ofcccat - Mathbox for Thierry Arnoux

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Theorem ofcccat 30620

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 6094	. . . 4 ⊢ (𝐾 ∈ 𝑇 → ((0..^(#‘𝐹)) × {𝐾}):(0..^(#‘𝐹))⟶𝑇)
5		iswrdi 13309	. . . 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 6094	. . . 4 ⊢ (𝐾 ∈ 𝑇 → ((0..^(#‘𝐺)) × {𝐾}):(0..^(#‘𝐺))⟶𝑇)
8		iswrdi 13309	. . . 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 12773	. . . . 5 ⊢ (0..^(#‘𝐹)) ∈ Fin
11		snfi 8038	. . . . 5 ⊢ {𝐾} ∈ Fin
12		hashxp 13221	. . . . 5 ⊢ (((0..^(#‘𝐹)) ∈ Fin ∧ {𝐾} ∈ Fin) → (#‘((0..^(#‘𝐹)) × {𝐾})) = ((#‘(0..^(#‘𝐹))) · (#‘{𝐾})))
13	10, 11, 12	mp2an 708	. . . 4 ⊢ (#‘((0..^(#‘𝐹)) × {𝐾})) = ((#‘(0..^(#‘𝐹))) · (#‘{𝐾}))
14		wrdfin 13323	. . . . . . . 8 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹 ∈ Fin)
15		hashcl 13147	. . . . . . . 8 ⊢ (𝐹 ∈ Fin → (#‘𝐹) ∈ ℕ₀)
16	1, 14, 15	3syl 18	. . . . . . 7 ⊢ (𝜑 → (#‘𝐹) ∈ ℕ₀)
17		hashfzo0 13217	. . . . . . 7 ⊢ ((#‘𝐹) ∈ ℕ₀ → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
18	16, 17	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
19		hashsng 13159	. . . . . . 7 ⊢ (𝐾 ∈ 𝑇 → (#‘{𝐾}) = 1)
20	3, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘{𝐾}) = 1)
21	18, 20	oveq12d 6668	. . . . 5 ⊢ (𝜑 → ((#‘(0..^(#‘𝐹))) · (#‘{𝐾})) = ((#‘𝐹) · 1))
22	16	nn0cnd 11353	. . . . . 6 ⊢ (𝜑 → (#‘𝐹) ∈ ℂ)
23	22	mulid1d 10057	. . . . 5 ⊢ (𝜑 → ((#‘𝐹) · 1) = (#‘𝐹))
24	21, 23	eqtrd 2656	. . . 4 ⊢ (𝜑 → ((#‘(0..^(#‘𝐹))) · (#‘{𝐾})) = (#‘𝐹))
25	13, 24	syl5req 2669	. . 3 ⊢ (𝜑 → (#‘𝐹) = (#‘((0..^(#‘𝐹)) × {𝐾})))
26		fzofi 12773	. . . . 5 ⊢ (0..^(#‘𝐺)) ∈ Fin
27		hashxp 13221	. . . . 5 ⊢ (((0..^(#‘𝐺)) ∈ Fin ∧ {𝐾} ∈ Fin) → (#‘((0..^(#‘𝐺)) × {𝐾})) = ((#‘(0..^(#‘𝐺))) · (#‘{𝐾})))
28	26, 11, 27	mp2an 708	. . . 4 ⊢ (#‘((0..^(#‘𝐺)) × {𝐾})) = ((#‘(0..^(#‘𝐺))) · (#‘{𝐾}))
29		wrdfin 13323	. . . . . . . 8 ⊢ (𝐺 ∈ Word 𝑆 → 𝐺 ∈ Fin)
30		hashcl 13147	. . . . . . . 8 ⊢ (𝐺 ∈ Fin → (#‘𝐺) ∈ ℕ₀)
31	2, 29, 30	3syl 18	. . . . . . 7 ⊢ (𝜑 → (#‘𝐺) ∈ ℕ₀)
32		hashfzo0 13217	. . . . . . 7 ⊢ ((#‘𝐺) ∈ ℕ₀ → (#‘(0..^(#‘𝐺))) = (#‘𝐺))
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘(0..^(#‘𝐺))) = (#‘𝐺))
34	33, 20	oveq12d 6668	. . . . 5 ⊢ (𝜑 → ((#‘(0..^(#‘𝐺))) · (#‘{𝐾})) = ((#‘𝐺) · 1))
35	31	nn0cnd 11353	. . . . . 6 ⊢ (𝜑 → (#‘𝐺) ∈ ℂ)
36	35	mulid1d 10057	. . . . 5 ⊢ (𝜑 → ((#‘𝐺) · 1) = (#‘𝐺))
37	34, 36	eqtrd 2656	. . . 4 ⊢ (𝜑 → ((#‘(0..^(#‘𝐺))) · (#‘{𝐾})) = (#‘𝐺))
38	28, 37	syl5req 2669	. . 3 ⊢ (𝜑 → (#‘𝐺) = (#‘((0..^(#‘𝐺)) × {𝐾})))
39	1, 2, 6, 9, 25, 38	ofccat 13708	. 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 13359	. . . . . 6 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐺 ∈ Word 𝑆) → (𝐹 ++ 𝐺) ∈ Word 𝑆)
41	1, 2, 40	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐹 ++ 𝐺) ∈ Word 𝑆)
42		wrdf 13310	. . . . 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		ovexd 6680	. . . 4 ⊢ (𝜑 → (0..^(#‘(𝐹 ++ 𝐺))) ∈ V)
