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Theorem shslubi 26501
Description: The least upper bound law for Hilbert subspace sum. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)
Hypotheses
Ref Expression
shslub.1  |-  A  e.  SH
shslub.2  |-  B  e.  SH
shslub.3  |-  C  e.  SH
Assertion
Ref Expression
shslubi  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  +H  B )  C_  C
)

Proof of Theorem shslubi
StepHypRef Expression
1 shslub.1 . . . . 5  |-  A  e.  SH
2 shslub.3 . . . . 5  |-  C  e.  SH
3 shslub.2 . . . . 5  |-  B  e.  SH
41, 2, 3shlessi 26493 . . . 4  |-  ( A 
C_  C  ->  ( A  +H  B )  C_  ( C  +H  B
) )
52, 3shscomi 26479 . . . 4  |-  ( C  +H  B )  =  ( B  +H  C
)
64, 5syl6sseq 3535 . . 3  |-  ( A 
C_  C  ->  ( A  +H  B )  C_  ( B  +H  C
) )
73, 2, 2shlessi 26493 . . . 4  |-  ( B 
C_  C  ->  ( B  +H  C )  C_  ( C  +H  C
) )
82shsidmi 26500 . . . 4  |-  ( C  +H  C )  =  C
97, 8syl6sseq 3535 . . 3  |-  ( B 
C_  C  ->  ( B  +H  C )  C_  C )
106, 9sylan9ss 3502 . 2  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( A  +H  B
)  C_  C )
111, 3shsub1i 26488 . . . 4  |-  A  C_  ( A  +H  B
)
12 sstr 3497 . . . 4  |-  ( ( A  C_  ( A  +H  B )  /\  ( A  +H  B )  C_  C )  ->  A  C_  C )
1311, 12mpan 668 . . 3  |-  ( ( A  +H  B ) 
C_  C  ->  A  C_  C )
143, 1shsub2i 26489 . . . 4  |-  B  C_  ( A  +H  B
)
15 sstr 3497 . . . 4  |-  ( ( B  C_  ( A  +H  B )  /\  ( A  +H  B )  C_  C )  ->  B  C_  C )
1614, 15mpan 668 . . 3  |-  ( ( A  +H  B ) 
C_  C  ->  B  C_  C )
1713, 16jca 530 . 2  |-  ( ( A  +H  B ) 
C_  C  ->  ( A  C_  C  /\  B  C_  C ) )
1810, 17impbii 188 1  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  +H  B )  C_  C
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    e. wcel 1823    C_ wss 3461  (class class class)co 6270   SHcsh 26043    +H cph 26046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-hilex 26114  ax-hfvadd 26115  ax-hvcom 26116  ax-hvass 26117  ax-hv0cl 26118  ax-hvaddid 26119  ax-hfvmul 26120  ax-hvmulid 26121  ax-hvdistr2 26124  ax-hvmul0 26125
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799  df-grpo 25391  df-ablo 25482  df-hvsub 26086  df-sh 26322  df-shs 26424
This theorem is referenced by:  shlesb1i  26502  shsval2i  26503
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