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Theorem addrval 42037
Description: Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
addrval ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
Distinct variable groups:   𝑣,𝐴   𝑣,𝐵
Allowed substitution hints:   𝐶(𝑣)   𝐷(𝑣)

Proof of Theorem addrval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3448 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3448 . 2 (𝐵𝐷𝐵 ∈ V)
3 fveq1 6767 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑣) = (𝐴𝑣))
4 fveq1 6767 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑣) = (𝐵𝑣))
53, 4oveqan12d 7287 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑣) + (𝑦𝑣)) = ((𝐴𝑣) + (𝐵𝑣)))
65mpteq2dv 5180 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥𝑣) + (𝑦𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
7 df-addr 42034 . . 3 +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥𝑣) + (𝑦𝑣))))
8 reex 10946 . . . 4 ℝ ∈ V
98mptex 7093 . . 3 (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))) ∈ V
106, 7, 9ovmpoa 7419 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
111, 2, 10syl2an 595 1 ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1541  wcel 2109  Vcvv 3430  cmpt 5161  cfv 6430  (class class class)co 7268  cr 10854   + caddc 10858  +𝑟cplusr 42028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-cnex 10911  ax-resscn 10912
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-addr 42034
This theorem is referenced by:  addrfv  42040  addrfn  42043
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