ofcccat - Mathbox for Thierry Arnoux

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Theorem ofcccat 29946

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 6007	. . . 4 ⊢ (𝐾 ∈ 𝑇 → ((0..^(#‘𝐹)) × {𝐾}):(0..^(#‘𝐹))⟶𝑇)
5		iswrdi 13164	. . . 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 6007	. . . 4 ⊢ (𝐾 ∈ 𝑇 → ((0..^(#‘𝐺)) × {𝐾}):(0..^(#‘𝐺))⟶𝑇)
8		iswrdi 13164	. . . 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 12635	. . . . 5 ⊢ (0..^(#‘𝐹)) ∈ Fin
11		snfi 7923	. . . . 5 ⊢ {𝐾} ∈ Fin
12		hashxp 13081	. . . . 5 ⊢ (((0..^(#‘𝐹)) ∈ Fin ∧ {𝐾} ∈ Fin) → (#‘((0..^(#‘𝐹)) × {𝐾})) = ((#‘(0..^(#‘𝐹))) · (#‘{𝐾})))
13	10, 11, 12	mp2an 704	. . . 4 ⊢ (#‘((0..^(#‘𝐹)) × {𝐾})) = ((#‘(0..^(#‘𝐹))) · (#‘{𝐾}))
14		wrdfin 13178	. . . . . . . 8 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹 ∈ Fin)
15		hashcl 13009	. . . . . . . 8 ⊢ (𝐹 ∈ Fin → (#‘𝐹) ∈ ℕ₀)
16	1, 14, 15	3syl 18	. . . . . . 7 ⊢ (𝜑 → (#‘𝐹) ∈ ℕ₀)
17		hashfzo0 13077	. . . . . . 7 ⊢ ((#‘𝐹) ∈ ℕ₀ → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
18	16, 17	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
19		hashsng 13020	. . . . . . 7 ⊢ (𝐾 ∈ 𝑇 → (#‘{𝐾}) = 1)
20	3, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘{𝐾}) = 1)
21	18, 20	oveq12d 6567	. . . . 5 ⊢ (𝜑 → ((#‘(0..^(#‘𝐹))) · (#‘{𝐾})) = ((#‘𝐹) · 1))
22	16	nn0cnd 11230	. . . . . 6 ⊢ (𝜑 → (#‘𝐹) ∈ ℂ)
23	22	mulid1d 9936	. . . . 5 ⊢ (𝜑 → ((#‘𝐹) · 1) = (#‘𝐹))
24	21, 23	eqtrd 2644	. . . 4 ⊢ (𝜑 → ((#‘(0..^(#‘𝐹))) · (#‘{𝐾})) = (#‘𝐹))
25	13, 24	syl5req 2657	. . 3 ⊢ (𝜑 → (#‘𝐹) = (#‘((0..^(#‘𝐹)) × {𝐾})))
26		fzofi 12635	. . . . 5 ⊢ (0..^(#‘𝐺)) ∈ Fin
27		hashxp 13081	. . . . 5 ⊢ (((0..^(#‘𝐺)) ∈ Fin ∧ {𝐾} ∈ Fin) → (#‘((0..^(#‘𝐺)) × {𝐾})) = ((#‘(0..^(#‘𝐺))) · (#‘{𝐾})))
28	26, 11, 27	mp2an 704	. . . 4 ⊢ (#‘((0..^(#‘𝐺)) × {𝐾})) = ((#‘(0..^(#‘𝐺))) · (#‘{𝐾}))
29		wrdfin 13178	. . . . . . . 8 ⊢ (𝐺 ∈ Word 𝑆 → 𝐺 ∈ Fin)
30		hashcl 13009	. . . . . . . 8 ⊢ (𝐺 ∈ Fin → (#‘𝐺) ∈ ℕ₀)
31	2, 29, 30	3syl 18	. . . . . . 7 ⊢ (𝜑 → (#‘𝐺) ∈ ℕ₀)
32		hashfzo0 13077	. . . . . . 7 ⊢ ((#‘𝐺) ∈ ℕ₀ → (#‘(0..^(#‘𝐺))) = (#‘𝐺))
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘(0..^(#‘𝐺))) = (#‘𝐺))
34	33, 20	oveq12d 6567	. . . . 5 ⊢ (𝜑 → ((#‘(0..^(#‘𝐺))) · (#‘{𝐾})) = ((#‘𝐺) · 1))
35	31	nn0cnd 11230	. . . . . 6 ⊢ (𝜑 → (#‘𝐺) ∈ ℂ)
36	35	mulid1d 9936	. . . . 5 ⊢ (𝜑 → ((#‘𝐺) · 1) = (#‘𝐺))
37	34, 36	eqtrd 2644	. . . 4 ⊢ (𝜑 → ((#‘(0..^(#‘𝐺))) · (#‘{𝐾})) = (#‘𝐺))
38	28, 37	syl5req 2657	. . 3 ⊢ (𝜑 → (#‘𝐺) = (#‘((0..^(#‘𝐺)) × {𝐾})))
39	1, 2, 6, 9, 25, 38	ofccat 13556	. 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 13212	. . . . . 6 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐺 ∈ Word 𝑆) → (𝐹 ++ 𝐺) ∈ Word 𝑆)
