ofcccat - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcccat		Structured version Visualization version GIF version

Theorem ofcccat 29748

Description: Letterwise operations on word concatenations. (Contributed by Thierry Arnoux, 5-Oct-2018.)

**Hypotheses**
Ref	Expression
ofcccat.1	⊢ (𝜑 → 𝐹 ∈ Word 𝑆)
ofcccat.2	⊢ (𝜑 → 𝐺 ∈ Word 𝑆)
ofcccat.3	⊢ (𝜑 → 𝐾 ∈ 𝑇)

**Assertion**
Ref	Expression
ofcccat	⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹∘_𝑓/𝑐𝑅𝐾) ++ (𝐺∘_𝑓/𝑐𝑅𝐾)))

**Proof of Theorem ofcccat**
Step	Hyp	Ref	Expression
1		ofcccat.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ Word 𝑆)
2		ofcccat.2	. . 3 ⊢ (𝜑 → 𝐺 ∈ Word 𝑆)
3		ofcccat.3	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑇)
4		fconst6g 5988	. . . 4 ⊢ (𝐾 ∈ 𝑇 → ((0..^(#‘𝐹)) × {𝐾}):(0..^(#‘𝐹))⟶𝑇)
5		iswrdi 13106	. . . 4 ⊢ (((0..^(#‘𝐹)) × {𝐾}):(0..^(#‘𝐹))⟶𝑇 → ((0..^(#‘𝐹)) × {𝐾}) ∈ Word 𝑇)
6	3, 4, 5	3syl 18	. . 3 ⊢ (𝜑 → ((0..^(#‘𝐹)) × {𝐾}) ∈ Word 𝑇)
7		fconst6g 5988	. . . 4 ⊢ (𝐾 ∈ 𝑇 → ((0..^(#‘𝐺)) × {𝐾}):(0..^(#‘𝐺))⟶𝑇)
8		iswrdi 13106	. . . 4 ⊢ (((0..^(#‘𝐺)) × {𝐾}):(0..^(#‘𝐺))⟶𝑇 → ((0..^(#‘𝐺)) × {𝐾}) ∈ Word 𝑇)
9	3, 7, 8	3syl 18	. . 3 ⊢ (𝜑 → ((0..^(#‘𝐺)) × {𝐾}) ∈ Word 𝑇)
10		fzofi 12586	. . . . 5 ⊢ (0..^(#‘𝐹)) ∈ Fin
11		snfi 7896	. . . . 5 ⊢ {𝐾} ∈ Fin
12		hashxp 13029	. . . . 5 ⊢ (((0..^(#‘𝐹)) ∈ Fin ∧ {𝐾} ∈ Fin) → (#‘((0..^(#‘𝐹)) × {𝐾})) = ((#‘(0..^(#‘𝐹))) · (#‘{𝐾})))
13	10, 11, 12	mp2an 703	. . . 4 ⊢ (#‘((0..^(#‘𝐹)) × {𝐾})) = ((#‘(0..^(#‘𝐹))) · (#‘{𝐾}))
14		wrdfin 13120	. . . . . . . 8 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹 ∈ Fin)
15		hashcl 12957	. . . . . . . 8 ⊢ (𝐹 ∈ Fin → (#‘𝐹) ∈ ℕ₀)
16	1, 14, 15	3syl 18	. . . . . . 7 ⊢ (𝜑 → (#‘𝐹) ∈ ℕ₀)
17		hashfzo0 13025	. . . . . . 7 ⊢ ((#‘𝐹) ∈ ℕ₀ → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
18	16, 17	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
19		hashsng 12968	. . . . . . 7 ⊢ (𝐾 ∈ 𝑇 → (#‘{𝐾}) = 1)
20	3, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘{𝐾}) = 1)
21	18, 20	oveq12d 6541	. . . . 5 ⊢ (𝜑 → ((#‘(0..^(#‘𝐹))) · (#‘{𝐾})) = ((#‘𝐹) · 1))
22	16	nn0cnd 11196	. . . . . 6 ⊢ (𝜑 → (#‘𝐹) ∈ ℂ)
23	22	mulid1d 9909	. . . . 5 ⊢ (𝜑 → ((#‘𝐹) · 1) = (#‘𝐹))
