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Theorem ltrniotavalbN 34240
Description: Value of the unique translation specified by a value. (Contributed by NM, 10-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
ltrniotavalb.l  |-  .<_  =  ( le `  K )
ltrniotavalb.a  |-  A  =  ( Atoms `  K )
ltrniotavalb.h  |-  H  =  ( LHyp `  K
)
ltrniotavalb.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrniotavalbN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( F `  P
)  =  Q  <->  F  =  ( iota_ f  e.  T  ( f `  P
)  =  Q ) ) )
Distinct variable groups:    .<_ , f    A, f    f, H    f, K    P, f    Q, f    T, f   
f, W
Allowed substitution hint:    F( f)

Proof of Theorem ltrniotavalbN
StepHypRef Expression
1 simpl1 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  ( F `  P
)  =  Q )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simpl3 993 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  ( F `  P
)  =  Q )  ->  F  e.  T
)
3 simpl2l 1041 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  ( F `  P
)  =  Q )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simpl2r 1042 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  ( F `  P
)  =  Q )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
5 ltrniotavalb.l . . . . 5  |-  .<_  =  ( le `  K )
6 ltrniotavalb.a . . . . 5  |-  A  =  ( Atoms `  K )
7 ltrniotavalb.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 ltrniotavalb.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
9 eqid 2443 . . . . 5  |-  ( iota_ f  e.  T  ( f `
 P )  =  Q )  =  (
iota_ f  e.  T  ( f `  P
)  =  Q )
105, 6, 7, 8, 9ltrniotacl 34235 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( iota_ f  e.  T  ( f `
 P )  =  Q )  e.  T
)
111, 3, 4, 10syl3anc 1218 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  ( F `  P
)  =  Q )  ->  ( iota_ f  e.  T  ( f `  P )  =  Q )  e.  T )
12 simpr 461 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  ( F `  P
)  =  Q )  ->  ( F `  P )  =  Q )
135, 6, 7, 8, 9ltrniotaval 34237 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( iota_ f  e.  T  ( f `  P )  =  Q ) `  P )  =  Q )
141, 3, 4, 13syl3anc 1218 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  ( F `  P
)  =  Q )  ->  ( ( iota_ f  e.  T  ( f `
 P )  =  Q ) `  P
)  =  Q )
1512, 14eqtr4d 2478 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  ( F `  P
)  =  Q )  ->  ( F `  P )  =  ( ( iota_ f  e.  T  ( f `  P
)  =  Q ) `
 P ) )
165, 6, 7, 8cdlemd 33863 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( iota_ f  e.  T  ( f `  P )  =  Q )  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F `  P )  =  ( ( iota_ f  e.  T  ( f `  P
)  =  Q ) `
 P ) )  ->  F  =  (
iota_ f  e.  T  ( f `  P
)  =  Q ) )
171, 2, 11, 3, 15, 16syl311anc 1232 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  ( F `  P
)  =  Q )  ->  F  =  (
iota_ f  e.  T  ( f `  P
)  =  Q ) )
18 fveq1 5702 . . 3  |-  ( F  =  ( iota_ f  e.  T  ( f `  P )  =  Q )  ->  ( F `  P )  =  ( ( iota_ f  e.  T  ( f `  P
)  =  Q ) `
 P ) )
19 simp1 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
20 simp2l 1014 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
21 simp2r 1015 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
2219, 20, 21, 13syl3anc 1218 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( iota_ f  e.  T  ( f `  P
)  =  Q ) `
 P )  =  Q )
2318, 22sylan9eqr 2497 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  /\  F  =  ( iota_ f  e.  T  ( f `  P )  =  Q ) )  ->  ( F `  P )  =  Q )
2417, 23impbida 828 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  F  e.  T )  ->  (
( F `  P
)  =  Q  <->  F  =  ( iota_ f  e.  T  ( f `  P
)  =  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4304   ` cfv 5430   iota_crio 6063   lecple 14257   Atomscatm 32920   HLchlt 33007   LHypclh 33640   LTrncltrn 33757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-riotaBAD 32616
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-undef 6804  df-map 7228  df-poset 15128  df-plt 15140  df-lub 15156  df-glb 15157  df-join 15158  df-meet 15159  df-p0 15221  df-p1 15222  df-lat 15228  df-clat 15290  df-oposet 32833  df-ol 32835  df-oml 32836  df-covers 32923  df-ats 32924  df-atl 32955  df-cvlat 32979  df-hlat 33008  df-llines 33154  df-lplanes 33155  df-lvols 33156  df-lines 33157  df-psubsp 33159  df-pmap 33160  df-padd 33452  df-lhyp 33644  df-laut 33645  df-ldil 33760  df-ltrn 33761  df-trl 33815
This theorem is referenced by: (None)
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