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Theorem mvmulval 21157
 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 21156 . 2 (𝜑× = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
8 oveq 7157 . . . . . . 7 (𝑥 = 𝑋 → (𝑖𝑥𝑗) = (𝑖𝑋𝑗))
9 fveq1 6662 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝑗) = (𝑌𝑗))
108, 9oveqan12d 7170 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑖𝑥𝑗) · (𝑦𝑗)) = ((𝑖𝑋𝑗) · (𝑌𝑗)))
1110adantl 485 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑖𝑥𝑗) · (𝑦𝑗)) = ((𝑖𝑋𝑗) · (𝑌𝑗)))
1211mpteq2dv 5149 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))) = (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))
1312oveq2d 7167 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗)))))
1413mpteq2dv 5149 . 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))
175mptexd 6980 . 2 (𝜑 → (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))) ∈ V)
187, 14, 15, 16, 17ovmpod 7297 1 (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3480  ⟨cop 4556   ↦ cmpt 5133   × cxp 5541  ‘cfv 6345  (class class class)co 7151   ↑m cmap 8404  Fincfn 8507  Basecbs 16485  .rcmulr 16568   Σg cgsu 16716   maVecMul cmvmul 21154 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-mvmul 21155 This theorem is referenced by:  mvmulfv  21158  mavmulval  21159
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