24	21, 23	eqtrd 2639	. . . 4 ⊢ (𝜑 → ((#‘(0..^(#‘𝐹))) · (#‘{𝐾})) = (#‘𝐹))
25	13, 24	syl5req 2652	. . 3 ⊢ (𝜑 → (#‘𝐹) = (#‘((0..^(#‘𝐹)) × {𝐾})))
26		fzofi 12586	. . . . 5 ⊢ (0..^(#‘𝐺)) ∈ Fin
27		hashxp 13029	. . . . 5 ⊢ (((0..^(#‘𝐺)) ∈ Fin ∧ {𝐾} ∈ Fin) → (#‘((0..^(#‘𝐺)) × {𝐾})) = ((#‘(0..^(#‘𝐺))) · (#‘{𝐾})))
28	26, 11, 27	mp2an 703	. . . 4 ⊢ (#‘((0..^(#‘𝐺)) × {𝐾})) = ((#‘(0..^(#‘𝐺))) · (#‘{𝐾}))
29		wrdfin 13120	. . . . . . . 8 ⊢ (𝐺 ∈ Word 𝑆 → 𝐺 ∈ Fin)
30		hashcl 12957	. . . . . . . 8 ⊢ (𝐺 ∈ Fin → (#‘𝐺) ∈ ℕ₀)
31	2, 29, 30	3syl 18	. . . . . . 7 ⊢ (𝜑 → (#‘𝐺) ∈ ℕ₀)
32		hashfzo0 13025	. . . . . . 7 ⊢ ((#‘𝐺) ∈ ℕ₀ → (#‘(0..^(#‘𝐺))) = (#‘𝐺))
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘(0..^(#‘𝐺))) = (#‘𝐺))
34	33, 20	oveq12d 6541	. . . . 5 ⊢ (𝜑 → ((#‘(0..^(#‘𝐺))) · (#‘{𝐾})) = ((#‘𝐺) · 1))
35	31	nn0cnd 11196	. . . . . 6 ⊢ (𝜑 → (#‘𝐺) ∈ ℂ)
36	35	mulid1d 9909	. . . . 5 ⊢ (𝜑 → ((#‘𝐺) · 1) = (#‘𝐺))
37	34, 36	eqtrd 2639	. . . 4 ⊢ (𝜑 → ((#‘(0..^(#‘𝐺))) · (#‘{𝐾})) = (#‘𝐺))
38	28, 37	syl5req 2652	. . 3 ⊢ (𝜑 → (#‘𝐺) = (#‘((0..^(#‘𝐺)) × {𝐾})))
39	1, 2, 6, 9, 25, 38	ofccat 13498	. 2 ⊢ (𝜑 → ((𝐹 ++ 𝐺) ∘_𝑓 𝑅(((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾}))) = ((𝐹 ∘_𝑓 𝑅((0..^(#‘𝐹)) × {𝐾})) ++ (𝐺 ∘_𝑓 𝑅((0..^(#‘𝐺)) × {𝐾}))))
40		ccatcl 13154	. . . . . 6 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐺 ∈ Word 𝑆) → (𝐹 ++ 𝐺) ∈ Word 𝑆)
41	1, 2, 40	syl2anc 690	. . . . 5 ⊢ (𝜑 → (𝐹 ++ 𝐺) ∈ Word 𝑆)
42		wrdf 13107	. . . . 5 ⊢ ((𝐹 ++ 𝐺) ∈ Word 𝑆 → (𝐹 ++ 𝐺):(0..^(#‘(𝐹 ++ 𝐺)))⟶𝑆)
43	41, 42	syl 17	. . . 4 ⊢ (𝜑 → (𝐹 ++ 𝐺):(0..^(#‘(𝐹 ++ 𝐺)))⟶𝑆)
44		ovex 6551	. . . . 5 ⊢ (0..^(#‘(𝐹 ++ 𝐺))) ∈ V
45	44	a1i 11	. . . 4 ⊢ (𝜑 → (0..^(#‘(𝐹 ++ 𝐺))) ∈ V)
46	43, 45, 3	ofcof 29298	. . 3 ⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹 ++ 𝐺) ∘_𝑓 𝑅((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾})))
47		ccatlen 13155	. . . . . . . 8 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐺 ∈ Word 𝑆) → (#‘(𝐹 ++ 𝐺)) = ((#‘𝐹) + (#‘𝐺)))
48	1, 2, 47	syl2anc 690	. . . . . . 7 ⊢ (𝜑 → (#‘(𝐹 ++ 𝐺)) = ((#‘𝐹) + (#‘𝐺)))
49	48	oveq2d 6539	. . . . . 6 ⊢ (𝜑 → (0..^(#‘(𝐹 ++ 𝐺))) = (0..^((#‘𝐹) + (#‘𝐺))))
