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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 ∈ Ring) |
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 2821 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | 2, 3, 4 | frlmfzowrd 39190 | . . . 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 39190 | . . . 4 ⊢ (𝑉 ∈ 𝐷 → 𝑉 ∈ Word (Base‘𝐾)) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑉 ∈ Word (Base‘𝐾)) |
12 | ccatcl 13926 | . . 3 ⊢ ((𝑈 ∈ Word (Base‘𝐾) ∧ 𝑉 ∈ Word (Base‘𝐾)) → (𝑈 ++ 𝑉) ∈ Word (Base‘𝐾)) | |
13 | 6, 11, 12 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑈 ++ 𝑉) ∈ Word (Base‘𝐾)) |
14 | ccatlen 13927 | . . . 4 ⊢ ((𝑈 ∈ Word (Base‘𝐾) ∧ 𝑉 ∈ Word (Base‘𝐾)) → (♯‘(𝑈 ++ 𝑉)) = ((♯‘𝑈) + (♯‘𝑉))) | |
15 | 6, 11, 14 | syl2anc 586 | . . 3 ⊢ (𝜑 → (♯‘(𝑈 ++ 𝑉)) = ((♯‘𝑈) + (♯‘𝑉))) |
16 | frlmfzoccat.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
17 | ovexd 7191 | . . . . . 6 ⊢ (𝜑 → (0..^𝑀) ∈ V) | |
18 | 2, 4, 3 | frlmbasf 20904 | . . . . . 6 ⊢ (((0..^𝑀) ∈ V ∧ 𝑈 ∈ 𝐶) → 𝑈:(0..^𝑀)⟶(Base‘𝐾)) |
19 | 17, 1, 18 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑈:(0..^𝑀)⟶(Base‘𝐾)) |
20 | fnfzo0hash 13809 | . . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑈:(0..^𝑀)⟶(Base‘𝐾)) → (♯‘𝑈) = 𝑀) | |
21 | 16, 19, 20 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (♯‘𝑈) = 𝑀) |
22 | frlmfzoccat.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
23 | ovexd 7191 | . . . . . 6 ⊢ (𝜑 → (0..^𝑁) ∈ V) | |
24 | 8, 4, 9 | frlmbasf 20904 | . . . . . 6 ⊢ (((0..^𝑁) ∈ V ∧ 𝑉 ∈ 𝐷) → 𝑉:(0..^𝑁)⟶(Base‘𝐾)) |
25 | 23, 7, 24 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑉:(0..^𝑁)⟶(Base‘𝐾)) |
26 | fnfzo0hash 13809 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑉:(0..^𝑁)⟶(Base‘𝐾)) → (♯‘𝑉) = 𝑁) | |
27 | 22, 25, 26 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (♯‘𝑉) = 𝑁) |
28 | 21, 27 | oveq12d 7174 | . . 3 ⊢ (𝜑 → ((♯‘𝑈) + (♯‘𝑉)) = (𝑀 + 𝑁)) |
29 | frlmfzoccat.l | . . 3 ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿) | |
30 | 15, 28, 29 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → (♯‘(𝑈 ++ 𝑉)) = 𝐿) |
31 | frlmfzoccat.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Ring) | |
32 | 16, 22 | nn0addcld 11960 | . . . 4 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℕ0) |
33 | 29, 32 | eqeltrrd 2914 | . . 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 39192 | . . 3 ⊢ ((𝐾 ∈ Ring ∧ 𝐿 ∈ ℕ0) → ((𝑈 ++ 𝑉) ∈ 𝐵 ↔ ((𝑈 ++ 𝑉) ∈ Word (Base‘𝐾) ∧ (♯‘(𝑈 ++ 𝑉)) = 𝐿))) |
37 | 31, 33, 36 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝑈 ++ 𝑉) ∈ 𝐵 ↔ ((𝑈 ++ 𝑉) ∈ Word (Base‘𝐾) ∧ (♯‘(𝑈 ++ 𝑉)) = 𝐿))) |
38 | 13, 30, 37 | mpbir2and 711 | 1 ⊢ (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 0cc0 10537 + caddc 10540 ℕ0cn0 11898 ..^cfzo 13034 ♯chash 13691 Word cword 13862 ++ cconcat 13922 Basecbs 16483 Ringcrg 19297 freeLMod cfrlm 20890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-sra 19944 df-rgmod 19945 df-dsmm 20876 df-frlm 20891 |
This theorem is referenced by: frlmvscadiccat 39194 |
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