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Theorem hvsubval 27857
Description: Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubval ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))

Proof of Theorem hvsubval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6654 . 2 (𝑥 = 𝐴 → (𝑥 + (-1 · 𝑦)) = (𝐴 + (-1 · 𝑦)))
2 oveq2 6655 . . 3 (𝑦 = 𝐵 → (-1 · 𝑦) = (-1 · 𝐵))
32oveq2d 6663 . 2 (𝑦 = 𝐵 → (𝐴 + (-1 · 𝑦)) = (𝐴 + (-1 · 𝐵)))
4 df-hvsub 27812 . 2 = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 + (-1 · 𝑦)))
5 ovex 6675 . 2 (𝐴 + (-1 · 𝐵)) ∈ V
61, 3, 4, 5ovmpt2 6793 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  (class class class)co 6647  1c1 9934  -cneg 10264  chil 27760   + cva 27761   · csm 27762   cmv 27766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-hvsub 27812
This theorem is referenced by:  hvsubcl  27858  hvsubvali  27861  hvsubid  27867  hvnegid  27868  hv2neg  27869  hvaddsubval  27874  hvsub4  27878  hvaddsub12  27879  hvpncan  27880  hvaddsubass  27882  hvsubass  27885  hvsubdistr1  27890  hvsubdistr2  27891  hvsubcan  27915  hvsub0  27917  his2sub  27933  hhph  28019  shsubcl  28061  shsel3  28158  honegsubi  28639  lnopsubi  28817  lnfnsubi  28889  superpos  29197  cdj1i  29276
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