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Theorem hvaddcl 28199
Description: Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
hvaddcl ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) ∈ ℋ)

Proof of Theorem hvaddcl
StepHypRef Expression
1 ax-hfvadd 28187 . 2 + :( ℋ × ℋ)⟶ ℋ
21fovcl 6931 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) ∈ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  (class class class)co 6814  chil 28106   + cva 28107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-hfvadd 28187
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817
This theorem is referenced by:  hvsubf  28202  hvsubcl  28204  hvaddcli  28205  hvadd4  28223  hvsub4  28224  hvpncan  28226  hvaddsubass  28228  hvsubass  28231  hv2times  28248  hvaddsub4  28265  his7  28277  normpyc  28333  hhph  28365  hlimadd  28380  helch  28430  ocsh  28472  spanunsni  28768  3oalem1  28851  pjcompi  28861  mayete3i  28917  hoscl  28934  hoaddcl  28947  unoplin  29109  hmoplin  29131  braadd  29134  0lnfn  29174  lnopmi  29189  lnophsi  29190  lnopcoi  29192  lnopeq0i  29196  nlelshi  29249  cnlnadjlem2  29257  cnlnadjlem6  29261  adjlnop  29275  superpos  29543  cdj3lem2b  29626  cdj3i  29630
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