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Theorem subrval 42839
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 3465 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3465 . 2 (𝐵𝐷𝐵 ∈ V)
3 fveq1 6845 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑣) = (𝐴𝑣))
4 fveq1 6845 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑣) = (𝐵𝑣))
53, 4oveqan12d 7380 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑣) − (𝑦𝑣)) = ((𝐴𝑣) − (𝐵𝑣)))
65mpteq2dv 5211 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥𝑣) − (𝑦𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
7 df-subr 42836 . . 3 -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥𝑣) − (𝑦𝑣))))
8 reex 11150 . . . 4 ℝ ∈ V
98mptex 7177 . . 3 (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))) ∈ V
106, 7, 9ovmpoa 7514 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
111, 2, 10syl2an 597 1 ((𝐴𝐶𝐵𝐷) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  cmpt 5192  cfv 6500  (class class class)co 7361  cr 11058  cmin 11393  -𝑟cminusr 42830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-cnex 11115  ax-resscn 11116
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-subr 42836
This theorem is referenced by:  subrfv  42842  subrfn  42845
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