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Mirrors > Home > MPE Home > Th. List > Mathboxes > frlmfzoccat | Structured version Visualization version GIF version |
Description: The concatenation of two vectors of dimension 𝑁 and 𝑀 forms a vector of dimension 𝑁 + 𝑀. (Contributed by SN, 31-Aug-2023.) |
Ref | Expression |
---|---|
frlmfzoccat.w | ⊢ 𝑊 = (𝐾 freeLMod (0..^𝐿)) |
frlmfzoccat.x | ⊢ 𝑋 = (𝐾 freeLMod (0..^𝑀)) |
frlmfzoccat.y | ⊢ 𝑌 = (𝐾 freeLMod (0..^𝑁)) |
frlmfzoccat.b | ⊢ 𝐵 = (Base‘𝑊) |
frlmfzoccat.c | ⊢ 𝐶 = (Base‘𝑋) |
frlmfzoccat.d | ⊢ 𝐷 = (Base‘𝑌) |
frlmfzoccat.k | ⊢ (𝜑 → 𝐾 ∈ 𝑍) |
frlmfzoccat.l | ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿) |
frlmfzoccat.m | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
frlmfzoccat.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
frlmfzoccat.u | ⊢ (𝜑 → 𝑈 ∈ 𝐶) |
frlmfzoccat.v | ⊢ (𝜑 → 𝑉 ∈ 𝐷) |
Ref | Expression |
---|---|
frlmfzoccat | ⊢ (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmfzoccat.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐶) | |
2 | frlmfzoccat.x | . . . . 5 ⊢ 𝑋 = (𝐾 freeLMod (0..^𝑀)) | |
3 | frlmfzoccat.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑋) | |
4 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | 2, 3, 4 | frlmfzowrd 39946 | . . . 4 ⊢ (𝑈 ∈ 𝐶 → 𝑈 ∈ Word (Base‘𝐾)) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ∈ Word (Base‘𝐾)) |
7 | frlmfzoccat.v | . . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝐷) | |
8 | frlmfzoccat.y | . . . . 5 ⊢ 𝑌 = (𝐾 freeLMod (0..^𝑁)) | |
9 | frlmfzoccat.d | . . . . 5 ⊢ 𝐷 = (Base‘𝑌) | |
10 | 8, 9, 4 | frlmfzowrd 39946 | . . . 4 ⊢ (𝑉 ∈ 𝐷 → 𝑉 ∈ Word (Base‘𝐾)) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑉 ∈ Word (Base‘𝐾)) |
12 | ccatcl 14129 | . . 3 ⊢ ((𝑈 ∈ Word (Base‘𝐾) ∧ 𝑉 ∈ Word (Base‘𝐾)) → (𝑈 ++ 𝑉) ∈ Word (Base‘𝐾)) | |
13 | 6, 11, 12 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑈 ++ 𝑉) ∈ Word (Base‘𝐾)) |
14 | ccatlen 14130 | . . . 4 ⊢ ((𝑈 ∈ Word (Base‘𝐾) ∧ 𝑉 ∈ Word (Base‘𝐾)) → (♯‘(𝑈 ++ 𝑉)) = ((♯‘𝑈) + (♯‘𝑉))) | |
15 | 6, 11, 14 | syl2anc 587 | . . 3 ⊢ (𝜑 → (♯‘(𝑈 ++ 𝑉)) = ((♯‘𝑈) + (♯‘𝑉))) |
16 | frlmfzoccat.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
17 | ovexd 7248 | . . . . . 6 ⊢ (𝜑 → (0..^𝑀) ∈ V) | |
18 | 2, 4, 3 | frlmbasf 20722 | . . . . . 6 ⊢ (((0..^𝑀) ∈ V ∧ 𝑈 ∈ 𝐶) → 𝑈:(0..^𝑀)⟶(Base‘𝐾)) |
19 | 17, 1, 18 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝑈:(0..^𝑀)⟶(Base‘𝐾)) |
20 | fnfzo0hash 14014 | . . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑈:(0..^𝑀)⟶(Base‘𝐾)) → (♯‘𝑈) = 𝑀) | |
21 | 16, 19, 20 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (♯‘𝑈) = 𝑀) |
22 | frlmfzoccat.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
23 | ovexd 7248 | . . . . . 6 ⊢ (𝜑 → (0..^𝑁) ∈ V) | |
24 | 8, 4, 9 | frlmbasf 20722 | . . . . . 6 ⊢ (((0..^𝑁) ∈ V ∧ 𝑉 ∈ 𝐷) → 𝑉:(0..^𝑁)⟶(Base‘𝐾)) |
25 | 23, 7, 24 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝑉:(0..^𝑁)⟶(Base‘𝐾)) |
26 | fnfzo0hash 14014 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑉:(0..^𝑁)⟶(Base‘𝐾)) → (♯‘𝑉) = 𝑁) | |
27 | 22, 25, 26 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (♯‘𝑉) = 𝑁) |
28 | 21, 27 | oveq12d 7231 | . . 3 ⊢ (𝜑 → ((♯‘𝑈) + (♯‘𝑉)) = (𝑀 + 𝑁)) |
29 | frlmfzoccat.l | . . 3 ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿) | |
30 | 15, 28, 29 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → (♯‘(𝑈 ++ 𝑉)) = 𝐿) |
31 | frlmfzoccat.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑍) | |
32 | 16, 22 | nn0addcld 12154 | . . . 4 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℕ0) |
33 | 29, 32 | eqeltrrd 2839 | . . 3 ⊢ (𝜑 → 𝐿 ∈ ℕ0) |
34 | frlmfzoccat.w | . . . 4 ⊢ 𝑊 = (𝐾 freeLMod (0..^𝐿)) | |
35 | frlmfzoccat.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
36 | 34, 35, 4 | frlmfzowrdb 39948 | . . 3 ⊢ ((𝐾 ∈ 𝑍 ∧ 𝐿 ∈ ℕ0) → ((𝑈 ++ 𝑉) ∈ 𝐵 ↔ ((𝑈 ++ 𝑉) ∈ Word (Base‘𝐾) ∧ (♯‘(𝑈 ++ 𝑉)) = 𝐿))) |
37 | 31, 33, 36 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝑈 ++ 𝑉) ∈ 𝐵 ↔ ((𝑈 ++ 𝑉) ∈ Word (Base‘𝐾) ∧ (♯‘(𝑈 ++ 𝑉)) = 𝐿))) |
38 | 13, 30, 37 | mpbir2and 713 | 1 ⊢ (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 0cc0 10729 + caddc 10732 ℕ0cn0 12090 ..^cfzo 13238 ♯chash 13896 Word cword 14069 ++ cconcat 14125 Basecbs 16760 freeLMod cfrlm 20708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-fzo 13239 df-hash 13897 df-word 14070 df-concat 14126 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-hom 16826 df-cco 16827 df-0g 16946 df-prds 16952 df-pws 16954 df-sra 20209 df-rgmod 20210 df-dsmm 20694 df-frlm 20709 |
This theorem is referenced by: frlmvscadiccat 39950 |
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