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Mirrors > Home > MPE Home > Th. List > mavmulfv | Structured version Visualization version GIF version |
Description: A cell/element in the vector resulting from a multiplication of a vector with a square matrix. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 18-Feb-2019.) (Revised by AV, 23-Feb-2019.) |
Ref | Expression |
---|---|
mavmulval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mavmulval.m | ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) |
mavmulval.b | ⊢ 𝐵 = (Base‘𝑅) |
mavmulval.t | ⊢ · = (.r‘𝑅) |
mavmulval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
mavmulval.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
mavmulval.x | ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) |
mavmulval.y | ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m 𝑁)) |
mavmulfv.i | ⊢ (𝜑 → 𝐼 ∈ 𝑁) |
Ref | Expression |
---|---|
mavmulfv | ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mavmulval.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | mavmulval.m | . . 3 ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
3 | mavmulval.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | mavmulval.t | . . 3 ⊢ · = (.r‘𝑅) | |
5 | mavmulval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
6 | mavmulval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
7 | mavmulval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) | |
8 | mavmulval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m 𝑁)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mavmulval 21442 | . 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
10 | oveq1 7220 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗)) | |
11 | 10 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗)) |
12 | 11 | oveq1d 7228 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) = ((𝐼𝑋𝑗) · (𝑌‘𝑗))) |
13 | 12 | mpteq2dv 5151 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))) |
14 | 13 | oveq2d 7229 | . 2 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) |
15 | mavmulfv.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑁) | |
16 | ovexd 7248 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))) ∈ V) | |
17 | 9, 14, 15, 16 | fvmptd 6825 | 1 ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 〈cop 4547 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 Fincfn 8626 Basecbs 16760 .rcmulr 16803 Σg cgsu 16945 Mat cmat 21304 maVecMul cmvmul 21437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-ot 4550 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-hom 16826 df-cco 16827 df-0g 16946 df-prds 16952 df-pws 16954 df-sra 20209 df-rgmod 20210 df-dsmm 20694 df-frlm 20709 df-mat 21305 df-mvmul 21438 |
This theorem is referenced by: mavmulass 21446 mulmarep1gsum2 21471 |
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