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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 20848 | . 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) |
10 | oveq1 6977 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗)) | |
11 | 10 | adantl 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑖𝑋𝑗) = (𝐼𝑋𝑗)) |
12 | 11 | oveq1d 6985 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) = ((𝐼𝑋𝑗) · (𝑌‘𝑗))) |
13 | 12 | mpteq2dv 5017 | . . 3 ⊢ ((𝜑 ∧ 𝑖 = 𝐼) → (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))) = (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))) |
14 | 13 | oveq2d 6986 | . 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 7004 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗)))) ∈ V) | |
17 | 9, 14, 15, 16 | fvmptd 6595 | 1 ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 Vcvv 3409 〈cop 4441 ↦ cmpt 5002 ‘cfv 6182 (class class class)co 6970 ↑𝑚 cmap 8198 Fincfn 8298 Basecbs 16329 .rcmulr 16412 Σg cgsu 16560 Mat cmat 20710 maVecMul cmvmul 20843 |
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 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
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 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-ot 4444 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-sup 8693 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-hom 16435 df-cco 16436 df-0g 16561 df-prds 16567 df-pws 16569 df-sra 19656 df-rgmod 19657 df-dsmm 20568 df-frlm 20583 df-mat 20711 df-mvmul 20844 |
This theorem is referenced by: mavmulass 20852 mulmarep1gsum2 20877 |
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