41	1, 2, 40	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝐹 ++ 𝐺) ∈ Word 𝑆)
42		wrdf 13165	. . . . 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 6577	. . . . 5 ⊢ (0..^(#‘(𝐹 ++ 𝐺))) ∈ V
45	44	a1i 11	. . . 4 ⊢ (𝜑 → (0..^(#‘(𝐹 ++ 𝐺))) ∈ V)
46	43, 45, 3	ofcof 29496	. . 3 ⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹 ++ 𝐺) ∘_𝑓 𝑅((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾})))
47		ccatlen 13213	. . . . . . . 8 ⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝐺 ∈ Word 𝑆) → (#‘(𝐹 ++ 𝐺)) = ((#‘𝐹) + (#‘𝐺)))
48	1, 2, 47	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (#‘(𝐹 ++ 𝐺)) = ((#‘𝐹) + (#‘𝐺)))
49	48	oveq2d 6565	. . . . . 6 ⊢ (𝜑 → (0..^(#‘(𝐹 ++ 𝐺))) = (0..^((#‘𝐹) + (#‘𝐺))))
50	49	xpeq1d 5062	. . . . 5 ⊢ (𝜑 → ((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾}) = ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾}))
51		eqid 2610	. . . . . 6 ⊢ ((0..^(#‘𝐹)) × {𝐾}) = ((0..^(#‘𝐹)) × {𝐾})
52		eqid 2610	. . . . . 6 ⊢ ((0..^(#‘𝐺)) × {𝐾}) = ((0..^(#‘𝐺)) × {𝐾})
53		eqid 2610	. . . . . 6 ⊢ ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾}) = ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾})
54	51, 52, 53, 3, 16, 31	ccatmulgnn0dir 29945	. . . . 5 ⊢ (𝜑 → (((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾})) = ((0..^((#‘𝐹) + (#‘𝐺))) × {𝐾}))
55	50, 54	eqtr4d 2647	. . . 4 ⊢ (𝜑 → ((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾}) = (((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾})))
56	55	oveq2d 6565	. . 3 ⊢ (𝜑 → ((𝐹 ++ 𝐺) ∘_𝑓 𝑅((0..^(#‘(𝐹 ++ 𝐺))) × {𝐾})) = ((𝐹 ++ 𝐺) ∘_𝑓 𝑅(((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾}))))
57	46, 56	eqtrd 2644	. 2 ⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹 ++ 𝐺) ∘_𝑓 𝑅(((0..^(#‘𝐹)) × {𝐾}) ++ ((0..^(#‘𝐺)) × {𝐾}))))
58		wrdf 13165	. . . . 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 29496	. . 3 ⊢ (𝜑 → (𝐹∘_𝑓/𝑐𝑅𝐾) = (𝐹 ∘_𝑓 𝑅((0..^(#‘𝐹)) × {𝐾})))
62		wrdf 13165	. . . . 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 29496	. . 3 ⊢ (𝜑 → (𝐺∘_𝑓/𝑐𝑅𝐾) = (𝐺 ∘_𝑓 𝑅((0..^(#‘𝐺)) × {𝐾})))
66	61, 65	oveq12d 6567	. 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 2654	1 ⊢ (𝜑 → ((𝐹 ++ 𝐺)∘_𝑓/𝑐𝑅𝐾) = ((𝐹∘_𝑓/𝑐𝑅𝐾) ++ (𝐺∘_𝑓/𝑐𝑅𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 Fincfn 7841 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℕ₀cn0 11169 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 ∘_𝑓/𝑐cofc 29484

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

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-of 6795 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-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-ofc 29485

This theorem is referenced by: ofcs2 29948

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