Theorem ldilcnv 35401

Description: The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)

Hypotheses

Ref	Expression
ldilcnv.h	|- H = ( LHyp ` K )
ldilcnv.d	|- D = ( ( LDil ` K ) ` W )

Assertion

Ref	Expression
ldilcnv	|- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> `' F e. D )

Proof of Theorem ldilcnv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 790	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> K e. HL )
2		ldilcnv.h	. . . 4 |- H = ( LHyp ` K )
3		eqid 2622	. . . 4 |- ( LAut ` K ) = ( LAut ` K )
4		ldilcnv.d	. . . 4 |- D = ( ( LDil ` K ) ` W )
5	2, 3, 4	ldillaut 35397	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> F e. ( LAut ` K ) )
6	3	lautcnv 35376	. . 3 |- ( ( K e. HL /\ F e. ( LAut ` K ) ) -> `' F e. ( LAut ` K ) )
7	1, 5, 6	syl2anc 693	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> `' F e. ( LAut ` K ) )
8		eqid 2622	. . . . . . . . 9 |- ( Base ` K ) = ( Base ` K )
9		eqid 2622	. . . . . . . . 9 |- ( le ` K ) = ( le ` K )
10	8, 9, 2, 4	ldilval 35399	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ F e. D /\ ( x e. ( Base ` K ) /\ x ( le ` K ) W ) ) -> ( F ` x ) = x )
11	10	3expa 1265	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ ( x e. ( Base ` K ) /\ x ( le ` K ) W ) ) -> ( F ` x ) = x )
12	11	3impb 1260	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( F ` x ) = x )
13	12	fveq2d 6195	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( `' F ` ( F ` x ) ) = ( `' F ` x ) )
14	8, 2, 4	ldil1o 35398	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
15	14	3ad2ant1 1082	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
16		simp2 1062	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> x e. ( Base ` K ) )
17		f1ocnvfv1 6532	. . . . . 6 |- ( ( F : ( Base ` K ) -1-1-onto-> ( Base ` K ) /\ x e. ( Base ` K ) ) -> ( `' F ` ( F ` x ) ) = x )
18	15, 16, 17	syl2anc 693	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( `' F ` ( F ` x ) ) = x )
19	13, 18	eqtr3d 2658	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. D ) /\ x e. ( Base ` K ) /\ x ( le ` K ) W ) -> ( `' F ` x ) = x )
20	19	3exp 1264	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> ( x e. ( Base ` K ) -> ( x ( le ` K ) W -> ( `' F ` x ) = x ) ) )
21	20	ralrimiv 2965	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> A. x e. ( Base ` K ) ( x ( le ` K ) W -> ( `' F ` x ) = x ) )
22	8, 9, 2, 3, 4	isldil 35396	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( `' F e. D <-> ( `' F e. ( LAut ` K ) /\ A. x e. ( Base ` K ) ( x ( le ` K ) W -> ( `' F ` x ) = x ) ) ) )
23	22	adantr 481	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> ( `' F e. D <-> ( `' F e. ( LAut ` K ) /\ A. x e. ( Base ` K ) ( x ( le ` K ) W -> ( `' F ` x ) = x ) ) ) )
24	7, 21, 23	mpbir2and 957	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. D ) -> `' F e. D )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888   Basecbs 15857   lecple 15948   HLchlt 34637   LHypclh 35270   LAutclaut 35271   LDilcldil 35386

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  df-ldil 35390

This theorem is referenced by:  ltrncnv  35432