45	43, 44, 3	ofcof 30169	. . 3 ⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹 ++ 𝐺) ∘_𝑓 𝑅((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾})))
46		ccatlen 13360	. . . . . . . 8 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐺 ∈ Word 𝑆) → (#‘(𝐹 ++ 𝐺)) = ((#‘𝐹) + (#‘𝐺)))
47	1, 2, 46	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (#‘(𝐹 ++ 𝐺)) = ((#‘𝐹) + (#‘𝐺)))
48	47	oveq2d 6666	. . . . . 6 ⊢ (𝜑 → (0..^(#‘(𝐹 ++ 𝐺))) = (0..^((#‘𝐹) + (#‘𝐺))))
49	48	xpeq1d 5138	. . . . 5 ⊢ (𝜑 → ((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾}) = ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾}))
50		eqid 2622	. . . . . 6 ⊢ ((0..^(#‘𝐹)) × {𝐾}) = ((0..^(#‘𝐹)) × {𝐾})
51		eqid 2622	. . . . . 6 ⊢ ((0..^(#‘𝐺)) × {𝐾}) = ((0..^(#‘𝐺)) × {𝐾})
52		eqid 2622	. . . . . 6 ⊢ ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾}) = ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾})
53	50, 51, 52, 3, 16, 31	ccatmulgnn0dir 30619	. . . . 5 ⊢ (𝜑 → (((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾})) = ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾}))
54	49, 53	eqtr4d 2659	. . . 4 ⊢ (𝜑 → ((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾}) = (((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾})))
55	54	oveq2d 6666	. . 3 ⊢ (𝜑 → ((𝐹 ++ 𝐺) ∘_𝑓 𝑅((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾})) = ((𝐹 ++ 𝐺) ∘_𝑓 𝑅(((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾}))))
56	45, 55	eqtrd 2656	. 2 ⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹 ++ 𝐺) ∘_𝑓 𝑅(((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾}))))
57		wrdf 13310	. . . . 5 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹:(0..^(#‘𝐹))⟶𝑆)
58	1, 57	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:(0..^(#‘𝐹))⟶𝑆)
59	10	a1i 11	. . . 4 ⊢ (𝜑 → (0..^(#‘𝐹)) ∈ Fin)
60	58, 59, 3	ofcof 30169	. . 3 ⊢ (𝜑 → (𝐹∘_𝑓/𝑐𝑅𝐾) = (𝐹 ∘_𝑓 𝑅((0..^(#‘𝐹)) × {𝐾})))
61		wrdf 13310	. . . . 5 ⊢ (𝐺 ∈ Word 𝑆 → 𝐺:(0..^(#‘𝐺))⟶𝑆)
62	2, 61	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:(0..^(#‘𝐺))⟶𝑆)
63	26	a1i 11	. . . 4 ⊢ (𝜑 → (0..^(#‘𝐺)) ∈ Fin)
64	62, 63, 3	ofcof 30169	. . 3 ⊢ (𝜑 → (𝐺∘_𝑓/𝑐𝑅𝐾) = (𝐺 ∘_𝑓 𝑅((0..^(#‘𝐺)) × {𝐾})))
65	60, 64	oveq12d 6668	. 2 ⊢ (𝜑 → ((𝐹∘_𝑓/𝑐𝑅𝐾) ++ (𝐺∘_𝑓/𝑐𝑅𝐾)) = ((𝐹 ∘_𝑓 𝑅((0..^(#‘𝐹)) × {𝐾})) ++ (𝐺 ∘_𝑓 𝑅((0..^(#‘𝐺)) × {𝐾}))))
66	39, 56, 65	3eqtr4d 2666	1 ⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹∘_𝑓/𝑐𝑅𝐾) ++ (𝐺∘_𝑓/𝑐𝑅𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 Fincfn 7955 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℕ₀cn0 11292 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 ∘_𝑓/𝑐cofc 30157

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-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-of 6897 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-cda 8990 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-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-ofc 30158

This theorem is referenced by: ofcs2 30622

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