50	49	xpeq1d 5048	. . . . 5 ⊢ (𝜑 → ((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾}) = ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾}))
51		eqid 2605	. . . . . 6 ⊢ ((0..^(#‘𝐹)) × {𝐾}) = ((0..^(#‘𝐹)) × {𝐾})
52		eqid 2605	. . . . . 6 ⊢ ((0..^(#‘𝐺)) × {𝐾}) = ((0..^(#‘𝐺)) × {𝐾})
53		eqid 2605	. . . . . 6 ⊢ ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾}) = ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾})
54	51, 52, 53, 3, 16, 31	ccatmulgnn0dir 29747	. . . . 5 ⊢ (𝜑 → (((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾})) = ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾}))
55	50, 54	eqtr4d 2642	. . . 4 ⊢ (𝜑 → ((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾}) = (((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾})))
56	55	oveq2d 6539	. . 3 ⊢ (𝜑 → ((𝐹 ++ 𝐺) ∘_𝑓 𝑅((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾})) = ((𝐹 ++ 𝐺) ∘_𝑓 𝑅(((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾}))))
57	46, 56	eqtrd 2639	. 2 ⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹 ++ 𝐺) ∘_𝑓 𝑅(((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾}))))
58		wrdf 13107	. . . . 5 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹:(0..^(#‘𝐹))⟶𝑆)
59	1, 58	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:(0..^(#‘𝐹))⟶𝑆)
60	10	a1i 11	. . . 4 ⊢ (𝜑 → (0..^(#‘𝐹)) ∈ Fin)
61	59, 60, 3	ofcof 29298	. . 3 ⊢ (𝜑 → (𝐹∘_𝑓/𝑐𝑅𝐾) = (𝐹 ∘_𝑓 𝑅((0..^(#‘𝐹)) × {𝐾})))
62		wrdf 13107	. . . . 5 ⊢ (𝐺 ∈ Word 𝑆 → 𝐺:(0..^(#‘𝐺))⟶𝑆)
63	2, 62	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:(0..^(#‘𝐺))⟶𝑆)
64	26	a1i 11	. . . 4 ⊢ (𝜑 → (0..^(#‘𝐺)) ∈ Fin)
65	63, 64, 3	ofcof 29298	. . 3 ⊢ (𝜑 → (𝐺∘_𝑓/𝑐𝑅𝐾) = (𝐺 ∘_𝑓 𝑅((0..^(#‘𝐺)) × {𝐾})))
66	61, 65	oveq12d 6541	. 2 ⊢ (𝜑 → ((𝐹∘_𝑓/𝑐𝑅𝐾) ++ (𝐺∘_𝑓/𝑐𝑅𝐾)) = ((𝐹 ∘_𝑓 𝑅((0..^(#‘𝐹)) × {𝐾})) ++ (𝐺 ∘_𝑓 𝑅((0..^(#‘𝐺)) × {𝐾}))))
67	39, 57, 66	3eqtr4d 2649	1 ⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹∘_𝑓/𝑐𝑅𝐾) ++ (𝐺∘_𝑓/𝑐𝑅𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1474 ∈ wcel 1975 Vcvv 3168 {csn 4120 × cxp 5022 ⟶wf 5782 ‘cfv 5786 (class class class)co 6523 ∘_𝑓 cof 6766 Fincfn 7814 0cc0 9788 1c1 9789 + caddc 9791 · cmul 9793 ℕ₀cn0 11135 ..^cfzo 12285 #chash 12930 Word cword 13088 ++ cconcat 13090 ∘_𝑓/𝑐cofc 29286

