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Theorem hladdid 22255
Description: Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
hladdid.1  |-  X  =  ( BaseSet `  U )
hladdid.2  |-  G  =  ( +v `  U
)
hladdid.5  |-  Z  =  ( 0vec `  U
)
Assertion
Ref Expression
hladdid  |-  ( ( U  e.  CHil OLD  /\  A  e.  X )  ->  ( A G Z )  =  A )

Proof of Theorem hladdid
StepHypRef Expression
1 hlnv 22243 . 2  |-  ( U  e.  CHil OLD  ->  U  e.  NrmCVec )
2 hladdid.1 . . 3  |-  X  =  ( BaseSet `  U )
3 hladdid.2 . . 3  |-  G  =  ( +v `  U
)
4 hladdid.5 . . 3  |-  Z  =  ( 0vec `  U
)
52, 3, 4nv0rid 21966 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G Z )  =  A )
61, 5sylan 458 1  |-  ( ( U  e.  CHil OLD  /\  A  e.  X )  ->  ( A G Z )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5396  (class class class)co 6022   NrmCVeccnv 21913   +vcpv 21914   BaseSetcba 21915   0veccn0v 21917   CHil
OLDchlo 22237
This theorem is referenced by:  axhvaddid-zf  22339
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-1st 6290  df-2nd 6291  df-riota 6487  df-grpo 21629  df-gid 21630  df-ablo 21720  df-vc 21875  df-nv 21921  df-va 21924  df-ba 21925  df-sm 21926  df-0v 21927  df-nmcv 21929  df-cbn 22215  df-hlo 22238
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