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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 2820 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | 2, 3, 4 | frlmfzowrd 39216 | . . . 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 39216 | . . . 4 ⊢ (𝑉 ∈ 𝐷 → 𝑉 ∈ Word (Base‘𝐾)) |
11 | 7, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑉 ∈ Word (Base‘𝐾)) |
12 | ccatcl 13922 | . . 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 13923 | . . . 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 7188 | . . . . . 6 ⊢ (𝜑 → (0..^𝑀) ∈ V) | |
18 | 2, 4, 3 | frlmbasf 20900 | . . . . . 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 13806 | . . . . 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 7188 | . . . . . 6 ⊢ (𝜑 → (0..^𝑁) ∈ V) | |
24 | 8, 4, 9 | frlmbasf 20900 | . . . . . 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 13806 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑉:(0..^𝑁)⟶(Base‘𝐾)) → (♯‘𝑉) = 𝑁) | |
27 | 22, 25, 26 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (♯‘𝑉) = 𝑁) |
28 | 21, 27 | oveq12d 7171 | . . 3 ⊢ (𝜑 → ((♯‘𝑈) + (♯‘𝑉)) = (𝑀 + 𝑁)) |
29 | frlmfzoccat.l | . . 3 ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿) | |
30 | 15, 28, 29 | 3eqtrd 2859 | . 2 ⊢ (𝜑 → (♯‘(𝑈 ++ 𝑉)) = 𝐿) |
31 | frlmfzoccat.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Ring) | |
32 | 16, 22 | nn0addcld 11957 | . . . 4 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℕ0) |
33 | 29, 32 | eqeltrrd 2913 | . . 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 39218 | . . 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 1536 ∈ wcel 2113 Vcvv 3493 ⟶wf 6348 ‘cfv 6352 (class class class)co 7153 0cc0 10534 + caddc 10537 ℕ0cn0 11895 ..^cfzo 13031 ♯chash 13688 Word cword 13859 ++ cconcat 13918 Basecbs 16479 Ringcrg 19293 freeLMod cfrlm 20886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-supp 7828 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-oadd 8103 df-er 8286 df-map 8405 df-ixp 8459 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-fsupp 8831 df-sup 8903 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-z 11980 df-dec 12097 df-uz 12242 df-fz 12891 df-fzo 13032 df-hash 13689 df-word 13860 df-concat 13919 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-sca 16577 df-vsca 16578 df-ip 16579 df-tset 16580 df-ple 16581 df-ds 16583 df-hom 16585 df-cco 16586 df-0g 16711 df-prds 16717 df-pws 16719 df-sra 19940 df-rgmod 19941 df-dsmm 20872 df-frlm 20887 |
This theorem is referenced by: frlmvscadiccat 39220 |
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