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Theorem ltrniotavalbN 35872
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 1064 . . 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 1066 . . 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 1114 . . . 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 1115 . . . 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 2622 . . . . 5 |- ( iota_ f e. T ( f ` P ) = Q ) = ( iota_ f e. T ( f ` P ) = Q )
105, 6, 7, 8, 9ltrniotacl 35867 . . . 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 1326 . . 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 477 . . . 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 35869 . . . . 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 1326 . . . 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 2659 . . 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 35494 . . 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 1340 . 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 6190 . . 3 |- ( F = ( iota_ f e. T ( f ` P ) = Q ) -> ( F ` P ) = ( ( iota_ f e. T ( f ` P ) = Q ) ` P ) )
19 simp1 1061 . . . 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 1087 . . . 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 1088 . . . 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 1326 . . 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 2678 . 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 877 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888   iota_crio 6610   lecple 15948   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446
This theorem is referenced by: (None)
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