cshwlen - Metamath Proof Explorer

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Theorem cshwlen 13396

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 6556	. . . . 5 ⊢ (𝑊 = ∅ → (𝑊 cyclShift 𝑁) = (∅ cyclShift 𝑁))
2		0csh0 13390	. . . . . 6 ⊢ (∅ cyclShift 𝑁) = ∅
3	2	a1i 11	. . . . 5 ⊢ (𝑊 = ∅ → (∅ cyclShift 𝑁) = ∅)
4		eqcom 2617	. . . . . 6 ⊢ (𝑊 = ∅ ↔ ∅ = 𝑊)
5	4	biimpi 205	. . . . 5 ⊢ (𝑊 = ∅ → ∅ = 𝑊)
6	1, 3, 5	3eqtrd 2648	. . . 4 ⊢ (𝑊 = ∅ → (𝑊 cyclShift 𝑁) = 𝑊)
7	6	fveq2d 6107	. . 3 ⊢ (𝑊 = ∅ → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))
8	7	a1d 25	. 2 ⊢ (𝑊 = ∅ → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊)))
9		cshword 13388	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
10	9	fveq2d 6107	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
11	10	adantr 480	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (#‘(𝑊 cyclShift 𝑁)) = (#‘((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
12		swrdcl 13271	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ∈ Word 𝑉)
13		swrdcl 13271	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩) ∈ Word 𝑉)
14		ccatlen 13213	. . . . . . 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 691	. . . . . 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 480	. . . . 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 480	. . . 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 13180	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ)
19		pm3.21 463	. . . . . . . . . . 11 ⊢ (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ))))
20	19	ex 449	. . . . . . . . . 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 449	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ → (𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ))))))
23	22	com24 93	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ∈ Word 𝑉 → (𝑁 ∈ ℤ → (𝑊 ≠ ∅ → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ))))))
24	23	pm2.43i 50	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑁 ∈ ℤ → (𝑊 ≠ ∅ → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))))
25	24	imp31 447	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)))
26		simpl 472	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → 𝑊 ∈ Word 𝑉)
27		pm3.22 464	. . . . . . . . . 10 ⊢ (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
28	27	adantl 481	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ))
29		zmodfzp1 12556	. . . . . . . . 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 13179	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ₀)
32		nn0fz0 12306	. . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ₀ ↔ (#‘𝑊) ∈ (0...(#‘𝑊)))
33	31, 32	sylib 207	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ (0...(#‘𝑊)))
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (#‘𝑊) ∈ (0...(#‘𝑊)))
35		swrdlen 13275	. . . . . . . 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 1318	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (#‘(𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩)) = ((#‘𝑊) − (𝑁 mod (#‘𝑊))))
37		zmodcl 12552	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → (𝑁 mod (#‘𝑊)) ∈ ℕ₀)
38	37	ancoms 468	. . . . . . . . . 10 ⊢ (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 mod (#‘𝑊)) ∈ ℕ₀)
39	38	adantl 481	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℕ₀)
40		0elfz 12305	. . . . . . . . 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 13275	. . . . . . . 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 1318	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (#‘(𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)) = ((𝑁 mod (#‘𝑊)) − 0))
44	36, 43	oveq12d 6567	. . . . . 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 11230	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ (#‘𝑊) ∈ ℕ) → (𝑁 mod (#‘𝑊)) ∈ ℂ)
46	45	ancoms 468	. . . . . . . . 9 ⊢ (((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ) → (𝑁 mod (#‘𝑊)) ∈ ℂ)
47	46	adantl 481	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (𝑁 mod (#‘𝑊)) ∈ ℂ)
48	47	subid1d 10260	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → ((𝑁 mod (#‘𝑊)) − 0) = (𝑁 mod (#‘𝑊)))
49	48	oveq2d 6565	. . . . . 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 11230	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℂ)
51		npcan 10169	. . . . . . 7 ⊢ (((#‘𝑊) ∈ ℂ ∧ (𝑁 mod (#‘𝑊)) ∈ ℂ) → (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + (𝑁 mod (#‘𝑊))) = (#‘𝑊))
52	50, 46, 51	syl2an 493	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) ∈ ℕ ∧ 𝑁 ∈ ℤ)) → (((#‘𝑊) − (𝑁 mod (#‘𝑊))) + (𝑁 mod (#‘𝑊))) = (#‘𝑊))
53	44, 49, 52	3eqtrd 2648	. . . . 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 2648	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) ∧ 𝑊 ≠ ∅) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))
56	55	expcom 450	. 2 ⊢ (𝑊 ≠ ∅ → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊)))
57	8, 56	pm2.61ine 2865	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 ⟨cop 4131 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 + caddc 9818 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 mod cmo 12530 #chash 12979 Word cword 13146 ++ cconcat 13148 substr csubstr 13150 cyclShift ccsh 13385

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 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 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-rmo 2904 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-sup 8231 df-inf 8232 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-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 df-csh 13386

This theorem is referenced by: cshwf 13397 2cshw 13410 lswcshw 13412 cshwleneq 13414 clwwisshclwwlem 26334 clwwnisshclwwn 26337 erclwwlkeqlen 26340 erclwwlkneqlen 26352 crctcshlem2 41021 clwwisshclwwslem 41234 clwwisshclwws 41235 clwwnisshclwwsn 41237 erclwwlkseqlen 41240 erclwwlksneqlen 41252 eucrct2eupth 41413

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