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Theorem hvaddcli 28451
Description: Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvaddcl.1 𝐴 ∈ ℋ
hvaddcl.2 𝐵 ∈ ℋ
Assertion
Ref Expression
hvaddcli (𝐴 + 𝐵) ∈ ℋ

Proof of Theorem hvaddcli
StepHypRef Expression
1 hvaddcl.1 . 2 𝐴 ∈ ℋ
2 hvaddcl.2 . 2 𝐵 ∈ ℋ
3 hvaddcl 28445 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 + 𝐵) ∈ ℋ)
41, 2, 3mp2an 682 1 (𝐴 + 𝐵) ∈ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  (class class class)co 6924  chba 28352   + cva 28353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-hfvadd 28433
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ov 6927
This theorem is referenced by:  hvsubsub4i  28492  hvsubaddi  28499  normlem0  28542  normlem8  28550  norm-ii-i  28570  normpythi  28575  norm3difi  28580  normpari  28587  normpar2i  28589  polidi  28591  nonbooli  29086  lnopunilem1  29445  lnophmlem2  29452
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