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Theorem hlateq 33406
Description: The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 25956 analog.) (Contributed by NM, 24-May-2012.)
Hypotheses
Ref Expression
hlatle.b  |-  B  =  ( Base `  K
)
hlatle.l  |-  .<_  =  ( le `  K )
hlatle.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlateq  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( A. p  e.  A  ( p  .<_  X  <-> 
p  .<_  Y )  <->  X  =  Y ) )
Distinct variable groups:    A, p    B, p    K, p    .<_ , p    X, p    Y, p

Proof of Theorem hlateq
StepHypRef Expression
1 hlatle.b . . . . 5  |-  B  =  ( Base `  K
)
2 hlatle.l . . . . 5  |-  .<_  =  ( le `  K )
3 hlatle.a . . . . 5  |-  A  =  ( Atoms `  K )
41, 2, 3hlatle 33405 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  A. p  e.  A  ( p  .<_  X  ->  p  .<_  Y ) ) )
51, 2, 3hlatle 33405 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<_  X  <->  A. p  e.  A  ( p  .<_  Y  ->  p  .<_  X ) ) )
653com23 1194 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  A. p  e.  A  ( p  .<_  Y  ->  p  .<_  X ) ) )
74, 6anbi12d 710 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
( A. p  e.  A  ( p  .<_  X  ->  p  .<_  Y )  /\  A. p  e.  A  ( p  .<_  Y  ->  p  .<_  X ) ) ) )
8 ralbiim 2960 . . 3  |-  ( A. p  e.  A  (
p  .<_  X  <->  p  .<_  Y )  <->  ( A. p  e.  A  ( p  .<_  X  ->  p  .<_  Y )  /\  A. p  e.  A  ( p  .<_  Y  ->  p  .<_  X ) ) )
97, 8syl6rbbr 264 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( A. p  e.  A  ( p  .<_  X  <-> 
p  .<_  Y )  <->  ( X  .<_  Y  /\  Y  .<_  X ) ) )
10 hllat 33371 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
111, 2latasymb 15347 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
1210, 11syl3an1 1252 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y  /\  Y  .<_  X )  <-> 
X  =  Y ) )
139, 12bitrd 253 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( A. p  e.  A  ( p  .<_  X  <-> 
p  .<_  Y )  <->  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   class class class wbr 4403   ` cfv 5529   Basecbs 14296   lecple 14368   Latclat 15338   Atomscatm 33271   HLchlt 33358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359
This theorem is referenced by:  lauteq  34102  ltrneq2  34155
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