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Theorem hsupss 22835
Description: Subset relation for supremum of Hilbert space subsets. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hsupss  |-  ( ( A  C_  ~P ~H  /\  B  C_  ~P ~H )  ->  ( A  C_  B  ->  (  \/H  `  A
)  C_  (  \/H  `  B ) ) )

Proof of Theorem hsupss
StepHypRef Expression
1 uniss 4028 . . 3  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
2 sspwuni 4168 . . . 4  |-  ( A 
C_  ~P ~H  <->  U. A  C_  ~H )
3 sspwuni 4168 . . . 4  |-  ( B 
C_  ~P ~H  <->  U. B  C_  ~H )
4 occon2 22782 . . . 4  |-  ( ( U. A  C_  ~H  /\ 
U. B  C_  ~H )  ->  ( U. A  C_ 
U. B  ->  ( _|_ `  ( _|_ `  U. A ) )  C_  ( _|_ `  ( _|_ `  U. B ) ) ) )
52, 3, 4syl2anb 466 . . 3  |-  ( ( A  C_  ~P ~H  /\  B  C_  ~P ~H )  ->  ( U. A  C_ 
U. B  ->  ( _|_ `  ( _|_ `  U. A ) )  C_  ( _|_ `  ( _|_ `  U. B ) ) ) )
61, 5syl5 30 . 2  |-  ( ( A  C_  ~P ~H  /\  B  C_  ~P ~H )  ->  ( A  C_  B  ->  ( _|_ `  ( _|_ `  U. A ) )  C_  ( _|_ `  ( _|_ `  U. B ) ) ) )
7 hsupval 22828 . . . 4  |-  ( A 
C_  ~P ~H  ->  (  \/H  `  A )  =  ( _|_ `  ( _|_ `  U. A ) ) )
87adantr 452 . . 3  |-  ( ( A  C_  ~P ~H  /\  B  C_  ~P ~H )  ->  (  \/H  `  A
)  =  ( _|_ `  ( _|_ `  U. A ) ) )
9 hsupval 22828 . . . 4  |-  ( B 
C_  ~P ~H  ->  (  \/H  `  B )  =  ( _|_ `  ( _|_ `  U. B ) ) )
109adantl 453 . . 3  |-  ( ( A  C_  ~P ~H  /\  B  C_  ~P ~H )  ->  (  \/H  `  B
)  =  ( _|_ `  ( _|_ `  U. B ) ) )
118, 10sseq12d 3369 . 2  |-  ( ( A  C_  ~P ~H  /\  B  C_  ~P ~H )  ->  ( (  \/H  `  A )  C_  (  \/H  `  B )  <->  ( _|_ `  ( _|_ `  U. A ) )  C_  ( _|_ `  ( _|_ `  U. B ) ) ) )
126, 11sylibrd 226 1  |-  ( ( A  C_  ~P ~H  /\  B  C_  ~P ~H )  ->  ( A  C_  B  ->  (  \/H  `  A
)  C_  (  \/H  `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   ` cfv 5446   ~Hchil 22414   _|_cort 22425    \/H chsup 22429
This theorem is referenced by:  chsupss  22836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-hilex 22494  ax-hfvadd 22495  ax-hv0cl 22498  ax-hfvmul 22500  ax-hvmul0 22505  ax-hfi 22573  ax-his2 22577  ax-his3 22578
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sh 22701  df-oc 22746  df-chsup 22805
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