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Theorem hvaddcli 22482
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 22476 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
41, 2, 3mp2an 654 1  |-  ( A  +h  B )  e. 
~H
Colors of variables: wff set class
Syntax hints:    e. wcel 1721  (class class class)co 6048   ~Hchil 22383    +h cva 22384
This theorem is referenced by:  hvsubsub4i  22522  hvsubaddi  22529  normlem0  22572  normlem8  22580  norm-ii-i  22600  normpythi  22605  norm3difi  22610  normpari  22617  normpar2i  22619  polidi  22621  nonbooli  23114  lnopunilem1  23474  lnophmlem2  23481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-hfvadd 22464
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051
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