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Theorem diameetN 31856
Description: Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diam.m  |-  ./\  =  ( meet `  K )
diam.h  |-  H  =  ( LHyp `  K
)
diam.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diameetN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `  Y ) ) )

Proof of Theorem diameetN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 732 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  K  e.  HL )
2 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3 diam.h . . . . . 6  |-  H  =  ( LHyp `  K
)
4 diam.i . . . . . 6  |-  I  =  ( ( DIsoA `  K
) `  W )
52, 3, 4diadmclN 31837 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  e.  ( Base `  K
) )
65adantrr 699 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  X  e.  (
Base `  K )
)
72, 3, 4diadmclN 31837 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  dom  I )  ->  Y  e.  ( Base `  K
) )
87adantrl 698 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  Y  e.  (
Base `  K )
)
9 eqid 2438 . . . . 5  |-  ( glb `  K )  =  ( glb `  K )
10 diam.m . . . . 5  |-  ./\  =  ( meet `  K )
112, 9, 10meetval 14454 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  =  ( ( glb `  K
) `  { X ,  Y } ) )
121, 6, 8, 11syl3anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  ( X  ./\  Y )  =  ( ( glb `  K ) `
 { X ,  Y } ) )
1312fveq2d 5734 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  ( I `  ( X  ./\  Y ) )  =  ( I `
 ( ( glb `  K ) `  { X ,  Y }
) ) )
14 simpl 445 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
15 prssi 3956 . . . 4  |-  ( ( X  e.  dom  I  /\  Y  e.  dom  I )  ->  { X ,  Y }  C_  dom  I )
1615adantl 454 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  { X ,  Y }  C_  dom  I
)
17 prnzg 3926 . . . 4  |-  ( X  e.  dom  I  ->  { X ,  Y }  =/=  (/) )
1817ad2antrl 710 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  { X ,  Y }  =/=  (/) )
199, 3, 4diaglbN 31855 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( { X ,  Y }  C_  dom  I  /\  { X ,  Y }  =/=  (/) ) )  ->  ( I `  ( ( glb `  K
) `  { X ,  Y } ) )  =  |^|_ x  e.  { X ,  Y } 
( I `  x
) )
2014, 16, 18, 19syl12anc 1183 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  ( I `  ( ( glb `  K
) `  { X ,  Y } ) )  =  |^|_ x  e.  { X ,  Y } 
( I `  x
) )
21 fveq2 5730 . . . 4  |-  ( x  =  X  ->  (
I `  x )  =  ( I `  X ) )
22 fveq2 5730 . . . 4  |-  ( x  =  Y  ->  (
I `  x )  =  ( I `  Y ) )
2321, 22iinxprg 4170 . . 3  |-  ( ( X  e.  dom  I  /\  Y  e.  dom  I )  ->  |^|_ x  e.  { X ,  Y }  ( I `  x )  =  ( ( I `  X
)  i^i  ( I `  Y ) ) )
2423adantl 454 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  |^|_ x  e.  { X ,  Y } 
( I `  x
)  =  ( ( I `  X )  i^i  ( I `  Y ) ) )
2513, 20, 243eqtrd 2474 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
dom  I  /\  Y  e.  dom  I ) )  ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601    i^i cin 3321    C_ wss 3322   (/)c0 3630   {cpr 3817   |^|_ciin 4096   dom cdm 4880   ` cfv 5456  (class class class)co 6083   Basecbs 13471   glbcglb 14402   meetcmee 14404   HLchlt 30150   LHypclh 30783   DIsoAcdia 31828
This theorem is referenced by:  diainN  31857  djajN  31937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-lhyp 30787  df-laut 30788  df-ldil 30903  df-ltrn 30904  df-trl 30958  df-disoa 31829
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