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Theorem subrval 43907
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 3490 . 2 (𝐴𝐶𝐴 ∈ V)
2 elex 3490 . 2 (𝐵𝐷𝐵 ∈ V)
3 fveq1 6899 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑣) = (𝐴𝑣))
4 fveq1 6899 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑣) = (𝐵𝑣))
53, 4oveqan12d 7443 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑣) − (𝑦𝑣)) = ((𝐴𝑣) − (𝐵𝑣)))
65mpteq2dv 5252 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥𝑣) − (𝑦𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
7 df-subr 43904 . . 3 -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥𝑣) − (𝑦𝑣))))
8 reex 11235 . . . 4 ℝ ∈ V
98mptex 7239 . . 3 (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))) ∈ V
106, 7, 9ovmpoa 7580 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
111, 2, 10syl2an 594 1 ((𝐴𝐶𝐵𝐷) → (𝐴-𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴𝑣) − (𝐵𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3471  cmpt 5233  cfv 6551  (class class class)co 7424  cr 11143  cmin 11480  -𝑟cminusr 43898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-cnex 11200  ax-resscn 11201
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-subr 43904
This theorem is referenced by:  subrfv  43910  subrfn  43913
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