HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvsubval Structured version   Visualization version   GIF version

Theorem hvsubval 28788
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 7145 . 2 (𝑥 = 𝐴 → (𝑥 + (-1 · 𝑦)) = (𝐴 + (-1 · 𝑦)))
2 oveq2 7146 . . 3 (𝑦 = 𝐵 → (-1 · 𝑦) = (-1 · 𝐵))
32oveq2d 7154 . 2 (𝑦 = 𝐵 → (𝐴 + (-1 · 𝑦)) = (𝐴 + (-1 · 𝐵)))
4 df-hvsub 28743 . 2 = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 + (-1 · 𝑦)))
5 ovex 7171 . 2 (𝐴 + (-1 · 𝐵)) ∈ V
61, 3, 4, 5ovmpo 7292 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  (class class class)co 7138  1c1 10523  -cneg 10856  chba 28691   + cva 28692   · csm 28693   cmv 28697
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-hvsub 28743
This theorem is referenced by:  hvsubcl  28789  hvsubvali  28792  hvsubid  28798  hvnegid  28799  hv2neg  28800  hvaddsubval  28805  hvsub4  28809  hvaddsub12  28810  hvpncan  28811  hvaddsubass  28813  hvsubass  28816  hvsubdistr1  28821  hvsubdistr2  28822  hvsubcan  28846  hvsub0  28848  his2sub  28864  hhph  28950  shsubcl  28992  shsel3  29087  honegsubi  29568  lnopsubi  29746  lnfnsubi  29818  superpos  30126  cdj1i  30205
  Copyright terms: Public domain W3C validator