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Mirrors > Home > MPE Home > Th. List > mavmulcl | Structured version Visualization version GIF version |
Description: Multiplication of an NxN matrix with an N-dimensional vector results in an N-dimensional vector. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 23-Feb-2019.) (Proof shortened by AV, 23-Jul-2019.) |
Ref | Expression |
---|---|
mavmulcl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mavmulcl.m | ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) |
mavmulcl.b | ⊢ 𝐵 = (Base‘𝑅) |
mavmulcl.t | ⊢ · = (.r‘𝑅) |
mavmulcl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mavmulcl.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
mavmulcl.x | ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) |
mavmulcl.y | ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) |
Ref | Expression |
---|---|
mavmulcl | ⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑𝑚 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mavmulcl.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | mavmulcl.m | . . 3 ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
3 | mavmulcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | mavmulcl.t | . . 3 ⊢ · = (.r‘𝑅) | |
5 | mavmulcl.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
6 | mavmulcl.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
7 | mavmulcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) | |
8 | mavmulcl.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mavmulval 20861 | . 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
10 | ringcmn 19057 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) | |
11 | 5, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd) |
12 | 11 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CMnd) |
13 | 6 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin) |
14 | 5 | ad2antrr 713 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring) |
15 | 1, 3 | matbas2 20737 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐵 ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴)) |
16 | 6, 5, 15 | syl2anc 576 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴)) |
17 | 7, 16 | eleqtrrd 2869 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑁))) |
18 | elmapi 8230 | . . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶𝐵) | |
19 | 17, 18 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(𝑁 × 𝑁)⟶𝐵) |
20 | 19 | ad2antrr 713 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑋:(𝑁 × 𝑁)⟶𝐵) |
21 | simpr 477 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁) | |
22 | 21 | adantr 473 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁) |
23 | simpr 477 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁) | |
24 | 20, 22, 23 | fovrnd 7138 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑋𝑗) ∈ 𝐵) |
25 | elmapi 8230 | . . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑𝑚 𝑁) → 𝑌:𝑁⟶𝐵) | |
26 | 8, 25 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌:𝑁⟶𝐵) |
27 | 26 | ad2antrr 713 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑌:𝑁⟶𝐵) |
28 | 27, 23 | ffvelrnd 6679 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵) |
29 | 3, 4 | ringcl 19037 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑌‘𝑗) ∈ 𝐵) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵) |
30 | 14, 24, 28, 29 | syl3anc 1351 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵) |
31 | 30 | ralrimiva 3132 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → ∀𝑗 ∈ 𝑁 ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵) |
32 | 3, 12, 13, 31 | gsummptcl 18843 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵) |
33 | 32 | ralrimiva 3132 | . . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵) |
34 | 3 | fvexi 6515 | . . . . 5 ⊢ 𝐵 ∈ V |
35 | elmapg 8221 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑁 ∈ Fin) → ((𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑𝑚 𝑁) ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))):𝑁⟶𝐵)) | |
36 | 34, 6, 35 | sylancr 578 | . . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑𝑚 𝑁) ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))):𝑁⟶𝐵)) |
37 | eqid 2778 | . . . . 5 ⊢ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) | |
38 | 37 | fmpt 6699 | . . . 4 ⊢ (∀𝑖 ∈ 𝑁 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵 ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))):𝑁⟶𝐵) |
39 | 36, 38 | syl6rbbr 282 | . . 3 ⊢ (𝜑 → (∀𝑖 ∈ 𝑁 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵 ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑𝑚 𝑁))) |
40 | 33, 39 | mpbid 224 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑𝑚 𝑁)) |
41 | 9, 40 | eqeltrd 2866 | 1 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑𝑚 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3088 Vcvv 3415 〈cop 4448 ↦ cmpt 5009 × cxp 5406 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ↑𝑚 cmap 8208 Fincfn 8308 Basecbs 16342 .rcmulr 16425 Σg cgsu 16573 CMndccmn 18669 Ringcrg 19023 Mat cmat 20723 maVecMul cmvmul 20856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-ot 4451 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-sup 8703 df-oi 8771 df-card 9164 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-z 11797 df-dec 11915 df-uz 12062 df-fz 12712 df-fzo 12853 df-seq 13188 df-hash 13509 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-hom 16448 df-cco 16449 df-0g 16574 df-gsum 16575 df-prds 16580 df-pws 16582 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-grp 17897 df-minusg 17898 df-cntz 18221 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-sra 19669 df-rgmod 19670 df-dsmm 20581 df-frlm 20596 df-mat 20724 df-mvmul 20857 |
This theorem is referenced by: mavmulass 20865 slesolex 20998 |
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