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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 22566 | . 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
10 | ringcmn 20295 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) | |
11 | 5, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd) |
12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CMnd) |
13 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin) |
14 | 5 | ad2antrr 726 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring) |
15 | 1, 3 | matbas2 22442 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐵 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
16 | 6, 5, 15 | syl2anc 584 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
17 | 7, 16 | eleqtrrd 2841 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑁 × 𝑁))) |
18 | elmapi 8887 | . . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑m (𝑁 × 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶𝐵) | |
19 | 17, 18 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(𝑁 × 𝑁)⟶𝐵) |
20 | 19 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑋:(𝑁 × 𝑁)⟶𝐵) |
21 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁) | |
22 | 21 | adantr 480 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁) |
23 | simpr 484 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁) | |
24 | 20, 22, 23 | fovcdmd 7604 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑋𝑗) ∈ 𝐵) |
25 | elmapi 8887 | . . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑m 𝑁) → 𝑌:𝑁⟶𝐵) | |
26 | 8, 25 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌:𝑁⟶𝐵) |
27 | 26 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑌:𝑁⟶𝐵) |
28 | 27, 23 | ffvelcdmd 7104 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵) |
29 | 3, 4 | ringcl 20267 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑌‘𝑗) ∈ 𝐵) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵) |
30 | 14, 24, 28, 29 | syl3anc 1370 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵) |
31 | 30 | ralrimiva 3143 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → ∀𝑗 ∈ 𝑁 ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵) |
32 | 3, 12, 13, 31 | gsummptcl 19999 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵) |
33 | 32 | ralrimiva 3143 | . . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵) |
34 | eqid 2734 | . . . . 5 ⊢ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) | |
35 | 34 | fmpt 7129 | . . . 4 ⊢ (∀𝑖 ∈ 𝑁 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵 ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))):𝑁⟶𝐵) |
36 | 3 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V |
37 | elmapg 8877 | . . . . 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)) | |
38 | 36, 6, 37 | sylancr 587 | . . . 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)) |
39 | 35, 38 | bitr4id 290 | . . 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 232 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑m 𝑁)) |
41 | 9, 40 | eqeltrd 2838 | 1 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑m 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 〈cop 4636 ↦ cmpt 5230 × cxp 5686 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ↑m cmap 8864 Fincfn 8983 Basecbs 17244 .rcmulr 17298 Σg cgsu 17486 CMndccmn 19812 Ringcrg 20250 Mat cmat 22426 maVecMul cmvmul 22561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-ur 20199 df-ring 20252 df-sra 21189 df-rgmod 21190 df-dsmm 21769 df-frlm 21784 df-mat 22427 df-mvmul 22562 |
This theorem is referenced by: mavmulass 22570 slesolex 22703 |
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