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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 20351 | . 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
10 | ringcmn 18581 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) | |
11 | 5, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd) |
12 | 11 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CMnd) |
13 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin) |
14 | 5 | ad2antrr 762 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring) |
15 | 1, 3 | matbas2 20227 | . . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐵 ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴)) |
16 | 6, 5, 15 | syl2anc 693 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↑𝑚 (𝑁 × 𝑁)) = (Base‘𝐴)) |
17 | 7, 16 | eleqtrrd 2704 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑁))) |
18 | elmapi 7879 | . . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶𝐵) | |
19 | 17, 18 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(𝑁 × 𝑁)⟶𝐵) |
20 | 19 | ad2antrr 762 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑋:(𝑁 × 𝑁)⟶𝐵) |
21 | simpr 477 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁) | |
22 | 21 | adantr 481 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁) |
23 | simpr 477 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁) | |
24 | 20, 22, 23 | fovrnd 6806 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑋𝑗) ∈ 𝐵) |
25 | elmapi 7879 | . . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑𝑚 𝑁) → 𝑌:𝑁⟶𝐵) | |
26 | 8, 25 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌:𝑁⟶𝐵) |
27 | 26 | ad2antrr 762 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑌:𝑁⟶𝐵) |
28 | 27, 23 | ffvelrnd 6360 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵) |
29 | 3, 4 | ringcl 18561 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑌‘𝑗) ∈ 𝐵) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵) |
30 | 14, 24, 28, 29 | syl3anc 1326 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵) |
31 | 30 | ralrimiva 2966 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → ∀𝑗 ∈ 𝑁 ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵) |
32 | 3, 12, 13, 31 | gsummptcl 18366 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵) |
33 | 32 | ralrimiva 2966 | . . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵) |
34 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
35 | 3, 34 | eqeltri 2697 | . . . . 5 ⊢ 𝐵 ∈ V |
36 | elmapg 7870 | . . . . 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)) | |
37 | 35, 6, 36 | sylancr 695 | . . . 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)) |
38 | eqid 2622 | . . . . 5 ⊢ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) | |
39 | 38 | fmpt 6381 | . . . 4 ⊢ (∀𝑖 ∈ 𝑁 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵 ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))):𝑁⟶𝐵) |
40 | 37, 39 | syl6rbbr 279 | . . 3 ⊢ (𝜑 → (∀𝑖 ∈ 𝑁 (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵 ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑𝑚 𝑁))) |
41 | 33, 40 | mpbid 222 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑𝑚 𝑁)) |
42 | 9, 41 | eqeltrd 2701 | 1 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑𝑚 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 〈cop 4183 ↦ cmpt 4729 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Fincfn 7955 Basecbs 15857 .rcmulr 15942 Σg cgsu 16101 CMndccmn 18193 Ringcrg 18547 Mat cmat 20213 maVecMul cmvmul 20346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-sra 19172 df-rgmod 19173 df-dsmm 20076 df-frlm 20091 df-mat 20214 df-mvmul 20347 |
This theorem is referenced by: mavmulass 20355 slesolex 20488 |
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