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Theorem hvsubvali 28730
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 28726 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))
41, 2, 3mp2an 688 1 (𝐴 𝐵) = (𝐴 + (-1 · 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2107  (class class class)co 7150  1c1 10532  -cneg 10865  chba 28629   + cva 28630   · csm 28631   cmv 28635
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-hvsub 28681
This theorem is referenced by:  hvsubsub4i  28769  hvnegdii  28772  hvsubeq0i  28773  hvsubcan2i  28774  hvsubaddi  28776  normlem0  28819  normlem9  28828  norm3difi  28857  normpar2i  28866  pjsubii  29388  pjssmii  29391  pjcji  29394  lnophmlem2  29727
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