqtrd 2660	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))
56	55	expcom 451	. 2 ⊢ (𝑊 ≠ ∅ → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊)))
57	8, 56	pm2.61ine 2877	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 ⟨cop 4183 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 + caddc 9939 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ...cfz 12326 mod cmo 12668 #chash 13117 Word cword 13291 ++ cconcat 13293 substr csubstr 13295 cyclShift ccsh 13534

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 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-rmo 2920 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-sup 8348 df-inf 8349 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-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-hash 13118 df-word 13299 df-concat 13301 df-substr 13303 df-csh 13535

This theorem is referenced by: cshwf 13546 2cshw 13559 lswcshw 13561 cshwleneq 13563 crctcshlem2 26710 clwwisshclwwslem 26927 clwwisshclwws 26928 clwwnisshclwwsn 26930 erclwwlkseqlen 26933 erclwwlksneqlen 26945 eucrct2eupth 27105

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