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Theorem mvmulval 20522
Description: Multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.)
Hypotheses
Ref Expression
mvmulfval.x × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
mvmulfval.b 𝐵 = (Base‘𝑅)
mvmulfval.t · = (.r𝑅)
mvmulfval.r (𝜑𝑅𝑉)
mvmulfval.m (𝜑𝑀 ∈ Fin)
mvmulfval.n (𝜑𝑁 ∈ Fin)
mvmulval.x (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))
mvmulval.y (𝜑𝑌 ∈ (𝐵𝑚 𝑁))
Assertion
Ref Expression
mvmulval (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
Distinct variable groups:   𝑖,𝑗,𝜑   𝑖,𝑀,𝑗   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   · ,𝑖   𝑖,𝑋,𝑗   𝑖,𝑌,𝑗
Allowed substitution hints:   𝐵(𝑖,𝑗)   · (𝑗)   × (𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem mvmulval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvmulfval.x . . 3 × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
2 mvmulfval.b . . 3 𝐵 = (Base‘𝑅)
3 mvmulfval.t . . 3 · = (.r𝑅)
4 mvmulfval.r . . 3 (𝜑𝑅𝑉)
5 mvmulfval.m . . 3 (𝜑𝑀 ∈ Fin)
6 mvmulfval.n . . 3 (𝜑𝑁 ∈ Fin)
71, 2, 3, 4, 5, 6mvmulfval 20521 . 2 (𝜑× = (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
8 oveq 6807 . . . . . . 7 (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗))
9 fveq1 6339 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝑗) = (𝑌𝑗))
108, 9oveqan12d 6820 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑦𝑗)) = ((𝑖𝑋𝑗) · (𝑌𝑗)))
1110adantl 473 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑦𝑗)) = ((𝑖𝑋𝑗) · (𝑌𝑗)))
1211mpteq2dv 4885 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))) = (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))
1312oveq2d 6817 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗)))))
1413mpteq2dv 4885 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
15 mvmulval.x . 2 (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))
16 mvmulval.y . 2 (𝜑𝑌 ∈ (𝐵𝑚 𝑁))
17 mptexg 6636 . . 3 (𝑀 ∈ Fin → (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))) ∈ V)
185, 17syl 17 . 2 (𝜑 → (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))) ∈ V)
197, 14, 15, 16, 18ovmpt2d 6941 1 (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  Vcvv 3328  cop 4315  cmpt 4869   × cxp 5252  cfv 6037  (class class class)co 6801  𝑚 cmap 8011  Fincfn 8109  Basecbs 16030  .rcmulr 16115   Σg cgsu 16274   maVecMul cmvmul 20519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-mvmul 20520
This theorem is referenced by:  mvmulfv  20523  mavmulval  20524
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