Theorem lautcnvle 35375

Description: Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)

Hypotheses

Ref	Expression
lautcnvle.b	|- B = ( Base ` K )
lautcnvle.l	|- .<_ = ( le ` K )
lautcnvle.i	|- I = ( LAut ` K )

Assertion

Ref	Expression
lautcnvle	|- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( X .<_ Y <-> ( `' F ` X ) .<_ ( `' F ` Y ) ) )

Proof of Theorem lautcnvle

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( K e. V /\ F e. I ) )
2		lautcnvle.b	. . . . . 6 |- B = ( Base ` K )
3		lautcnvle.i	. . . . . 6 |- I = ( LAut ` K )
4	2, 3	laut1o 35371	. . . . 5 |- ( ( K e. V /\ F e. I ) -> F : B -1-1-onto-> B )
5	4	adantr 481	. . . 4 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> F : B -1-1-onto-> B )
6		simprl 794	. . . 4 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> X e. B )
7		f1ocnvdm 6540	. . . 4 |- ( ( F : B -1-1-onto-> B /\ X e. B ) -> ( `' F ` X ) e. B )
8	5, 6, 7	syl2anc 693	. . 3 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( `' F ` X ) e. B )
9		simprr 796	. . . 4 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
10		f1ocnvdm 6540	. . . 4 |- ( ( F : B -1-1-onto-> B /\ Y e. B ) -> ( `' F ` Y ) e. B )
11	5, 9, 10	syl2anc 693	. . 3 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( `' F ` Y ) e. B )
12		lautcnvle.l	. . . 4 |- .<_ = ( le ` K )
13	2, 12, 3	lautle 35370	. . 3 |- ( ( ( K e. V /\ F e. I ) /\ ( ( `' F ` X ) e. B /\ ( `' F ` Y ) e. B ) ) -> ( ( `' F ` X ) .<_ ( `' F ` Y ) <-> ( F ` ( `' F ` X ) ) .<_ ( F ` ( `' F ` Y ) ) ) )
14	1, 8, 11, 13	syl12anc 1324	. 2 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( ( `' F ` X ) .<_ ( `' F ` Y ) <-> ( F ` ( `' F ` X ) ) .<_ ( F ` ( `' F ` Y ) ) ) )
15		f1ocnvfv2 6533	. . . 4 |- ( ( F : B -1-1-onto-> B /\ X e. B ) -> ( F ` ( `' F ` X ) ) = X )
16	5, 6, 15	syl2anc 693	. . 3 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( F ` ( `' F ` X ) ) = X )
17		f1ocnvfv2 6533	. . . 4 |- ( ( F : B -1-1-onto-> B /\ Y e. B ) -> ( F ` ( `' F ` Y ) ) = Y )
18	5, 9, 17	syl2anc 693	. . 3 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( F ` ( `' F ` Y ) ) = Y )
19	16, 18	breq12d 4666	. 2 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( ( F ` ( `' F ` X ) ) .<_ ( F ` ( `' F ` Y ) ) <-> X .<_ Y ) )
20	14, 19	bitr2d 269	1 |- ( ( ( K e. V /\ F e. I ) /\ ( X e. B /\ Y e. B ) ) -> ( X .<_ Y <-> ( `' F ` X ) .<_ ( `' F ` Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888   Basecbs 15857   lecple 15948   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-laut 35275

This theorem is referenced by:  lautcnv  35376  lautj  35379  lautm  35380  ltrncnvleN  35416  ltrneq2  35434