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Theorem dihf11lem 34299
Description: Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)
Hypotheses
Ref Expression
dihf11.b  |-  B  =  ( Base `  K
)
dihf11.h  |-  H  =  ( LHyp `  K
)
dihf11.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihf11.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihf11.s  |-  S  =  ( LSubSp `  U )
Assertion
Ref Expression
dihf11lem  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : B --> S )

Proof of Theorem dihf11lem
Dummy variables  x  y  u  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5861 . . . . . . 7  |-  ( ( ( DIsoB `  K ) `  W ) `  x
)  e.  _V
2 riotaex 6246 . . . . . . 7  |-  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) )  e. 
_V
31, 2ifex 3955 . . . . . 6  |-  if ( x ( le `  K ) W , 
( ( ( DIsoB `  K ) `  W
) `  x ) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) )  e.  _V
43rgenw 2767 . . . . 5  |-  A. x  e.  B  if (
x ( le `  K ) W , 
( ( ( DIsoB `  K ) `  W
) `  x ) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) )  e.  _V
54a1i 11 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  A. x  e.  B  if ( x ( le
`  K ) W ,  ( ( (
DIsoB `  K ) `  W ) `  x
) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) )  e.  _V )
6 eqid 2404 . . . . 5  |-  ( x  e.  B  |->  if ( x ( le `  K ) W , 
( ( ( DIsoB `  K ) `  W
) `  x ) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) ) )  =  ( x  e.  B  |->  if ( x ( le `  K ) W , 
( ( ( DIsoB `  K ) `  W
) `  x ) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) ) )
76mptfng 5691 . . . 4  |-  ( A. x  e.  B  if ( x ( le
`  K ) W ,  ( ( (
DIsoB `  K ) `  W ) `  x
) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) )  e.  _V  <->  ( x  e.  B  |->  if ( x ( le `  K ) W , 
( ( ( DIsoB `  K ) `  W
) `  x ) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) ) )  Fn  B )
85, 7sylib 198 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( x  e.  B  |->  if ( x ( le `  K ) W ,  ( ( ( DIsoB `  K ) `  W ) `  x
) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) ) )  Fn  B )
9 dihf11.b . . . . 5  |-  B  =  ( Base `  K
)
10 eqid 2404 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
11 eqid 2404 . . . . 5  |-  ( join `  K )  =  (
join `  K )
12 eqid 2404 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
13 eqid 2404 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
14 dihf11.h . . . . 5  |-  H  =  ( LHyp `  K
)
15 dihf11.i . . . . 5  |-  I  =  ( ( DIsoH `  K
) `  W )
16 eqid 2404 . . . . 5  |-  ( (
DIsoB `  K ) `  W )  =  ( ( DIsoB `  K ) `  W )
17 eqid 2404 . . . . 5  |-  ( (
DIsoC `  K ) `  W )  =  ( ( DIsoC `  K ) `  W )
18 dihf11.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
19 dihf11.s . . . . 5  |-  S  =  ( LSubSp `  U )
20 eqid 2404 . . . . 5  |-  ( LSSum `  U )  =  (
LSSum `  U )
219, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20dihfval 34264 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  =  ( x  e.  B  |->  if ( x ( le `  K ) W , 
( ( ( DIsoB `  K ) `  W
) `  x ) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) ) ) )
2221fneq1d 5654 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I  Fn  B  <->  ( x  e.  B  |->  if ( x ( le
`  K ) W ,  ( ( (
DIsoB `  K ) `  W ) `  x
) ,  ( iota_ u  e.  S  A. q  e.  ( Atoms `  K )
( ( -.  q
( le `  K
) W  /\  (
q ( join `  K
) ( x (
meet `  K ) W ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  W ) `  q
) ( LSSum `  U
) ( ( (
DIsoB `  K ) `  W ) `  (
x ( meet `  K
) W ) ) ) ) ) ) )  Fn  B ) )
238, 22mpbird 234 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  B )
249, 14, 15, 18, 19dihlss 34283 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  y  e.  B
)  ->  ( I `  y )  e.  S
)
2524ralrimiva 2820 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  A. y  e.  B  ( I `  y
)  e.  S )
26 fnfvrnss 6040 . . 3  |-  ( ( I  Fn  B  /\  A. y  e.  B  ( I `  y )  e.  S )  ->  ran  I  C_  S )
2723, 25, 26syl2anc 661 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ran  I  C_  S
)
28 df-f 5575 . 2  |-  ( I : B --> S  <->  ( I  Fn  B  /\  ran  I  C_  S ) )
2923, 27, 28sylanbrc 664 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : B --> S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   _Vcvv 3061    C_ wss 3416   ifcif 3887   class class class wbr 4397    |-> cmpt 4455   ran crn 4826    Fn wfn 5566   -->wf 5567   ` cfv 5571   iota_crio 6241  (class class class)co 6280   Basecbs 14843   lecple 14918   joincjn 15899   meetcmee 15900   LSSumclsm 16980   LSubSpclss 17900   Atomscatm 32294   HLchlt 32381   LHypclh 33014   DVecHcdvh 34111   DIsoBcdib 34171   DIsoCcdic 34205   DIsoHcdih 34261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-riotaBAD 31990
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-tpos 6960  df-undef 7007  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-sca 14927  df-vsca 14928  df-0g 15058  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-grp 16383  df-minusg 16384  df-sbg 16385  df-subg 16524  df-cntz 16681  df-lsm 16982  df-cmn 17126  df-abl 17127  df-mgp 17464  df-ur 17476  df-ring 17522  df-oppr 17594  df-dvdsr 17612  df-unit 17613  df-invr 17643  df-dvr 17654  df-drng 17720  df-lmod 17836  df-lss 17901  df-lsp 17940  df-lvec 18071  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-llines 32528  df-lplanes 32529  df-lvols 32530  df-lines 32531  df-psubsp 32533  df-pmap 32534  df-padd 32826  df-lhyp 33018  df-laut 33019  df-ldil 33134  df-ltrn 33135  df-trl 33190  df-tendo 33787  df-edring 33789  df-disoa 34062  df-dvech 34112  df-dib 34172  df-dic 34206  df-dih 34262
This theorem is referenced by:  dihf11  34300
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