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Theorem diainN 31792
Description: Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-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
diainN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  ( X  i^i  Y )  =  ( I `
 ( ( `' I `  X ) 
./\  ( `' I `  Y ) ) ) )

Proof of Theorem diainN
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 diam.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 diam.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
42, 3diacnvclN 31786 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  ( `' I `  X )  e.  dom  I )
54adantrr 698 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  ( `' I `  X )  e.  dom  I )
62, 3diacnvclN 31786 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  ran  I )  ->  ( `' I `  Y )  e.  dom  I )
76adantrl 697 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  ( `' I `  Y )  e.  dom  I )
8 diam.m . . . 4  |-  ./\  =  ( meet `  K )
98, 2, 3diameetN 31791 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( `' I `  X )  e.  dom  I  /\  ( `' I `  Y )  e.  dom  I ) )  ->  ( I `  ( ( `' I `  X )  ./\  ( `' I `  Y ) ) )  =  ( ( I `  ( `' I `  X ) )  i^i  ( I `
 ( `' I `  Y ) ) ) )
101, 5, 7, 9syl12anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  ( I `  ( ( `' I `  X )  ./\  ( `' I `  Y ) ) )  =  ( ( I `  ( `' I `  X ) )  i^i  ( I `
 ( `' I `  Y ) ) ) )
112, 3diaf11N 31784 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
1211adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  I : dom  I
-1-1-onto-> ran  I )
13 simprl 733 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  X  e.  ran  I )
14 f1ocnvfv2 6007 . . . 4  |-  ( ( I : dom  I -1-1-onto-> ran  I  /\  X  e.  ran  I )  ->  (
I `  ( `' I `  X )
)  =  X )
1512, 13, 14syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  ( I `  ( `' I `  X ) )  =  X )
16 simprr 734 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  Y  e.  ran  I )
17 f1ocnvfv2 6007 . . . 4  |-  ( ( I : dom  I -1-1-onto-> ran  I  /\  Y  e.  ran  I )  ->  (
I `  ( `' I `  Y )
)  =  Y )
1812, 16, 17syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  ( I `  ( `' I `  Y ) )  =  Y )
1915, 18ineq12d 3535 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  ( ( I `
 ( `' I `  X ) )  i^i  ( I `  ( `' I `  Y ) ) )  =  ( X  i^i  Y ) )
2010, 19eqtr2d 2468 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e. 
ran  I  /\  Y  e.  ran  I ) )  ->  ( X  i^i  Y )  =  ( I `
 ( ( `' I `  X ) 
./\  ( `' I `  Y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3311   `'ccnv 4869   dom cdm 4870   ran crn 4871   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   meetcmee 14394   HLchlt 30085   LHypclh 30718   DIsoAcdia 31763
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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-reu 2704  df-rmo 2705  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-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  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-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-disoa 31764
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