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Theorem ltrniotavalbN 35597
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 999 . . 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 1001 . . 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 1049 . . . 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 1050 . . . 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 2467 . . . . 5  |-  ( iota_ f  e.  T  ( f `
 P )  =  Q )  =  (
iota_ f  e.  T  ( f `  P
)  =  Q )
105, 6, 7, 8, 9ltrniotacl 35592 . . . 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 1228 . . 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 35594 . . . . 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 1228 . . . 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 2511 . . 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 35220 . . 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 1242 . 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 5865 . . 3  |-  ( F  =  ( iota_ f  e.  T  ( f `  P )  =  Q )  ->  ( F `  P )  =  ( ( iota_ f  e.  T  ( f `  P
)  =  Q ) `
 P ) )
19 simp1 996 . . . 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 1022 . . . 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 1023 . . . 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 1228 . . 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 2530 . 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 830 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588   iota_crio 6245   lecple 14565   Atomscatm 34277   HLchlt 34364   LHypclh 34997   LTrncltrn 35114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-riotaBAD 33973
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-undef 7003  df-map 7423  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-p1 15530  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512  df-lvols 34513  df-lines 34514  df-psubsp 34516  df-pmap 34517  df-padd 34809  df-lhyp 35001  df-laut 35002  df-ldil 35117  df-ltrn 35118  df-trl 35172
This theorem is referenced by: (None)
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