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Theorem addrval 39195
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 3364 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3364 . 2 (𝐵𝐷𝐵 ∈ V)
3 fveq1 6331 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑣) = (𝐴𝑣))
4 fveq1 6331 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑣) = (𝐵𝑣))
53, 4oveqan12d 6812 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑣) + (𝑦𝑣)) = ((𝐴𝑣) + (𝐵𝑣)))
65mpteq2dv 4879 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥𝑣) + (𝑦𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
7 df-addr 39192 . . 3 +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥𝑣) + (𝑦𝑣))))
8 reex 10229 . . . 4 ℝ ∈ V
98mptex 6630 . . 3 (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))) ∈ V
106, 7, 9ovmpt2a 6938 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
111, 2, 10syl2an 583 1 ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cmpt 4863  cfv 6031  (class class class)co 6793  cr 10137   + caddc 10141  +𝑟cplusr 39186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-cnex 10194  ax-resscn 10195
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-addr 39192
This theorem is referenced by:  addrfv  39198  addrfn  39201
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