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1711 ax-4 1726 ax-5 1825 ax-6 1873 ax-7 1920 ax-8 1977 ax-9 1984 ax-10 2004 ax-11 2019 ax-12 2031 ax-13 2228 ax-ext 2585 ax-rep 4689 ax-sep 4699 ax-nul 4708 ax-pow 4760 ax-pr 4824 ax-un 6820 ax-cnex 9844 ax-resscn 9845 ax-1cn 9846 ax-icn 9847 ax-addcl 9848 ax-addrcl 9849 ax-mulcl 9850 ax-mulrcl 9851 ax-mulcom 9852 ax-addass 9853 ax-mulass 9854 ax-distr 9855 ax-i2m1 9856 ax-1ne0 9857 ax-1rid 9858 ax-rnegex 9859 ax-rrecex 9860 ax-cnre 9861 ax-pre-lttri 9862 ax-pre-lttrn 9863 ax-pre-ltadd 9864 ax-pre-mulgt0 9865

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-ex 1695 df-nf 1700 df-sb 1866 df-eu 2457 df-mo 2458 df-clab 2592 df-cleq 2598 df-clel 2601 df-nfc 2735 df-ne 2777 df-nel 2778 df-ral 2896 df-rex 2897 df-reu 2898 df-rmo 2899 df-rab 2900 df-v 3170 df-sbc 3398 df-csb 3495 df-dif 3538 df-un 3540 df-in 3542 df-ss 3549 df-pss 3551 df-nul 3870 df-if 4032 df-pw 4105 df-sn 4121 df-pr 4123 df-tp 4125 df-op 4127 df-uni 4363 df-int 4401 df-iun 4447 df-br 4574 df-opab 4634 df-mpt 4635 df-tr 4671 df-eprel 4935 df-id 4939 df-po 4945 df-so 4946 df-fr 4983 df-we 4985 df-xp 5030 df-rel 5031 df-cnv 5032 df-co 5033 df-dm 5034 df-rn 5035 df-res 5036 df-ima 5037 df-pred 5579 df-ord 5625 df-on 5626 df-lim 5627 df-suc 5628 df-iota 5750 df-fun 5788 df-fn 5789 df-f 5790 df-f1 5791 df-fo 5792 df-f1o 5793 df-fv 5794 df-riota 6485 df-ov 6526 df-oprab 6527 df-mpt2 6528 df-of 6768 df-om 6931 df-1st 7032 df-2nd 7033 df-wrecs 7267 df-recs 7328 df-rdg 7366 df-1o 7420 df-oadd 7424 df-er 7602 df-en 7815 df-dom 7816 df-sdom 7817 df-fin 7818 df-card 8621 df-cda 8846 df-pnf 9928 df-mnf 9929 df-xr 9930 df-ltxr 9931 df-le 9932 df-sub 10115 df-neg 10116 df-nn 10864 df-n0 11136 df-z 11207 df-uz 11516 df-fz 12149 df-fzo 12286 df-hash 12931 df-word 13096 df-concat 13098 df-ofc 29287

This theorem is referenced by: ofcs2 29750

W3C validator