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Theorem addrval 38149
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 3198 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3198 . 2 (𝐵𝐷𝐵 ∈ V)
3 fveq1 6147 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑣) = (𝐴𝑣))
4 fveq1 6147 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑣) = (𝐵𝑣))
53, 4oveqan12d 6623 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑣) + (𝑦𝑣)) = ((𝐴𝑣) + (𝐵𝑣)))
65mpteq2dv 4705 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥𝑣) + (𝑦𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
7 df-addr 38146 . . 3 +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥𝑣) + (𝑦𝑣))))
8 reex 9971 . . . 4 ℝ ∈ V
98mptex 6440 . . 3 (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))) ∈ V
106, 7, 9ovmpt2a 6744 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
111, 2, 10syl2an 494 1 ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) + (𝐵𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cmpt 4673  cfv 5847  (class class class)co 6604  cr 9879   + caddc 9883  +𝑟cplusr 38140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-addr 38146
This theorem is referenced by:  addrfv  38152  addrfn  38155
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