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Theorem subrval 38180
Description: Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
subrval ((𝐴𝐶𝐵𝐷) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
Distinct variable groups:   𝑣,𝐴   𝑣,𝐵
Allowed substitution hints:   𝐶(𝑣)   𝐷(𝑣)

Proof of Theorem subrval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3201 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3201 . 2 (𝐵𝐷𝐵 ∈ V)
3 fveq1 6152 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑣) = (𝐴𝑣))
4 fveq1 6152 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑣) = (𝐵𝑣))
53, 4oveqan12d 6629 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑣) − (𝑦𝑣)) = ((𝐴𝑣) − (𝐵𝑣)))
65mpteq2dv 4710 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥𝑣) − (𝑦𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
7 df-subr 38177 . . 3 -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥𝑣) − (𝑦𝑣))))
8 reex 9978 . . . 4 ℝ ∈ V
98mptex 6446 . . 3 (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))) ∈ V
106, 7, 9ovmpt2a 6751 . 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 3189  cmpt 4678  cfv 5852  (class class class)co 6610  cr 9886  cmin 10217  -𝑟cminusr 38171
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 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-cnex 9943  ax-resscn 9944
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-subr 38177
This theorem is referenced by:  subrfv  38183  subrfn  38186
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