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Theorem hvsubvali 28217
Description: Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvaddcl.1 𝐴 ∈ ℋ
hvaddcl.2 𝐵 ∈ ℋ
Assertion
Ref Expression
hvsubvali (𝐴 𝐵) = (𝐴 + (-1 · 𝐵))

Proof of Theorem hvsubvali
StepHypRef Expression
1 hvaddcl.1 . 2 𝐴 ∈ ℋ
2 hvaddcl.2 . 2 𝐵 ∈ ℋ
3 hvsubval 28213 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))
41, 2, 3mp2an 672 1 (𝐴 𝐵) = (𝐴 + (-1 · 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  (class class class)co 6793  1c1 10139  -cneg 10469  chil 28116   + cva 28117   · csm 28118   cmv 28122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-hvsub 28168
This theorem is referenced by:  hvsubsub4i  28256  hvnegdii  28259  hvsubeq0i  28260  hvsubcan2i  28261  hvsubaddi  28263  normlem0  28306  normlem9  28315  norm3difi  28344  normpar2i  28353  pjsubii  28877  pjssmii  28880  pjcji  28883  lnophmlem2  29216
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