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Theorem hvaddcli 24573
Description: Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvaddcl.1  |-  A  e. 
~H
hvaddcl.2  |-  B  e. 
~H
Assertion
Ref Expression
hvaddcli  |-  ( A  +h  B )  e. 
~H

Proof of Theorem hvaddcli
StepHypRef Expression
1 hvaddcl.1 . 2  |-  A  e. 
~H
2 hvaddcl.2 . 2  |-  B  e. 
~H
3 hvaddcl 24567 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
41, 2, 3mp2an 672 1  |-  ( A  +h  B )  e. 
~H
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758  (class class class)co 6201   ~Hchil 24474    +h cva 24475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-hfvadd 24555
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204
This theorem is referenced by:  hvsubsub4i  24614  hvsubaddi  24621  normlem0  24664  normlem8  24672  norm-ii-i  24692  normpythi  24697  norm3difi  24702  normpari  24709  normpar2i  24711  polidi  24713  nonbooli  25207  lnopunilem1  25567  lnophmlem2  25574
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