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Theorem subrval 41164
 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 3462 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3462 . 2 (𝐵𝐷𝐵 ∈ V)
3 fveq1 6648 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑣) = (𝐴𝑣))
4 fveq1 6648 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑣) = (𝐵𝑣))
53, 4oveqan12d 7158 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑣) − (𝑦𝑣)) = ((𝐴𝑣) − (𝐵𝑣)))
65mpteq2dv 5129 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥𝑣) − (𝑦𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
7 df-subr 41161 . . 3 -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥𝑣) − (𝑦𝑣))))
8 reex 10621 . . . 4 ℝ ∈ V
98mptex 6967 . . 3 (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))) ∈ V
106, 7, 9ovmpoa 7288 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
111, 2, 10syl2an 598 1 ((𝐴𝐶𝐵𝐷) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ↦ cmpt 5113  ‘cfv 6328  (class class class)co 7139  ℝcr 10529   − cmin 10863  -𝑟cminusr 41155 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-cnex 10586  ax-resscn 10587 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-subr 41161 This theorem is referenced by:  subrfv  41167  subrfn  41170
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