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Theorem lautlt 35377
Description: Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
lautlt.b |- B = ( Base ` K )
lautlt.s |- .< = ( lt ` K )
lautlt.i |- I = ( LAut ` K )
Assertion
Ref Expression
lautlt |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( X .< Y <-> ( F ` X ) .< ( F ` Y ) ) )
Proof of Theorem lautlt
StepHypRef Expression
1 simpl 473 . . . 4 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> K e. A )
2 simpr1 1067 . . . 4 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> F e. I )
3 simpr2 1068 . . . 4 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> X e. B )
4 simpr3 1069 . . . 4 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> Y e. B )
5 lautlt.b . . . . 5 |- B = ( Base ` K )
6 eqid 2622 . . . . 5 |- ( le ` K ) = ( le ` K )
7 lautlt.i . . . . 5 |- I = ( LAut ` K )
85, 6, 7lautle 35370 . . . 4 |- ( ( ( K e. A /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( X ( le ` K ) Y <-> ( F ` X ) ( le ` K ) ( F ` Y ) ) )
91, 2, 3, 4, 8syl22anc 1327 . . 3 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( X ( le ` K ) Y <-> ( F ` X ) ( le ` K ) ( F ` Y ) ) )
105, 7laut11 35372 . . . . . 6 |- ( ( ( K e. A /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( ( F ` X ) = ( F ` Y ) <-> X = Y ) )
111, 2, 3, 4, 10syl22anc 1327 . . . . 5 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( ( F ` X ) = ( F ` Y ) <-> X = Y ) )
1211bicomd 213 . . . 4 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( X = Y <-> ( F ` X ) = ( F ` Y ) ) )
1312necon3bid 2838 . . 3 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( X =/= Y <-> ( F ` X ) =/= ( F ` Y ) ) )
149, 13anbi12d 747 . 2 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( ( X ( le ` K ) Y /\ X =/= Y ) <-> ( ( F ` X ) ( le ` K ) ( F ` Y ) /\ ( F ` X ) =/= ( F ` Y ) ) ) )
15 lautlt.s . . . 4 |- .< = ( lt ` K )
166, 15pltval 16960 . . 3 |- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X ( le ` K ) Y /\ X =/= Y ) ) )
17163adant3r1 1274 . 2 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( X .< Y <-> ( X ( le ` K ) Y /\ X =/= Y ) ) )
185, 7lautcl 35373 . . . 4 |- ( ( ( K e. A /\ F e. I ) /\ X e. B ) -> ( F ` X ) e. B )
191, 2, 3, 18syl21anc 1325 . . 3 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( F ` X ) e. B )
205, 7lautcl 35373 . . . 4 |- ( ( ( K e. A /\ F e. I ) /\ Y e. B ) -> ( F ` Y ) e. B )
211, 2, 4, 20syl21anc 1325 . . 3 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( F ` Y ) e. B )
226, 15pltval 16960 . . 3 |- ( ( K e. A /\ ( F ` X ) e. B /\ ( F ` Y ) e. B ) -> ( ( F ` X ) .< ( F ` Y ) <-> ( ( F ` X ) ( le ` K ) ( F ` Y ) /\ ( F ` X ) =/= ( F ` Y ) ) ) )
231, 19, 21, 22syl3anc 1326 . 2 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( ( F ` X ) .< ( F ` Y ) <-> ( ( F ` X ) ( le ` K ) ( F ` Y ) /\ ( F ` X ) =/= ( F ` Y ) ) ) )
2414, 17, 233bitr4d 300 1 |- ( ( K e. A /\ ( F e. I /\ X e. B /\ Y e. B ) ) -> ( X .< Y <-> ( F ` X ) .< ( F ` Y ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   ltcplt 16941   LAutclaut 35271
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
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-plt 16958  df-laut 35275
This theorem is referenced by:  lautcvr  35378
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