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Theorem shsval 21883
Description: Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65. (Contributed by NM, 16-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
shsval  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  (  +h  " ( A  X.  B ) ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem shsval
StepHypRef Expression
1 xpeq12 4707 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  X.  y
)  =  ( A  X.  B ) )
21imaeq2d 5011 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  (  +h  " (
x  X.  y ) )  =  (  +h  " ( A  X.  B ) ) )
3 df-shs 21879 . 2  |-  +H  =  ( x  e.  SH ,  y  e.  SH  |->  (  +h  " ( x  X.  y ) ) )
4 hilablo 21731 . . 3  |-  +h  e.  AbelOp
5 imaexg 5025 . . 3  |-  (  +h  e.  AbelOp  ->  (  +h  " ( A  X.  B ) )  e.  _V )
64, 5ax-mp 10 . 2  |-  (  +h  " ( A  X.  B ) )  e. 
_V
72, 3, 6ovmpt2a 5939 1  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  (  +h  " ( A  X.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2789    X. cxp 4686   "cima 4691  (class class class)co 5819   AbelOpcablo 20940    +h cva 21492   SHcsh 21500    +H cph 21503
This theorem is referenced by:  shsss  21884  shsel  21885
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-hilex 21571  ax-hfvadd 21572  ax-hvcom 21573  ax-hvass 21574  ax-hv0cl 21575  ax-hvaddid 21576  ax-hfvmul 21577  ax-hvmulid 21578  ax-hvdistr2 21581  ax-hvmul0 21582
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-ltxr 8867  df-sub 9034  df-neg 9035  df-grpo 20850  df-ablo 20941  df-hvsub 21543  df-shs 21879
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