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Theorem ltrniotavalbN 36050
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 1000 . . 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 1002 . . 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 1050 . . . 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 1051 . . . 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 36045 . . . 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 1229 . . 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 36047 . . . . 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 1229 . . . 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 2487 . . 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 35672 . . 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 1243 . 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 5855 . . 3  |-  ( F  =  ( iota_ f  e.  T  ( f `  P )  =  Q )  ->  ( F `  P )  =  ( ( iota_ f  e.  T  ( f `  P
)  =  Q ) `
 P ) )
19 simp1 997 . . . 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 1023 . . . 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 1024 . . . 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 1229 . . 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 2506 . 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 832 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 974    = wceq 1383    e. wcel 1804   class class class wbr 4437   ` cfv 5578   iota_crio 6241   lecple 14581   Atomscatm 34728   HLchlt 34815   LHypclh 35448   LTrncltrn 35565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-riotaBAD 34424
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-undef 7004  df-map 7424  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-llines 34962  df-lplanes 34963  df-lvols 34964  df-lines 34965  df-psubsp 34967  df-pmap 34968  df-padd 35260  df-lhyp 35452  df-laut 35453  df-ldil 35568  df-ltrn 35569  df-trl 35624
This theorem is referenced by: (None)
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